id int64 | problem_number string | title string | statement string | background string | difficulty_level_id int64 | status string | proposed_by string | proposed_year int64 | category_id int64 | set_id int64 | view_count int64 | favorite_count int64 | created_at timestamp[s] | updated_at timestamp[s] | published bool | category dict | difficulty dict | set dict | duplicates list | status_review unknown | archive_reason string | duplicate_of list | old_problem_number string | notes string | tags list | literature_assessment string | literature_sources list | literature_checked_at timestamp[s] | solution_reference string | source_url string | source string | prize string | references list | clean_statement string | original_statement string | statement_status string | statement_verification string | research_classification string | research_summary string | research_difficulty_suggested string | historical_research_summary string | research_summary_reviewed_at timestamp[s] | source_citation string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
700,254 | KOU-4.47 | Kourovka Notebook Problem 4.47 | Does there exist a countable family of groups such that every variety
is generated by a subfamily of it? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.47.
Proposed by: A. Yu. Olshanskii
Published resolution:
No (Yu. G. Kleiman, *Math. USSR--Izv.*, **22** (1984),
33--65). | 2 | solved | A. Yu. Olshanskii | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No (Yu. G. Kleiman, *Math. USSR--Izv.*, **22** (1984),\n33--65)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,255 | KOU-4.48 | Kourovka Notebook Problem 4.48 | A locally finite group is said to be an *$A
$-group* if all of its Sylow subgroups are abelian. Does every
variety of $A $-groups possess a finite basis of identities? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.48.
Proposed by: A. Yu. Olshanskii | 2 | open | A. Yu. Olshanskii | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,256 | KOU-4.49 | Kourovka Notebook Problem 4.49 | Let $G$ and $H$ be finitely generated torsion-free nilpotent
groups such that $\mathrm{Aut} G\cong \mathrm{Aut} H$. Does it
follow that $G\cong H$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.49.
Proposed by: V. N. Remeslennikov
Published resolution:
No (G. A. Noskov, V. A. Roman'kov, *Algebra and
Logic*, **13** (1974), 306--311). | 2 | solved | V. N. Remeslennikov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No (G. A. Noskov, V. A. Roman'kov, *Algebra and\nLogic*, **13** (1974), 306--311)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,257 | KOU-4.50 | Kourovka Notebook Problem 4.50 | What are the soluble varieties of groups all of
whose finitely generated subgroups are residually finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.50.
Proposed by: V. N. Remeslennikov | 2 | open | V. N. Remeslennikov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,258 | KOU-4.51 | Kourovka Notebook Problem 4.51 | (Well-known problem). Are knot groups residually finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.51.
Proposed by: V. N. Remeslennikov
Published resolution:
Yes, they are (W. P. Thurston, *Bull. Amer. Math. Soc.*, **6**
(1982), 357--381). | 2 | solved | V. N. Remeslennikov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, they are (W. P. Thurston, *Bull. Amer. Math. Soc.*, **6**\n(1982), 357--381)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,259 | KOU-4.52 | Kourovka Notebook Problem 4.52 | Let $G$ be a finitely generated torsion-free group that is an
extension of an abelian group by a nilpotent group. Then is $G$
almost a residually finite $p $-group for almost all primes $p$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.52.
Proposed by: V. N. Remeslennikov
Published resolution:
Yes, it is (G. A. Noskov, *Algebra and Logic*, **13**
(1974), 388--393; \ D. Segal, *J. Algebra*, **32**
(1974), 389--3... | 2 | solved | V. N. Remeslennikov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (G. A. Noskov, *Algebra and Logic*, **13**\n(1974), 388--393; \\ D. Segal, *J. Algebra*, **32**\n(1974), 389--399)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,260 | KOU-4.53 | Kourovka Notebook Problem 4.53 | P. F. Pickel has proved that there are only finitely many
non-isomorphic finitely-generated nilpotent groups having the
same family of finite homomorphic images. Can Pickel's theorem
be extended to polycyclic groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.53.
Proposed by: V. N. Remeslennikov
Published resolution:
Yes, it can (F. J. Grunewald, P. F. Pickel, D. Segal, *Ann. of
Math. (2)*, **111** (1980), 155--195). | 2 | solved | V. N. Remeslennikov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it can (F. J. Grunewald, P. F. Pickel, D. Segal, *Ann. of\nMath. (2)*, **111** (1980), 155--195)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,261 | KOU-4.54 | Kourovka Notebook Problem 4.54 | Are any two minimal relation-modules of a finite group
isomorphic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.54.
Proposed by: K. W. Roggenkamp
Published resolution:
Not always (P. A. Linnell, *Ph. D. Thesis*, Cambridge,
1979; A. J. Sieradski, M. N. Dyer, *J. Pure Appl. Algebra*,
**15** ... | 2 | solved | K. W. Roggenkamp | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Not always (P. A. Linnell, *Ph. D. Thesis*, Cambridge,\n1979; A. J. Sieradski, M. N. Dyer, *J. Pure Appl. Algebra*,\n**15** (1979), 199--217)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,262 | KOU-4.55 | Kourovka Notebook Problem 4.55 | Let $G$ be a finite group, and ${\mathbb{Z}_{(p)}$ the
localization at $p$. Can every projective $\mathbb{Z}_{(p)}G$-module be uniquely, up to permutations and isomorphisms, written as a direct sum of
indecomposable projective $\mathbb{Z}_{(p)}G$-modules?
It is known that the category of finitely generated projective ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.55.
Proposed by: K. W. Roggenkamp
Discussion and literature:
[Solved] No, not every (AI solution confirmed by D. Rumynin: \ Achyuth Jayadevan, *Preprint*, 2026... | 2 | solved | K. W. Roggenkamp | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not every (AI solution confirmed by D. Rumynin: \\ Achyuth Jayadevan, *Preprint*, 2026,\nhttps://zen... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,263 | KOU-4.56 | Kourovka Notebook Problem 4.56 | Let $R$ be a commutative Noetherian ring with 1, and
$\Lambda$ an $ R $-algebra, which is finitely generated as $R
$-module. Put $T=\{ U \in \mathrm{Mod} \Lambda \mid \exists \mbox{ an
exact}\; \Lambda \mbox{-sequence} \; 0 \rightarrow P \rightarrow
\Lambda ^{(n)} \rightarrow U \rightarrow 0\mbox{ for some}\;n,\;
\mbox... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.56.
Proposed by: K. W. Roggenkamp | 2 | open | K. W. Roggenkamp | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,264 | KOU-4.57 | Kourovka Notebook Problem 4.57 | Let a group $G$ be the product of two of its abelian minimax
subgroups $A$ and $B$. Prove or refute the following statements:
a) $A_0B_0\ne 1$, where $A_0=\bigcap_{x\in G}A^x$ and similarly
for $ B_0$;
b) the derived subgroup of $G$ is a minimax subgroup. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.57.
Proposed by: N. F. Sesekin
Published resolution:
Both parts were proved (D. I. Zaitsev, *Algebra and
Logic*, **19** (1980), 94--106). | 2 | solved | N. F. Sesekin | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Both parts were proved (D. I. Zaitsev, *Algebra and\nLogic*, **19** (1980), 94--106)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,265 | KOU-4.58 | Kourovka Notebook Problem 4.58 | Let a finite group $G$ be the product of two subgroups $A$ and
$B$, where $A$ is abelian and $B$ is nilpotent. Find the
dependence of the derived length of $G$ on the nilpotency class
of $B$ and the order of its derived subgroup. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.58.
Proposed by: N. F. Sesekin
Published resolution:
This was found (D. I. Zaitsev, *Math. Notes*, **33** (1983), 414--419). | 2 | solved | N. F. Sesekin | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This was found (D. I. Zaitsev, *Math. Notes*, **33** (1983), 414--419)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,266 | KOU-4.59 | Kourovka Notebook Problem 4.59 | (P. Hall). Find the smallest positive integer $n$ such that
every countable group can be embedded in a simple group with $n$
generators. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.59.
Proposed by: D. M. Smirnov
Published resolution:
It is proved that $ n = 2$ (A. P. Goryushkin, *Math. Notes*, **16**
(1974), 725--727). | 2 | solved | D. M. Smirnov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: It is proved that $ n = 2$ (A. P. Goryushkin, *Math. Notes*, **16**\n(1974), 725--727)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,267 | KOU-4.60 | Kourovka Notebook Problem 4.60 | (P. Hall). What is the cardinality of the set of simple groups
generated by two elements, one of order 2 and the other of order
3? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.60.
Proposed by: D. M. Smirnov
Published resolution:
The cardinality of the continuum. Every 2-generated group $G$ is
embeddable into a simple group $H$ with two generators of or... | 2 | solved | D. M. Smirnov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: The cardinality of the continuum. Every 2-generated group $G$ is\nembeddable into a simple group $H$ with two generators of orders\n$2$ and $3$ (P. E. Schupp, *J... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,268 | KOU-4.61 | Kourovka Notebook Problem 4.61 | Does there exist a linear function $f$ with the following
property: if every abelian subgroup of a finite 2-group $G$ is
generated by $n$ elements, then $G$ is generated by $f(n)$
elements? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.61.
Proposed by: S. A. Syskin
Published resolution:
No (A. Yu. Olshanskii, *Math. Notes*, **23** (1978),
183--185). | 2 | solved | S. A. Syskin | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No (A. Yu. Olshanskii, *Math. Notes*, **23** (1978),\n183--185)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,269 | KOU-4.62 | Kourovka Notebook Problem 4.62 | Does there exist a finitely based variety of groups whose
universal theory is undecidable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.62.
Proposed by: A. Tarski
Published resolution:
Yes, there does; for example, ${\mathfrak{A}}^5$. Indeed, it is shown
in (V. N. Remeslennikov, *Algebra and Logic*, **12**,
no. 5... | 2 | solved | A. Tarski | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there does; for example, ${\\mathfrak{A}}^5$. Indeed, it is shown\nin (V. N. Remeslennikov, *Algebra and Logic*, **12**,\nno. 5 (1975), 327--346) that there... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,270 | KOU-4.63 | Kourovka Notebook Problem 4.63 | Does there exist a non-abelian variety of groups (in
particular, one that contains the variety of all abelian groups)
whose elementary theory is decidable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.63.
Proposed by: A. Tarski
Published resolution:
No (A. P. Zamyatin, *Algebra and Logic*, **27** (1978), 13--17). | 2 | solved | A. Tarski | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No (A. P. Zamyatin, *Algebra and Logic*, **27** (1978), 13--17)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,271 | KOU-4.64 | Kourovka Notebook Problem 4.64 | Does there exist a variety of groups that does not admit an independent
system of defining identities? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.64.
Proposed by: A. Tarski
Published resolution:
Yes, there does (Yu. G. Kleiman, *Math. USSR--Izv.*, **22** (1984),
33--65). | 2 | solved | A. Tarski | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there does (Yu. G. Kleiman, *Math. USSR--Izv.*, **22** (1984),\n33--65)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,272 | KOU-4.65 | Kourovka Notebook Problem 4.65 | *Conjecture:*
$\displaystyle{\frac{p^q-1}{p-1}}$ never divides
$\displaystyle{\frac{q^p-1}{q-1}}$ if $p$, $q$ are distinct
primes. The validity of this conjecture would simplify the proof
of solvability of groups of odd order (W. Feit, J. G. Thompson,
*Pacific J. Math.*, **13**, no. 3 (1963), 775--1029),
rendering unne... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.65.
Proposed by: J. G. Thompson | 2 | open | J. G. Thompson | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,273 | KOU-4.66 | Kourovka Notebook Problem 4.66 | Let $P$ be a presentation of a finite group $G$
on $m_p$ generators and $r_p$ relations. The *deficiency*
$\mathrm{def} (G)$ is the maximum of $m_p - r_p$ over all
presentations $P$. Let $G$ be a finite group such that $G=G'\ne 1$
and the multiplicator $M(G)=1$. Prove that $\mathrm{def} (G^n)
\rightarrow - \infty $ as ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.66.
Proposed by: J. Wiegold | 2 | open | J. Wiegold | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,274 | KOU-4.67 | Kourovka Notebook Problem 4.67 | Let $G$ be a finite $p $-group. Show that the rank of the
multiplicator $M(G)$ of $G$ is bounded in terms of the rank of
$G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.67.
Proposed by: J. Wiegold
Published resolution:
This was done (A. Lubotzky, A. Mann, *J. Algebra*, **105** (1987), 484--505). | 2 | solved | J. Wiegold | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This was done (A. Lubotzky, A. Mann, *J. Algebra*, **105** (1987), 484--505)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,275 | KOU-4.68 | Kourovka Notebook Problem 4.68 | Construct a finitely generated (infinite) characteristically
simple group that is not a direct power of a simple group. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.68.
Proposed by: J. Wiegold
Published resolution:
This was done (J. S. Wilson, *Math. Proc. Cambridge
Phil. Soc.*, **80** (1976), 19--35). | 2 | solved | J. Wiegold | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This was done (J. S. Wilson, *Math. Proc. Cambridge\nPhil. Soc.*, **80** (1976), 19--35)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,276 | KOU-4.69 | Kourovka Notebook Problem 4.69 | Let $G$ be a finite $p
$-group, and
suppose that $|G'|
> p^{n(n-1)/2}$ for some non-negative integer $n$. Prove that
$G$ is generated by the elements of breadths $\geq n$. The *breadth* of an element $x$ of $G$ is $b(x)$ where
$|G:C_G(x)|=p^{b(x)}$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.69.
Proposed by: J. Wiegold
Published resolution:
This has been proved (A. Skutin, *J. Algebra*, **526** (2019), 1--5). | 2 | solved | J. Wiegold | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This has been proved (A. Skutin, *J. Algebra*, **526** (2019), 1--5)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,277 | KOU-4.70 | Kourovka Notebook Problem 4.70 | Let $k$ be a field of characteristic different from $2$, and
$G_k$ the group of transformations $A = (a, \alpha) : x
\rightarrow ax +\alpha$ \ ($a, \alpha \in k$, \ $a\ne 0$). Extend
$G_k$ to the projective plane by adjoining the symbols $ (0,
\alpha )$ and a line at infinity. Then the lines are just the
centralizers $... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.70.
Proposed by: H. Schwerdtfeger
Published resolution:
No (E. A. Kuznetsov, Dep. no. 7028-V89, VINITI, Moscow, 1989
(Russian)). | 2 | solved | H. Schwerdtfeger | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No (E. A. Kuznetsov, Dep. no. 7028-V89, VINITI, Moscow, 1989\n(Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,278 | KOU-4.71 | Kourovka Notebook Problem 4.71 | Let $A$ be a group of automorphisms of a finite group $G$ which
has a series of $A$-invariant subgroups $G = G_0 > \cdots > G_k =
1$ such that every $|G_i:G_{i+1}|$ is prime. Prove that $A$ is
supersoluble. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.71.
Proposed by: L. A. Shemetkov
Published resolution:
This was proved (L. A. Shemetkov, *Math. USSR--Sb.*,
**23** (1974), 593--611). | 2 | solved | L. A. Shemetkov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This was proved (L. A. Shemetkov, *Math. USSR--Sb.*,\n**23** (1974), 593--611)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,279 | KOU-4.72 | Kourovka Notebook Problem 4.72 | Is it true that every variety of groups whose
free groups are residually nilpotent torsion-free, is either
soluble or coincides with the variety of all groups? For an
affirmative answer, it is sufficient to show that every variety of
Lie algebras over the field of rational numbers which does not
contain any finite-dime... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.72.
Proposed by: A. L. Shmel'kin | 2 | open | A. L. Shmel'kin | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,280 | KOU-4.73 | Kourovka Notebook Problem 4.73 | (Well-known problem). Does there exist a
non-abelian variety of groups
a) all of whose finite groups are abelian?
b) all of whose periodic groups are abelian? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.73.
Proposed by: A. L. Shmel'kin
Published resolution:
a) Yes, there does (A. Yu. Olshanskii, *Math.
USSR--Sb.*, **54** (1986), 57--80).
b) Yes, there does (P. A. Kozhevnikov, *... | 2 | solved | A. L. Shmel'kin | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) Yes, there does (A. Yu. Olshanskii, *Math.\nUSSR--Sb.*, **54** (1986), 57--80).\n\nb) Yes, there does (P. A. Kozhevnikov, *On varieties of\ngroups of large od... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,281 | KOU-4.74 | Kourovka Notebook Problem 4.74 | b) Is every binary-finite 2-group of order
greater than 2 non-simple?
[Archived solved part(s) of 4.74:]
a) Is every 2-group of order greater than 2 non-simple? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 4.74.
Proposed by: V. P. Shunkov
Archive of Solved Problems entry:
No, the existence of simple infinite 2-groups follows from the
existence of finitely genera... | 2 | partially_solved | V. P. Shunkov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (No, the existence of simple infinite 2-groups follows from the\nexistence of finitely generated infinite groups of exponent $2^n$\n(S. V. Ivanov, *Int. J. Algebr... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,282 | KOU-4.75 | Kourovka Notebook Problem 4.75 | Let $G$ be a periodic group containing an
involution $i$ and suppose that the Sylow 2-subgroups of $G$ are
either locally cyclic or generalized quaternion. Does the element
$ iO_{2'}(G)$ of the factor-group $G/O_{2'}(G)$ always lie in its
centre? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.75.
Proposed by: V. P. Shunkov
Discussion and literature:
[Solved] No, not always (AI solution confirmed by M. Amelio: \ M. Kinyon, J. Urban, *Preprint*, 2026,... | 2 | solved | V. P. Shunkov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (AI solution confirmed by M. Amelio: \\ M. Kinyon, J. Urban, *Preprint*, 2026,\nhttps://k... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,283 | KOU-4.76 | Kourovka Notebook Problem 4.76 | Let $G$ be a locally finite group containing an element $a$ of
prime order such that the centralizer $C_G(a)$ is finite. Is $G$
almost soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.76.
Proposed by: V. P. Shunkov
Published resolution:
Yes, it is. P. Fong (*Osaka J. Math.*, **13** (1976),
483--489) has shown (mod CFSG) that $G$ is almost locally soluble.
Give... | 2 | solved | V. P. Shunkov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is. P. Fong (*Osaka J. Math.*, **13** (1976),\n483--489) has shown (mod CFSG) that $G$ is almost locally soluble.\nGiven this, B. Hartley and T. Meixner ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,284 | KOU-4.77 | Kourovka Notebook Problem 4.77 | In 1972, A. Rudvalis discovered a new simple group $R$ of order
$2^{14}\cdot 3^3 \cdot 5^3
\cdot 7\cdot 13\cdot 29$. He has shown that $R$ possesses an
involution $i$ such that $C_R(i)=V \times F$, where $V$ is a
4-group (an elementary abelian group of order $4$) and $ F\cong
Sz(8)$.
a) Show that $R$ is the only finit... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.77.
Proposed by: Z. Janko
Published resolution:
a) This has been shown (V. D. Mazurov, *Math. Notes*, **31** (1982), 165--173).
b) This follows from the CFSG. | 2 | solved | Z. Janko | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) This has been shown (V. D. Mazurov, *Math. Notes*, **31** (1982), 165--173).\n\nb) This follows from the CFSG."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,285 | KOU-5.1 | Kourovka Notebook Problem 5.1 | b) Is every
locally finite minimal non-$FC $-group distinct from its derived
subgroup? \ The
question has an affirmative answer for minimal non-$BFC $-groups.
[Archived solved part(s) of 5.1:]
a) Is every locally finite minimal non-$FC
$-group non-simple? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 5.1.
Proposed by: V. V. Belyaev, N. F. Sesekin
Archive of Solved Problems entry:
Yes, it is (M. Kuzucuo\=glu, R. E. Phillips, *Proc.
Cambridge Philos. Soc.*, ... | 2 | partially_solved | V. V. Belyaev, N. F. Sesekin | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (Yes, it is (M. Kuzucuo\\=glu, R. E. Phillips, *Proc.\nCambridge Philos. Soc.*, **105**, no. 3 (1989), 417--420).); remaining part(s) in main section."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,286 | KOU-5.2 | Kourovka Notebook Problem 5.2 | Is it true that the growth function $f$ of any infinite
finitely generated group satisfies the inequality $f(n)\leq
(f(n-1)+f(n+1))/2$ for all sufficiently
large $n$ (for a fixed finite system of
generators)? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.2.
Proposed by: V. V. Belyaev, N. F. Sesekin
Published resolution:
No (P. de la Harpe, R. I. Grigorchuk, *Algebra and
Logic*, **37**, no. 6 (1998), 353--356). | 2 | solved | V. V. Belyaev, N. F. Sesekin | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No (P. de la Harpe, R. I. Grigorchuk, *Algebra and\nLogic*, **37**, no. 6 (1998), 353--356)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,287 | KOU-5.3 | Kourovka Notebook Problem 5.3 | (Well-known problem). Can every finite lattice $L$ be embedded
in the lattice of subgroups of a finite group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.3.
Proposed by: G. M. Bergman
Published resolution:
Yet, it can (P. Pudl\'ak, J. T$\stackrel{\circ{\mathrm{u}}$}ma, *Algebra Universalis*, **10** (1980), 74--95). | 2 | solved | G. M. Bergman | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yet, it can (P. Pudl\\'ak, J. T$\\stackrel{\\circ{\\mathrm{u}}$}ma, *Algebra Universalis*, **10** (1980), 74--95)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,288 | KOU-5.4 | Kourovka Notebook Problem 5.4 | Let
$g$ and $h$ be positive elements of a linearly ordered group $G$.
Can one always embed $G$ in a linearly ordered group
$ \overline{\! G}$ in such a way that $g$ and $h$ are conjugate
in $ \overline{\! G}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.4.
Proposed by: V. V. Bludov
Published resolution:
No, not always
(V. V. Bludov, *Algebra and Logic*, **44**, no. 6
(2005), 370--380). | 2 | solved | V. V. Bludov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always\n(V. V. Bludov, *Algebra and Logic*, **44**, no. 6\n(2005), 370--380)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,289 | KOU-5.5 | Kourovka Notebook Problem 5.5 | If
$G$ is a finitely generated abelian-by-polycyclic-by-finite group,
does there exist a finitely generated metabelian group $M$ such
that $G$ is isomorphic to a subgroup of the automorphism group of
$M$? If so, many of the tricky properties of $G$ like its residual
finiteness would become transparent. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.5.
Proposed by: B. A. F. Wehrfritz | 2 | open | B. A. F. Wehrfritz | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,290 | KOU-5.6 | Kourovka Notebook Problem 5.6 | If $R$ is a finitely generated integral domain of
characteristic $ p > 0$, does the profinite (ideal) topology of
$R$ induce the profinite topology on the group of units of $R$?
It does for $ p = 0$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.6.
Proposed by: B. A. F. Wehrfritz
Published resolution:
Yes, it does (D. Segal, *Bull. London Math. Soc.*, **11**
(1979), 186--190). | 2 | solved | B. A. F. Wehrfritz | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does (D. Segal, *Bull. London Math. Soc.*, **11**\n(1979), 186--190)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,291 | KOU-5.7 | Kourovka Notebook Problem 5.7 | An
algebraic variety $X$ over a field $k$ is called *rational*
if the field of functions $k(X)$ is purely transcendental over
$k$, and it is called *stably rational* if $k(X)$ becomes
purely transcendental after adjoining finitely many independent
variables. Let $T$ be a stably rational torus over a field $k$. Is
$ T$ ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.7.
Proposed by: V. E. Voskresenski\u{\i
Published resolution:
Yes, it is (V. E. Voskresenskii, *J. Math. Sci. (N. Y.)*, **161**, no. 1 (2009), 176--180). | 2 | solved | V. E. Voskresenski\u{\i | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (V. E. Voskresenskii, *J. Math. Sci. (N. Y.)*, **161**, no. 1 (2009), 176--180)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,292 | KOU-5.8 | Kourovka Notebook Problem 5.8 | Let $G$ be a torsion-free soluble group of finite cohomological
dimension $\mathrm{cd} G$. If $hG$ denotes the Hirsch number of
$G$, then it is known that $ hG\leq \mathrm{cd} G\leq hG+1$. Find
a purely group-theoretic criterion for $hG=\mathrm{cd} G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.8.
Proposed by: K. W. Gruenberg
Published resolution:
This has been found (P. H. Kropholler, *J. Pure Appl. Algebra*,
**43** (1986), 281--287). | 2 | solved | K. W. Gruenberg | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This has been found (P. H. Kropholler, *J. Pure Appl. Algebra*,\n**43** (1986), 281--287)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,293 | KOU-5.9 | Kourovka Notebook Problem 5.9 | Let $1\rightarrow R_i \stackrel{\pi _i}{\longrightarrow}F\rightarrow G\rightarrow 1$, \
$i=1, 2$, be two exact sequences of groups with $G$ finite and
$F$ free of finite rank $d(F)$. If we assume that $d(F)=d(G)+1$
(where $d(G)$ is the minimum number of generators of $G$), are
the corresponding abelianized extensions i... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.9.
Proposed by: K. W. Gruenberg
Published resolution:
Yes, they are (P. A. Linnell, *J. Pure Appl. Algebra*, **22**
(1981), 143--166). | 2 | solved | K. W. Gruenberg | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, they are (P. A. Linnell, *J. Pure Appl. Algebra*, **22**\n(1981), 143--166)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,294 | KOU-5.10 | Kourovka Notebook Problem 5.10 | Let $E=H*K$ and denote the augmentation ideals of
the groups $E$, $H$, $K$ by $ {\mathfrak{e}}$, ${\mathfrak{h}}$, ${\mathfrak{k}}$, respectively. If $I$ is a right ideal in ${\mathbb{Z}E$, let
$d_E(I)$ denote the minimum number of generators of $I$ as a right
ideal. Assuming $H$ and $K$ finitely generated, is it true ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.10.
Proposed by: K. W. Gruenberg
Published resolution:
Not always (P. A. Linnell, *unpublished*). A simple
example: $H$ is elementary abelian group of order $4$, $K$ is
elementar... | 2 | solved | K. W. Gruenberg | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Not always (P. A. Linnell, *unpublished*). A simple\nexample: $H$ is elementary abelian group of order $4$, $K$ is\nelementary abelian of order $9$. Here $d_E({\... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,295 | KOU-5.11 | Kourovka Notebook Problem 5.11 | Let $G$ be a finite group and suppose that there exists a
non-empty proper subset $\pi$ of the set of all primes dividing
$|G|$ such that the centralizer of every non-trivial
$\pi$-element is a $\pi$-subgroup. Does it follow that $G$
contains a subgroup $U$ such that $U^g\cap U=1$ or $U$ for
every $g\in G$, and the cen... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.11.
Proposed by: K. W. Gruenberg
Published resolution:
Yes, it does (mod CFSG) (J. S. Williams, *J. Algebra*, **69** (1981), 487--513). | 2 | solved | K. W. Gruenberg | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does (mod CFSG) (J. S. Williams, *J. Algebra*, **69** (1981), 487--513)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,296 | KOU-5.12 | Kourovka Notebook Problem 5.12 | Let $G$ be a finite group with trivial soluble
radical in which there are Sylow 2-subgroups having non-trivial
intersection. Suppose that, for any two Sylow 2-subgroups $P$ and
$Q$ of $G$ with $P \cap Q \ne 1$, the index $|P:P \cap Q|$ does
not exceed $2^n$. Is it then true that $|P| \leq 2^{2n}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.12.
Proposed by: V. V. Kabanov
Published resolution:
Yes, it is, mod CFSG (V. I. Zenkov, *Algebra and Logic*,
**36**, no. 2 (1997), 93--98). | 2 | solved | V. V. Kabanov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is, mod CFSG (V. I. Zenkov, *Algebra and Logic*,\n**36**, no. 2 (1997), 93--98)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,297 | KOU-5.13 | Kourovka Notebook Problem 5.13 | Suppose that $K$ and $L$ are distinct conjugacy classes of
involutions in a finite group $G$ and $\left< x,y\right>$ is a
$2$-group for all $x\in K$ and $y\in L$. Does it follow that
$G\ne [K,L]$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.13.
Proposed by: V. V. Kabanov
Published resolution:
Not always; for example, $G=Sp_4(2^n)$, $n\geq 2$, $K,L$ being
classes of involutions that have non-trivial intersections wit... | 2 | solved | V. V. Kabanov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Not always; for example, $G=Sp_4(2^n)$, $n\\geq 2$, $K,L$ being\nclasses of involutions that have non-trivial intersections with\nthe centres of $N_G(M)'$ and $N... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,298 | KOU-5.14 | Kourovka Notebook Problem 5.14 | An
intersection of some Sylow 2-subgroups is called a *Sylow
intersection* and an intersection of a pair of Sylow
2-subgroups is called a *paired Sylow intersection*.
a) Describe the finite
groups all of whose 2-local subgroups have odd indices.
b) Describe the finite groups all of whose
normalizers of
Sylow intersec... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.14.
Proposed by: V. V. Kabanov,
A. A. Makhn\"ev,
A. I. Starostin | 2 | open | V. V. Kabanov,
A. A. Makhn\"ev,
A. I. Starostin | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,299 | KOU-5.15 | Kourovka Notebook Problem 5.15 | Do
there exist finitely presented residually finite groups with
recursive, but not primitive recursive, solution of the word
problem? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.15.
Proposed by: F. B. Cannonito | 2 | open | F. B. Cannonito | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,300 | KOU-5.16 | Kourovka Notebook Problem 5.16 | Is
every countable locally linear group embeddable in a finitely
presented group? In (G. Baumslag, F. B. Cannonito,
C. F. Miller III, *Math. Z.*, **153** (1977), 117--134) it
is proved that every countable group which is locally linear of
bounded degree can be embedded in a finitely presented group with
solvable word p... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.16.
Proposed by: F. B. Cannonito,
C. F. Miller III | 2 | open | F. B. Cannonito,
C. F. Miller III | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,301 | KOU-5.17 | Kourovka Notebook Problem 5.17 | If the finite group $G$ has the form $G=AB$
where $A$ and $B$ are nilpotent of classes $\alpha $ and $\beta $,
respectively, then $G$ is soluble. Is $G^{(\alpha +\beta )}=1$?
One can show that $G^{(\alpha +\beta )}$ is nilpotent
(E. Pennington, 1973). One can ask the same question for
infinite groups (or Lie algebras),... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.17.
Proposed by: O. H. Kegel
Published resolution:
Not always (J. Cossey, S. Stonehewer, *Bull. London Math.
Soc.*, **30**, no. 144 (1998), 247--250). See also new problem
14.43. | 2 | solved | O. H. Kegel | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Not always (J. Cossey, S. Stonehewer, *Bull. London Math.\nSoc.*, **30**, no. 144 (1998), 247--250). See also new problem\n14.43."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,302 | KOU-5.18 | Kourovka Notebook Problem 5.18 | Let $G$ be an infinite locally finite simple group. Is
the centralizer of every element of $G$ infinite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.18.
Proposed by: O. H. Kegel
Published resolution:
Yes, it is (B. Hartley, M. Kuzucuo\=glu, *Proc. London Math.
Soc. (3)*, **62**, no. 2 (1991), 301--324). | 2 | solved | O. H. Kegel | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (B. Hartley, M. Kuzucuo\\=glu, *Proc. London Math.\nSoc. (3)*, **62**, no. 2 (1991), 301--324)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,303 | KOU-5.19 | Kourovka Notebook Problem 5.19 | a) Let $G$ be an infinite locally finite simple group
satisfying the minimum condition for 2-subgroups. Is $ G =
PSL_2(F)$, $F$ some locally finite field of odd characteristic, if
the centralizer of every involution of $G$ is almost locally
soluble?
b) Can one characterize the simple locally finite groups with
the min... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.19.
Proposed by: O. H. Kegel
Published resolution:
a) Yes, it is (N. S. Chernikov, *Ukrain. Math. J.*, **35** (1983), 230--231).
b) Yes, one can (mod CFSG) by Theorem 4.8 of (O.... | 2 | solved | O. H. Kegel | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) Yes, it is (N. S. Chernikov, *Ukrain. Math. J.*, **35** (1983), 230--231).\n\nb) Yes, one can (mod CFSG) by Theorem 4.8 of (O. H. Kegel,\nB. A. F. Wehrfritz, ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,304 | KOU-5.20 | Kourovka Notebook Problem 5.20 | Is the elementary theory of lattices of $l $-ideals of
lattice-ordered abelian groups decidable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.20.
Proposed by: A. I. Kokorin
Published resolution:
No, it is undecidable (N. Ya. Medvedev, *Algebra and Logic*,
**44**, no. 5 (2005), 302--312). | 2 | solved | A. I. Kokorin | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it is undecidable (N. Ya. Medvedev, *Algebra and Logic*,\n**44**, no. 5 (2005), 302--312)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,305 | KOU-5.21 | Kourovka Notebook Problem 5.21 | Can every torsion-free group with solvable word problem be
embedded in a group with solvable conjugacy problem? An example
due to A. Macintyre shows that this question has a negative
answer when the condition of torsion-freeness is omitted. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.21.
Proposed by: D. J. Collins
Published resolution:
No, not every
(A. Darbinyan, *Invent. Math.*, **224** (2021), 987--997). | 2 | solved | D. J. Collins | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not every\n(A. Darbinyan, *Invent. Math.*, **224** (2021), 987--997)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,306 | KOU-5.22 | Kourovka Notebook Problem 5.22 | Does there exist a version of the Higman embedding theorem in
which the degree of unsolvability of the conjugacy problem is
preserved? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.22.
Proposed by: D. J. Collins
Published resolution:
Yes, there does (A. Yu. Olshanskii, M. V. Sapir, *The
conjugacy problem and Higman embeddings* (*Memoirs AMS*,
**804**), 2004... | 2 | solved | D. J. Collins | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there does (A. Yu. Olshanskii, M. V. Sapir, *The\nconjugacy problem and Higman embeddings* (*Memoirs AMS*,\n**804**), 2004, 133 p.)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,307 | KOU-5.23 | Kourovka Notebook Problem 5.23 | Is
it true that a free lattice-ordered group of the variety of
lattice-ordered groups defined by the law $x^{-1}|y|x \ll |y|^2$
(or, equivalently, by the law $|[x,y]| \ll |x|$), is residually
linearly ordered nilpotent? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.23.
Proposed by: V. M. Kopytov
Published resolution:
No, it is not
(V. V. Bludov, A. M. W. Glass, *Trans. Amer. Math. Soc.*,
**358** (2006), 5179--5192). | 2 | solved | V. M. Kopytov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it is not\n(V. V. Bludov, A. M. W. Glass, *Trans. Amer. Math. Soc.*,\n**358** (2006), 5179--5192)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,308 | KOU-5.24 | Kourovka Notebook Problem 5.24 | Is it true that a free lattice-ordered group of
the variety of the lattice-ordered groups which are residually
linearly ordered, is residually soluble linearly ordered? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.24.
Proposed by: V. M. Kopytov
Published resolution:
No, it is not (N. Ya. Medvedev, *Algebra and Logic*, **44**, no. 3 (2005), 197--204). | 2 | solved | V. M. Kopytov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it is not (N. Ya. Medvedev, *Algebra and Logic*, **44**, no. 3 (2005), 197--204)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,309 | KOU-5.25 | Kourovka Notebook Problem 5.25 | Prove that the factor-group of any soluble linearly ordered group
by its derived subgroup is non-periodic. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.25.
Proposed by: V. M. Kopytov | 2 | open | V. M. Kopytov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,310 | KOU-5.26 | Kourovka Notebook Problem 5.26 | Let $G$ be a finite $p
$-group with the minimal number
of generators $d$, and let $r_1$ (respectively, $r_2$) be the
minimal number of defining relations on $d$ generators in the
sense of representing $G$ as a factor-group of a free discrete
group (pro-$p
$-group). It is well known that always
$r_2 > d^2/4$. For each p... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.26.
Proposed by: H. Koch | 2 | open | H. Koch | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,311 | KOU-5.27 | Kourovka Notebook Problem 5.27 | Prove that if $G$ is a torsion-free pro-$p
$-group with
a single defining relation, then $\mathrm{cd} G=2$. This has been
proved for a large class of groups in (J. P. Labute, *Inv.
Math.*, **4**, no. 2 (1967), 142--158). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.27.
Proposed by: H. Koch | 2 | open | H. Koch | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,312 | KOU-5.28 | Kourovka Notebook Problem 5.28 | Let $G$ be a group and $H$ a torsion-free subgroup of $G$ such
that the augmentation ideal $I_G$ of the integral group-ring $\mathbb{Z}
G$ can be decomposed as $ I_G=I_H \mathbb{Z} G\oplus M$
for some $\mathbb{Z}
G$-submodule $ M$. Prove that $G$ is a free product of the form
$ G = H*K$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.28.
Proposed by: D. E. Cohen
Published resolution:
This has been proved (W. Dicks, M. J. Dunwoody, *Groups acting on
graphs*, Cambridge Univ. Press, Cambridge--New York, 1989). | 2 | solved | D. E. Cohen | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This has been proved (W. Dicks, M. J. Dunwoody, *Groups acting on\ngraphs*, Cambridge Univ. Press, Cambridge--New York, 1989)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,313 | KOU-5.29 | Kourovka Notebook Problem 5.29 | Consider a group $G$ given by a presentation with $m$
generators and $n$ defining relations, where $m\geq n$. Do some
$m-n$ of the given generators generate a free subgroup of
$G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.29.
Proposed by: R. C. Lyndon
Published resolution:
Yes, they do (N. S. Romanovskiĭ, *Algebra and Logic*,
**16** (1977), 62--68). | 2 | solved | R. C. Lyndon | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, they do (N. S. Romanovskiĭ, *Algebra and Logic*,\n**16** (1977), 62--68)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,314 | KOU-5.30 | Kourovka Notebook Problem 5.30 | (Well-known problem). Suppose that $G$ is a finite soluble group, $A
\leq \mathrm{Aut} G$, $C_G(A)=1$, the orders of $G$ and $A$ are
coprime, and let $|A|$ be the product of $n$ not necessarily distinct
prime numbers. Is the nilpotent length of $G$ bounded above by $n$?
This is proved for large classes of groups (E. Sh... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.30.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,315 | KOU-5.32 | Kourovka Notebook Problem 5.32 | Let $p$ be a prime, $C$ a conjugacy class of $
p
$-elements of a finite group $G$ and suppose that
for any two elements $x$ and $y$ of $C$ the product $xy^{-1}$ is a
$p
$-element. Is the subgroup generated by the class $C$ a
$p
$-group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.32.
Proposed by: V. D. Mazurov
Published resolution:
Yes, it is (for $p>2$: W. Xiao, *Sci. in China*, **34**,
no. 9 (1991), 1025--1031; \ for $p=2$: V. I. Zenkov, in: *Algebra. P... | 2 | solved | V. D. Mazurov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (for $p>2$: W. Xiao, *Sci. in China*, **34**,\nno. 9 (1991), 1025--1031; \\ for $p=2$: V. I. Zenkov, in: *Algebra. Proc. IIIrd Int. Conf. on Algebra, ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,316 | KOU-5.33 | Kourovka Notebook Problem 5.33 | (Y. Ihara). Consider the quaternion algebra $Q$ with norm $f=x^2
- \tau y^2 - \rho z^2 + \rho \tau u^2$, $\rho , \tau \in {\mathbb{Z}$. Assume that $f$ is indefinite and of ${\mathbb{Q}$-rank 0, i. e.
$f=0$ for $x,y,z,u \in {\mathbb{Q}$ implies $ x=y=z=u=0$. Consider
$Q$ as the algebra of the matrices
$$X=\left(\!\!\be... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.33.
Proposed by: J. Mennicke | 2 | open | J. Mennicke | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,317 | KOU-5.34 | Kourovka Notebook Problem 5.34 | Let ${\mathfrak{o}}$ be a commutative ring with identity in which 2 is
invertible and which is not generated by zero divisors. Do there
exist non--standard automorphisms of $GL_n({\mathfrak{o}})$ for $ n \geq
3$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.34.
Proposed by: Yu. I. Merzlyakov
Published resolution:
No (V. Ya. Bloshchitsyn, *Algebra and Logic*, **17**
(1978), 415--417). | 2 | solved | Yu. I. Merzlyakov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No (V. Ya. Bloshchitsyn, *Algebra and Logic*, **17**\n(1978), 415--417)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,318 | KOU-5.35 | Kourovka Notebook Problem 5.35 | Let $V$ be a vector space of dimension $n$ over a field. A
subgroup $G$ of $GL_n(V)$ is said to be *rich in
transvections* if $n \geq 2$ and for every hyperplane $H
\subseteq V$ and every line $L \subseteq H$ there is at least one
transvection in $G$ with residual line $ L$ and fixed space $H$.
Describe the automorphis... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.35.
Proposed by: Yu. I. Merzlyakov | 2 | open | Yu. I. Merzlyakov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,319 | KOU-5.36 | Kourovka Notebook Problem 5.36 | What profinite groups satisfy the maximum condition for closed
subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.36.
Proposed by: Yu. N. Mukhin | 2 | open | Yu. N. Mukhin | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,320 | KOU-5.38 | Kourovka Notebook Problem 5.38 | Is
it the case that if $A, B$ are finitely generated soluble Hopfian
groups then $A \times B$ is Hopfian? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.38.
Proposed by: P. M. Neumann
Discussion and literature:
[Unverified AI claim] Yes, it is (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026,
https... | 2 | open | P. M. Neumann | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] Yes, it is (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026,\n\nhttps... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,321 | KOU-5.39 | Kourovka Notebook Problem 5.39 | Prove that every countable group can act faithfully as a group of
automorphisms of a finitely generated soluble group (of derived
length at most 4). For background to this problem, in particular
its relationship with problem 8.50, see (P. M. Neumann, *in: Groups--Korea, Pusan, 1988 $($Lect. Notes Math.*, **1398**),
Spr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.39.
Proposed by: P. M. Neumann
Discussion and literature:
[Unverified AI claim] This is proved (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026, ht... | 2 | open | P. M. Neumann | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] This is proved (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026, http... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,322 | KOU-5.40 | Kourovka Notebook Problem 5.40 | Let $G$ be a countable group acting on a set $\Omega$. Suppose
that $ G$ is $k$-fold transitive for every finite $k$, and $G$
contains no non-trivial permutations of finite support. Is it
true that $\Omega$ can be identified with the rational line $\mathbb{Q}$ in
such a way that $G$ becomes a group of autohomeomorphism... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.40.
Proposed by: P. M. Neumann
Published resolution:
Not always (A. H. Mekler, *J. London Math. Soc.*, **33** (1986),
49--58). | 2 | solved | P. M. Neumann | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Not always (A. H. Mekler, *J. London Math. Soc.*, **33** (1986),\n49--58)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,323 | KOU-5.41 | Kourovka Notebook Problem 5.41 | Does every non-trivial finite group, which is
free in some variety, contain a non-trivial abelian normal
subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.41.
Proposed by: A. Yu. Olshanskii
Published resolution:
Yes, it does (mod CFSG). Otherwise, if $G$ is a counterexample,
the centralizer of the product $E$ of all minimal normal ... | 2 | solved | A. Yu. Olshanskii | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does (mod CFSG). Otherwise, if $G$ is a counterexample,\nthe centralizer of the product $E$ of all minimal normal subgroups\nof $G$ is trivial and there ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,324 | KOU-5.42 | Kourovka Notebook Problem 5.42 | Does the free group of rank 2 have an infinite ascending chain of
verbal subgroups each being generated as a verbal subgroup by a
single element? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.42.
Proposed by: A. Yu. Ol'shanski\u{\i | 2 | open | A. Yu. Ol'shanski\u{\i | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,325 | KOU-5.43 | Kourovka Notebook Problem 5.43 | Does there exist a soluble variety of groups that is not
generated by its finite groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.43.
Proposed by: A. Yu. Olshanskii
Published resolution:
Yes, there does (Yu. G. Kleiman, *Trans. Moscow Math. Soc.*,
**1983**, no. 2, 63--110). | 2 | solved | A. Yu. Olshanskii | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there does (Yu. G. Kleiman, *Trans. Moscow Math. Soc.*,\n**1983**, no. 2, 63--110)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,326 | KOU-5.44 | Kourovka Notebook Problem 5.44 | A
union ${\mathfrak{A}}=\bigcup\limits_{\alpha} {\mathfrak{V}}_{\alpha}$ of
varieties of groups (in the lattice of varieties) is called *irreducible* if $\bigcup\limits_{\beta\ne \alpha} {\mathfrak{V}}_{\beta} \ne {\mathfrak{A}}$ for each index $\alpha $. Is every
variety an irreducible union of (finitely or infinitely... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.44.
Proposed by: A. Yu. Ol'shanski\u{\i | 2 | open | A. Yu. Ol'shanski\u{\i | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,327 | KOU-5.46 | Kourovka Notebook Problem 5.46 | Is every recursively presented soluble group embeddable in a
group finitely presented in the variety of all soluble groups of
derived length $n$, for suitable $n$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.46.
Proposed by: V. N. Remeslennikov
Published resolution:
Yes, it is (G. P. Kukin, *Soviet Math. Dokl.*, **21**
(1980), 378--381). | 2 | solved | V. N. Remeslennikov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (G. P. Kukin, *Soviet Math. Dokl.*, **21**\n(1980), 378--381)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,328 | KOU-5.47 | Kourovka Notebook Problem 5.47 | Is
every countable abelian group embeddable in the center of some
finitely presented group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.47.
Proposed by: V. N. Remeslennikov
Discussion and literature:
[Solved] Yes, it is (A. O. Houcine, *J. Algebra*, **307**, no. 1 (2007), 1--23). | 2 | solved | V. N. Remeslennikov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is (A. O. Houcine, *J. Algebra*, **307**, no. 1 (2007), 1--23)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,329 | KOU-5.48 | Kourovka Notebook Problem 5.48 | Suppose that $G$ and $H$ are finitely generated residually finite
groups having the same set of finite homomorphic images. Are $G$ and
$H$ isomorphic if one of them is a free (free soluble) group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.48.
Proposed by: V. N. Remeslennikov | 2 | open | V. N. Remeslennikov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,330 | KOU-5.49 | Kourovka Notebook Problem 5.49 | Let $A_{m,n}$ be the group of all
automorphisms of
the free soluble group of derived length $n$ and rank $m$. Is it
true that
a) $ A_{m,n}$ is finitely generated for any $m$ and $n$?
b) every automorphism in $A_{m,n}$ is induced by some
automorphism in $A_{m,n+1}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.49.
Proposed by: V. N. Remeslennikov
Published resolution:
a) No: $A_{3,2}$ is not finitely generated, although $A_{m,2}$ is
whenever $ m\ne 3$ (S. Bachmuth, H. Y. Mochizuki, *Tr... | 2 | solved | V. N. Remeslennikov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) No: $A_{3,2}$ is not finitely generated, although $A_{m,2}$ is\nwhenever $ m\\ne 3$ (S. Bachmuth, H. Y. Mochizuki, *Trans. Amer.\nMath. Soc.*, **270** (1982),... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,331 | KOU-5.50 | Kourovka Notebook Problem 5.50 | Is there a finite group whose set of quasi-laws does not have
an independent basis? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.50.
Proposed by: D. M. Smirnov
Published resolution:
Yes, there is (A. N. Fedorov, *Siberian Math. J.*, **21** (1980), 840--850). | 2 | solved | D. M. Smirnov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there is (A. N. Fedorov, *Siberian Math. J.*, **21** (1980), 840--850)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,332 | KOU-5.51 | Kourovka Notebook Problem 5.51 | Does a non-abelian free group have an independent basis for its
quasi-laws? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.51.
Proposed by: D. M. Smirnov
Published resolution:
Yes, it does (A. I. Budkin, *Math. Notes*, **31** (1982),
413--417). | 2 | solved | D. M. Smirnov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does (A. I. Budkin, *Math. Notes*, **31** (1982),\n413--417)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,333 | KOU-5.52 | Kourovka Notebook Problem 5.52 | It
is not hard to show that a finite perfect group is the normal
closure of a single element. Is the same true for infinite
finitely generated groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.52.
Proposed by: J. Wiegold
Discussion and literature:
[Solved] There exist finitely presented perfect groups that are not normal closures of a single element ... | 2 | solved | J. Wiegold | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] There exist finitely presented perfect groups that are not normal closures of a single element (L. Chen,... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,334 | KOU-5.53 | Kourovka Notebook Problem 5.53 | (P. Scott). Let $p,q,r$ be distinct prime numbers. Prove that the
free product $G=C_p*C_q*C_r$ of cyclics of orders $p,q,r$ is not
the normal closure of a single element. By Lemma 3.1 of
(J. Wiegold, *J. Austral. Math. Soc.*, **17**, no. 2
(1974), 133--141), every soluble image, and every finite image, of
$G$ is the no... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.53.
Proposed by: J. Wiegold
Published resolution:
This is proved in (J. Howie, *J. Pure Appl. Algebra*, **173**, no. 2 (2002), 167--176). | 2 | solved | J. Wiegold | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This is proved in (J. Howie, *J. Pure Appl. Algebra*, **173**, no. 2 (2002), 167--176)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,335 | KOU-5.54 | Kourovka Notebook Problem 5.54 | Let $p,q,r$ be distinct primes and $ u(x,y,z)$ a
commutator word in three variables. Prove that there exist
(infinitely many?) natural numbers $n$ such that the alternating
group ${\mathbb{A}_n$ can be generated by three elements $\xi , \eta ,
\zeta$ satisfying $\xi ^p= \eta ^q= \zeta ^r=\xi \eta \zeta \cdot
u(\xi , \e... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.54.
Proposed by: J. Wiegold | 2 | open | J. Wiegold | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,336 | KOU-5.55 | Kourovka Notebook Problem 5.55 | Find a finite $p
$-group that cannot be embedded in a
finite $p
$-group with trivial multiplicator. Notice
that every finite group can be embedded in a group with trivial
multiplicator. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.55.
Proposed by: J. Wiegold | 2 | open | J. Wiegold | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,337 | KOU-5.56 | Kourovka Notebook Problem 5.56 | a)
Let $p$ be a prime greater than 3. Is it true that every finite
group of exponent $p$ can be embedded in the commutator subgroup
of a finite group of exponent $p$?
[Archived solved part(s) of 5.56:]
b)
Does there exist a locally nilpotent group of prime exponent that
coincides with its derived subgroup (and hence h... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 5.56.
Proposed by: J. Wiegold
Archive of Solved Problems entry:
Yes, there does. Every non-soluble variety ${\mathfrak{V}}$ contains
a non-trivial group coinc... | 2 | partially_solved | J. Wiegold | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (Yes, there does. Every non-soluble variety ${\\mathfrak{V}}$ contains\na non-trivial group coinciding with the derived subgroup, the\ndirect limit of the spectru... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,338 | KOU-5.59 | Kourovka Notebook Problem 5.59 | Let $G$ be a locally finite group which is the product of a
$p
$-subgroup and a $q $-subgroup, where $p$ and
$q$ are distinct primes. Is $G$ a $\{ p,q\} $-group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.59.
Proposed by: B. Hartley | 2 | open | B. Hartley | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,339 | KOU-5.60 | Kourovka Notebook Problem 5.60 | Is an arbitrary soluble group that satisfies the minimum condition for
normal subgroups countable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.60.
Proposed by: B. Hartley
Published resolution:
Not always (B. Hartley, *Proc. London Math. Soc.*, **33** (1977),
55--75). | 2 | solved | B. Hartley | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Not always (B. Hartley, *Proc. London Math. Soc.*, **33** (1977),\n55--75)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,340 | KOU-5.61 | Kourovka Notebook Problem 5.61 | (Well-known problem). Does an arbitrary uncountable locally
finite group have only one end? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.61.
Proposed by: B. Hartley
Published resolution:
Yes, it does (D. Holt, *Bull. London Math. Soc.*, **13** (1981),
557--560). | 2 | solved | B. Hartley | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does (D. Holt, *Bull. London Math. Soc.*, **13** (1981),\n557--560)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,341 | KOU-5.62 | Kourovka Notebook Problem 5.62 | Is a group locally finite if it contains infinite abelian
subgroups and all of them are complementable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.62.
Proposed by: S. N. Chernikov
Published resolution:
Not always (N. S. Chernikov, *Math. Notes*, **28**
(1980), 788--792). | 2 | solved | S. N. Chernikov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Not always (N. S. Chernikov, *Math. Notes*, **28**\n(1980), 788--792)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,342 | KOU-5.63 | Kourovka Notebook Problem 5.63 | Prove that a finite group is not simple if it contains two
non-identity elements whose centralizers have coprime indices. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.63.
Proposed by: S. A. Chunikhin
Published resolution:
This has been proved mod CFSG (L. S. Kazarin, in: *Studies in
Group Theory*, Sverdlovsk, 1984, 81--99 (Russian)). | 2 | solved | S. A. Chunikhin | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This has been proved mod CFSG (L. S. Kazarin, in: *Studies in\nGroup Theory*, Sverdlovsk, 1984, 81--99 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,343 | KOU-5.64 | Kourovka Notebook Problem 5.64 | Suppose that a finite group $G$ is the product of two subgroups
$A_1$ and $A_2$. Prove that if $A_i$ contains a nilpotent
subgroup of index $\leq 2$ for $ i = 1, 2$, then $G$ is
soluble. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.64.
Proposed by: L. A. Shemetkov
Published resolution:
This has been proved (L. S. Kazarin, *Math. USSR--Sb.*, **38** (1981), 47--59). | 2 | solved | L. A. Shemetkov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This has been proved (L. S. Kazarin, *Math. USSR--Sb.*, **38** (1981), 47--59)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,344 | KOU-5.65 | Kourovka Notebook Problem 5.65 | Is the class of all finite groups that have Hall
$\pi $-subgroups closed under taking finite subdirect products? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.65.
Proposed by: L. A. Shemetkov
Published resolution:
Yes, it is (D. O. Revin, E. P. Vdovin, *J. Group
Theory*, **14**, no. 1 (2011), 93--101). | 2 | solved | L. A. Shemetkov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (D. O. Revin, E. P. Vdovin, *J. Group\nTheory*, **14**, no. 1 (2011), 93--101)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,345 | KOU-5.67 | Kourovka Notebook Problem 5.67 | Is a periodic
residually finite group finite if it satisfies the weak minimum
condition for subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 5.67.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,346 | KOU-5.68 | Kourovka Notebook Problem 5.68 | Let $G$ be a finitely-presented group, and assume that $G$ has
polynomial growth in the sense of Milnor. Show that $G$ has
soluble word problem. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.68.
Proposed by: P. E. Schupp
Published resolution:
This has been shown (M. Gromov, *Publ. Math. IHES*, **53** (1981),
53--73). | 2 | solved | P. E. Schupp | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This has been shown (M. Gromov, *Publ. Math. IHES*, **53** (1981),\n53--73)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,347 | KOU-5.69 | Kourovka Notebook Problem 5.69 | Is every lattice isomorphism between torsion-free groups having
no non-trivial cyclic normal subgroups induced by a group
isomorphism? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 5th issue (1976). Original problem number: 5.69.
Proposed by: B. V. Yakovlev
Published resolution:
No (A. Yu. Olshanskii, *C. R. Acad. Bulgare Sci.*, **32**
(1979), 1165--1166). | 2 | solved | B. V. Yakovlev | 1,976 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No (A. Yu. Olshanskii, *C. R. Acad. Bulgare Sci.*, **32**\n(1979), 1165--1166)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,348 | KOU-6.1 | Kourovka Notebook Problem 6.1 | A subgroup $H$
of an arbitrary group $G$ is said to be *$ C $-closed* if
$H=C^2(H)=C(C(H))$, and *weakly $C $-closed* if $C^2(x) \leq
H$ for any element $x$ of $H$. The structure of the finite groups
all of whose proper subgroups are $ C $-closed was studied by
Gasch\"utz. Describe the structure of the locally finite g... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.1.
Proposed by: V. A. Antonov, N. F. Sesekin | 2 | open | V. A. Antonov, N. F. Sesekin | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,349 | KOU-6.2 | Kourovka Notebook Problem 6.2 | The
totality of $C $-closed subgroups of an arbitrary group $G$ is a
complete lattice with respect to the operations
$\; A \wedge B=A \cap B,\; \; \; A\vee B=C(C(A) \cap
C(B)).$
Describe the groups whose lattice of $C $-closed subgroups is a
sublattice of the lattice of all subgroups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.2.
Proposed by: V. A. Antonov,
N. F. Sesekin | 2 | open | V. A. Antonov,
N. F. Sesekin | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,350 | KOU-6.3 | Kourovka Notebook Problem 6.3 | A
group $G$ is called {\it of type $(FP)_{\infty}$} if the trivial
$G $-module ${\mathbb{Z}$ has a resolution by finitely generated
projective $G $-modules. The class of all groups of type
$(FP)_{\infty}$ has a couple of excellent closure properties with
respect to extensions and amalgamated products (R. Bieri, *Homolo... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.3.
Proposed by: R. Bieri | 2 | open | R. Bieri | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,351 | KOU-6.4 | Kourovka Notebook Problem 6.4 | A
group $G$ is called {\it of type $(FP)_{\infty}$} if the trivial
$G $-module ${\mathbb{Z}$ has a resolution by finitely generated
projective $G $-modules.
Is it true that every torsion-free group of type
$(FP)_{\infty}$ has finite cohomological dimension? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.4.
Proposed by: R. Bieri
Published resolution:
No (K. S. Brown, R. Geoghegan, *Invent. Math.*, **77**
(1984), 367--381). | 2 | solved | R. Bieri | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No (K. S. Brown, R. Geoghegan, *Invent. Math.*, **77**\n(1984), 367--381)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,352 | KOU-6.5 | Kourovka Notebook Problem 6.5 | Is
it the case that every soluble group $G$ of type $(FP)_{\infty}$
is *constructible* in the sense of (G. Baumslag, R. Bieri,
*Math. Z.*, **151**, no. 3 (1976), 249--257)? In other
words, can $G$ be built up from the trivial group in terms of
finite extensions and $HNN $-extensions? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.5.
Proposed by: R. Bieri | 2 | open | R. Bieri | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,353 | KOU-6.6 | Kourovka Notebook Problem 6.6 | Let $P, Q$ be
permutation representations of a finite group $G$ with the same
character. Suppose $P(G)$ is a primitive permutation group. Is
$Q(G)$ necessarily primitive? The answer is known to be
affirmative if $G$ is soluble. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.6.
Proposed by: H. Wielandt
Published resolution:
No, not necessarily (mod CFSG) (R. M. Guralnick,
J. Saxl, in: *Groups, Combinatorics, Geometry, Proc. Durham, 1990*, Cambridge
U... | 2 | solved | H. Wielandt | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not necessarily (mod CFSG) (R. M. Guralnick,\nJ. Saxl, in: *Groups, Combinatorics, Geometry, Proc. Durham, 1990*, Cambridge\nUniv. Press, 1992, 364--367)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
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