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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
700,854 | KOU-12.35 | Kourovka Notebook Problem 12.35 | Suppose that ${\mathfrak{F}}$ is a radical composition
formation of finite groups. Prove that $\left< H,K\right> ^{{\mathfrak{F}}} =\left< H^{{\mathfrak{F}}}, K^{{\mathfrak{F}}}\right>$ for every finite
group $G$ and any subnormal subgroups $H$ and $K$ of $G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.35.
Proposed by: S. F. Kamornikov | 2 | open | S. F. Kamornikov | 1,992 | 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 |
700,855 | KOU-12.36 | Kourovka Notebook Problem 12.36 | Let $p$ be a prime, $V$ an $n $-dimensional
vector space over the field of $p$ elements, and let $G$ be a
subgroup of $GL(V)$. Let $S=S[V ^{\displaystyle{*}} ]$ be the
symmetric algebra on $V ^{\displaystyle{*}} $, the dual of $V$. Let
$T=S^G$ be the ring of invariants and let $b_m$ be the dimension of
the homogeneous ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.36.
Proposed by: D. Carlisle, P. H. Kropholler
Published resolution:
The conjecture is proved (D. J. Benson,
W. W. Crawley-Boevey, *Bull. London Math. Soc.*, **27**, no. 5 (199... | 2 | solved | D. Carlisle, P. H. Kropholler | 1,992 | 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 conjecture is proved (D. J. Benson,\nW. W. Crawley-Boevey, *Bull. London Math. Soc.*, **27**, no. 5 (1995),\n435--440); another proof based on the Grothendie... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,856 | KOU-12.37 | Kourovka Notebook Problem 12.37 | (J. G. Thompson). For a finite group $G$ and
natural number $n$, set $G(n)=\{ x \in G\mid x^n=1\} $ and define
the *type* of $G$ to be the function whose value at $n$ is
the order of $G(n)$. Is it true that a group is soluble if its
type is the same as that of a soluble one? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.37.
Proposed by: A. S. Kondratiev, W. J. Shi
Discussion and literature:
[Solved] No, not always (P. Piwek, *Algebra Number Theory*, **19**, no. 8 (2025), 166... | 2 | solved | A. S. Kondratiev, W. J. Shi | 1,992 | 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 (P. Piwek, *Algebra Number Theory*, **19**, no. 8 (2025), 1663--1670)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,857 | KOU-12.38 | Kourovka Notebook Problem 12.38 | (J. G. Thompson). For a finite group $G$, we
denote by $N(G)$ the set of all orders of the conjugacy classes of
$G$. Is it true that if $G$ is a finite non-abelian simple group,
$H$ a finite group with trivial centre and $N(G)=N(H)$, then $G$
and $H$ are isomorphic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.38.
Proposed by: A. S. Kondratiev, W. J. Shi
Published resolution:
Yes, it is true. The final step of the proof is in the paper (I. B. Gorshkov, *Commun. Algebra*, **47**, no. ... | 2 | solved | A. S. Kondratiev, W. J. Shi | 1,992 | 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 true. The final step of the proof is in the paper (I. B. Gorshkov, *Commun. Algebra*, **47**, no. 12 (2019), 5192--5206),\nwhich contains references t... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,858 | KOU-12.39 | Kourovka Notebook Problem 12.39 | (W. J. Shi). Must a finite group
and a finite simple group be isomorphic if they have equal orders
and the same set of orders of elements? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.39.
Proposed by: A. S. Kondratiev
Published resolution:
Yes, they must
(M. C. Xu, W. J. Shi, *Algebra Colloq.*, **10** (2003), 427--443; A.V. Vasil'ev, M.A. Grechkoseeva, V.D. ... | 2 | solved | A. S. Kondratiev | 1,992 | 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 must\n(M. C. Xu, W. J. Shi, *Algebra Colloq.*, **10** (2003), 427--443; A.V. Vasil'ev, M.A. Grechkoseeva, V.D. Mazurov, *Algebra Logic*, **48** (2009),... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,859 | KOU-12.40 | Kourovka Notebook Problem 12.40 | Let $\varphi$ be an irreducible $p
$-modular character of a finite group $G$. Find the best-possible
estimate of the form $\varphi (1)_p \leq f(|G|_p)$, where $n_p$ is
the $p
$-part of a positive integer $n$,
a) for $p=2$;
b) for $p\ne 2$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.40.
Proposed by: A. S. Kondratiev
Discussion and literature:
[Solved] a) There cannot be any estimate of this kind for $p=2$ (AI solution confirmed by A. S. ... | 2 | partially_solved | A. S. Kondratiev | 1,992 | 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": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] a) There cannot be any estimate of this kind for $p=2$ (AI solution confirmed by A. S. Kondrat... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,860 | KOU-12.41 | Kourovka Notebook Problem 12.41 | Let $F$ be a free group on two generators $x,y$
and let $\varphi$ be the automorphism of $F$ defined by $x
\rightarrow y$, $y \rightarrow xy$. Let $G$ be a semidirect
product of $F/(F''(F')^2)$ and $\left< \varphi \right>$. Then $G$
is just-non-polycyclic. What is the cohomological dimension of $G$
over ${\mathbb{Q}$? ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.41.
Proposed by: P. H. Kropholler
Discussion and literature:
[Solved] It is 4, because $G$ is not constructible, and then its cohomological dimension is equa... | 2 | solved | P. H. Kropholler | 1,992 | 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] It is 4, because $G$ is not constructible, and then its cohomological dimension is equal to the homologi... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,861 | KOU-12.42 | Kourovka Notebook Problem 12.42 | Describe the automorphisms of the Sylow $p
$-subgroup
of a Chevalley group of normal type over ${\mathbb{Z}/p^m{\mathbb{Z} $,
$m \geq 2$, where $p$ is a prime. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.42.
Proposed by: V. M. Levchuk
Published resolution:
Described in
(S. G. Kolesnikov, *Algebra and Logic*, **43**, no. 1
(2004), 17--33); *Izv. Gomel' Univ.*, **36**, no. 3 (200... | 2 | solved | V. M. Levchuk | 1,992 | 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: Described in\n(S. G. Kolesnikov, *Algebra and Logic*, **43**, no. 1\n(2004), 17--33); *Izv. Gomel' Univ.*, **36**, no. 3 (2006),\n137--146 (Russian); *J. Math. S... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,862 | KOU-12.43 | Kourovka Notebook Problem 12.43 | (Well-known problem). Does there exist an
infinite finitely-generated residually finite $p
$-group such that each subgroup is either finite or of finite
index? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.43.
Proposed by: J. C. Lennox | 2 | open | J. C. Lennox | 1,992 | 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 |
700,863 | KOU-12.44 | Kourovka Notebook Problem 12.44 | (P. Hall). Is there a non-trivial group which is isomorphic with
every proper extension of itself by itself? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.44.
Proposed by: J. C. Lennox
Published resolution:
Yes, such groups exist of cardinality any regular cardinal
(R. G\"obel, S. Shelah, *Math. Proc. Cambridge Philos.
Society (2... | 2 | solved | J. C. Lennox | 1,992 | 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, such groups exist of cardinality any regular cardinal\n(R. G\\\"obel, S. Shelah, *Math. Proc. Cambridge Philos.\nSociety (2)*, **134**, no. 1 (2003), 23--31... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,864 | KOU-12.45 | Kourovka Notebook Problem 12.45 | (P. Hall). Must a non-trivial group, which is
isomorphic to each of its non-trivial normal subgroups, be either
free of infinite rank, simple, or infinite cyclic? (Lennox, Smith
and Wiegold, 1992, have shown that a finitely generated group of
this kind which has a proper normal subgroup of finite index is
infinite cycl... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.45.
Proposed by: J. C. Lennox
Published resolution:
No (V. N. Obraztsov, *Proc. London Math. Soc.*, **75**
(1997), 79--98). | 2 | solved | J. C. Lennox | 1,992 | 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. N. Obraztsov, *Proc. London Math. Soc.*, **75**\n(1997), 79--98)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,865 | KOU-12.46 | Kourovka Notebook Problem 12.46 | Let $F$ be the nonabelian free group on two
generators $x,y$. For $a,b \in {\mathbb{C}$, $|a|=|b|=1$, let $\vartheta
_{a,b}$ be the automorphism of ${\mathbb{C}F$ defined by $\vartheta
_{a,b}(x)=ax$, $\vartheta _{a,b}(y)=by$. Given $0 \ne \alpha \in
{\mathbb{C}F$, can we always find $a,b \in {\mathbb{C}\setminus \{ 1\}... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.46.
Proposed by: P. A. Linnell
Published resolution:
No, not always (A. V. Tushev, *Ukrain. Mat. J.*, **47**, no. 4 (1995), 571--572). | 2 | solved | P. A. Linnell | 1,992 | 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 (A. V. Tushev, *Ukrain. Mat. J.*, **47**, no. 4 (1995), 571--572)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,866 | KOU-12.47 | Kourovka Notebook Problem 12.47 | Let $k$ be a field, let $p$ be a prime, and let
$G$ be the Wreath product ${\mathbb{Z}_p\wr {\mathbb{Z}$ (so the base
group has exponent $p$). Does $kG$ have a classical quotient ring?
(i. e. do the non-zero-divisors of $kG$ form an Ore set?) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.47.
Proposed by: P. A. Linnell
Published resolution:
No, it does not (P. A. Linnell, W. L\"uck, T. Schick, in: *High-dimensional manifold topology. Proc. of the school, ICTP
(T... | 2 | solved | P. A. Linnell | 1,992 | 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 does not (P. A. Linnell, W. L\\\"uck, T. Schick, in: *High-dimensional manifold topology. Proc. of the school, ICTP\n(Trieste, 2001)*, World Scientific, R... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,867 | KOU-12.48 | Kourovka Notebook Problem 12.48 | Let $G$ be a sharply doubly transitive
permutation group on a set $\Omega$ (see Archive 11.52 for a definition).
(a) Does $G$ possess a regular normal
subgroup if a point
stabilizer is locally finite?
(b) Does $G$ possess a regular normal
subgroup if a point
stabilizer has an abelian subgroup of finite index?
*Comme... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.48.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 1,992 | 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 |
700,868 | KOU-12.49 | Kourovka Notebook Problem 12.49 | Construct all non-split extensions of elementary
abelian 2-groups $V$ by $H=PSL_2(q)$ for which $H$ acts irreducibly
on $V$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.49.
Proposed by: V. D. Mazurov
Published resolution:
They are constructed (V. P. Burichenko, *Algebra and Logic*,
**39** (2000),
160--183). | 2 | solved | V. D. Mazurov | 1,992 | 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: They are constructed (V. P. Burichenko, *Algebra and Logic*,\n**39** (2000),\n160--183)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,869 | KOU-12.50 | Kourovka Notebook Problem 12.50 | (Well-known problem). Find an algorithm which
decides, by a given finite set of matrices in $SL_3({\mathbb{Z})$,
whether the first matrix of this set is contained in the subgroup
generated by the remaining matrices. An analogous problem for
$SL_4({\mathbb{Z})$ is insoluble since a direct product of two free
groups of r... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.50.
Proposed by: G. S. Makanin | 2 | open | G. S. Makanin | 1,992 | 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 |
700,870 | KOU-12.51 | Kourovka Notebook Problem 12.51 | Does the free group $F_{\eta}$ ($1 < \eta <
\infty$) have a finite subset $S$ for which there is a unique
total order of $F_{\eta}$ making all elements of $S$ positive?
Equivalently, does the free representable $l $-group of rank
$\eta$ have a basic element? (The analogs for right orders of
$F_{\eta}$ and free $l $-gro... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.51.
Proposed by: S. McCleary
Discussion and literature:
[Solved] No, it does not (S. Dovhyi, K. Muliarchyk, *Groups Geom. Dyn.*, **17** (2023), 613--632; \ K... | 2 | solved | S. McCleary | 1,992 | 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, it does not (S. Dovhyi, K. Muliarchyk, *Groups Geom. Dyn.*, **17** (2023), 613--632; \\ K. Muliarchy... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,871 | KOU-12.52 | Kourovka Notebook Problem 12.52 | Is the free $l $-group ${\mathcal{F}}_{\eta}$ of rank
$\eta $ ($1 < \eta < \infty$) Hopfian? That is, are $l
$-homomorphisms from ${\mathcal{F}}_{\eta}$ onto itself necessarily
one-to-one? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.52.
Proposed by: S. McCleary | 2 | open | S. McCleary | 1,992 | 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 |
700,872 | KOU-12.53 | Kourovka Notebook Problem 12.53 | Is it decidable whether or not two elements of a
free $l $-group are conjugate? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.53.
Proposed by: S. McCleary | 2 | open | S. McCleary | 1,992 | 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 |
700,873 | KOU-12.54 | Kourovka Notebook Problem 12.54 | Is there a normal valued $l $-group $G$ for
which there is no abelian $l $-group $A$ with $C(A) \cong C(G)$?
(Here $C(G)$ denotes the lattice of convex $l $-subgroups of
$G$. If $G$ is not required to be normal valued, this question has
an affirmative answer.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.54.
Proposed by: S. McCleary | 2 | open | S. McCleary | 1,992 | 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 |
700,874 | KOU-12.55 | Kourovka Notebook Problem 12.55 | Let $f(n,p)$ be the number of groups of order
$p^n$. Is $f(n,p)$ an increasing function of $p$ for any fixed $n
\geq 5$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.55.
Proposed by: A. Mann | 2 | open | A. Mann | 1,992 | 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 |
700,875 | KOU-12.56 | Kourovka Notebook Problem 12.56 | Let $\left< X\mid R\right>$ be a finite
presentation (the words in $R$ are assumed cyclically reduced).
Define the *length* of the presentation to be the sum of the
number of generators and the lengths of the relators. Let $F(n)$
be the number of (isomorphism types of) groups that have a
presentation of length at most ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.56.
Proposed by: A. Mann | 2 | open | A. Mann | 1,992 | 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 |
700,876 | KOU-12.57 | Kourovka Notebook Problem 12.57 | Suppose that a cyclic group $A$ of order 4 is a
$TI $-subgroup of a finite group $G$.
a) Does $A$ centralize a component $L$
of $G$ if $A$
intersects $L\cdot C(L)$ trivially?
b) What is the structure of the normal
closure of $A$ in the
case where $A$ centralizes each component of $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.57.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,992 | 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 |
700,877 | KOU-12.58 | Kourovka Notebook Problem 12.58 | A *generalized quadrangle* $GQ(s,t)$ with
parameters $s, t$ is by definition an incidence system consisting of
points and lines in which every line consists of $s + 1$ points,
every two different lines have at most one common point, each point
belongs to $t + 1$ lines, and for any point $a$ not lying on a line
$L$ ther... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.58.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,992 | 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 |
700,878 | KOU-12.59 | Kourovka Notebook Problem 12.59 | Does there exist a strongly regular graph with
parameters (85, 14, 3, 2) and a non-connected neighborhood of some
vertex? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.59.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,992 | 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 |
700,879 | KOU-12.60 | Kourovka Notebook Problem 12.60 | Describe the strongly regular graphs in which
the neighborhoods of vertices are generalized quadrangles (see 12.58). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.60.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,992 | 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 |
700,880 | KOU-12.62 | Kourovka Notebook Problem 12.62 | A *Frobenius group* is a transitive
permutation group in which the stabilizer of any two points is
trivial. Does there exist a Frobenius group of infinite degree
which is primitive as permutation group, and in which the
stabilizer of a point is cyclic and has only finitely many orbits?
*Conjecture:* No. Note that such... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.62.
Proposed by: P. M. Neumann | 2 | open | P. M. Neumann | 1,992 | 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 |
700,881 | KOU-12.63 | Kourovka Notebook Problem 12.63 | Does there exist a soluble permutation group of
infinite degree that has only finitely many orbits on triples?
*Conjecture:* No.
*Comment of 2001:* It is proved
(D. Macpherson, *in: Advances in algebra and model theory
$($Algebra Logic Appl.*, **9**), Gordon \& Breach, Amsterdam,
1997, 87--92) that every infinite solu... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.63.
Proposed by: P. M. Neumann | 2 | open | P. M. Neumann | 1,992 | 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 |
700,882 | KOU-12.64 | Kourovka Notebook Problem 12.64 | Is it true that for a given number $k \geq 2$
and for any (prime) number $n$, there exists a number $N=N(k,n)$
such that every finite group with generators $A=\{ a_1,\ldots
,a_k\} $ has exponent $\leq n$ if $(x_1\cdots x_N)^n=1$ for any
$x_1,\ldots ,x_N \in A \cup \{ 1\}$?
For given $k$ and $n$, a negative answer
impl... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.64.
Proposed by: A. Yu. Olshanskii | 2 | open | A. Yu. Olshanskii | 1,992 | 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 |
700,883 | KOU-12.65 | Kourovka Notebook Problem 12.65 | Let ${\mathcal{P}}=(P_0,P_1,P_2)$ be a parabolic system
in a finite group $G$, belonging to the $C_3$ Coxeter diagram \
$\parbox[t]{2ex}{ $ \circ $ \\ \mathstrut $
{}^1$}
\rule[0.5ex]{8ex}{0.1ex} \;
\parbox[t]{2ex}{ $ \circ $ \\ \mathstrut $
{}^2$}
\underline{\rule[0.5ex]{8ex{0.1ex}}} \;
\parbox[t]{2ex}{ $ \circ $ \\ \... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.65.
Proposed by: A. Pasini
Published resolution:
Yes, we can
(S. Yoshiara, *J. Algebraic Combin.*, **5**, no. 3 (1996),
251--284). | 2 | solved | A. Pasini | 1,992 | 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, we can\n(S. Yoshiara, *J. Algebraic Combin.*, **5**, no. 3 (1996),\n251--284)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,884 | KOU-12.66 | Kourovka Notebook Problem 12.66 | Describe the finite translation planes whose
collineation groups act doubly transitively on the affine points. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.66.
Proposed by: N. D. Podufalov | 2 | open | N. D. Podufalov | 1,992 | 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 |
700,885 | KOU-12.67 | Kourovka Notebook Problem 12.67 | Describe the structure of the locally compact
soluble groups with the maximum (minimum) condition for closed
non-compact subgroups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.67.
Proposed by: V. M. Poletskikh | 2 | open | V. M. Poletskikh | 1,992 | 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 |
700,886 | KOU-12.68 | Kourovka Notebook Problem 12.68 | Describe the locally compact abelian groups in
which any two closed subgroups of finite rank generate a subgroup
of finite rank. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.68.
Proposed by: V. M. Poletskikh | 2 | open | V. M. Poletskikh | 1,992 | 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 |
700,887 | KOU-12.69 | Kourovka Notebook Problem 12.69 | Let $F$ be a countably infinite field and $G$ a
finite group of automorphisms of $F$. The group ring ${\mathbb{Z}G$
acts on $F$ in a natural way. Suppose that $S$ is a subfield of
$F$ satisfying the following property: for any $x \in F$ there is
a non-zero element $f \in {\mathbb{Z}G$ such that $x^f \in S$. Is it
true ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.69.
Proposed by: K. N. Ponomar\"ev | 2 | open | K. N. Ponomar\"ev | 1,992 | 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 |
700,888 | KOU-12.70 | Kourovka Notebook Problem 12.70 | Let $p$ be a
prime number, $F$ a free pro-$p
$-group of finite rank,
and $\Delta \ne 1$ an automorphism of $F$ whose order is a power of
$p$. Is the rank of $\mathrm{Fix}_F(\Delta )=\{ x \in F\mid \Delta
(x)=x\}$ finite?
If the order of $\Delta$ is prime to $p$, then
$\mathrm{Fix}_F(\Delta )$ has infinite rank (W. Herf... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.70.
Proposed by: L. Ribes
Published resolution:
Yes, it is finite (C. Scheiderer, *Proc. Amer. Math. Soc.*,
**127**, no. 3 (1999),
695--700).
Moreover, in (W. N. Herfort, L. Ri... | 2 | solved | L. Ribes | 1,992 | 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 finite (C. Scheiderer, *Proc. Amer. Math. Soc.*,\n**127**, no. 3 (1999),\n695--700).\nMoreover, in (W. N. Herfort, L. Ribes, P. A. Zalesskii, *Forum M... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,889 | KOU-12.71 | Kourovka Notebook Problem 12.71 | Let $d(G)$ denote the smallest cardinality of a
generating set of the group $G$. Let $A$ and $B$ be finite groups.
Is there a finite group $G$ such that $A,B \leq G$, $G=\left<
A,B\right>$, and $d(G)=d(A) + d(B)$? The corresponding question
has negative answer in the class of solvable groups
(L. G. Kov\'acs, H.-S. Sim,... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.71.
Proposed by: L. Ribes
Published resolution:
No, not always (A. Lucchini, *J. Group Theory*, **4**,
no. 1 (2001), 53--58). | 2 | solved | L. Ribes | 1,992 | 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 (A. Lucchini, *J. Group Theory*, **4**,\nno. 1 (2001), 53--58)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,890 | KOU-12.72 | Kourovka Notebook Problem 12.72 | Let ${\mathfrak{F}}$ be a soluble local hereditary
formation of finite groups. Prove that ${\mathfrak{F}}$ is radical if
every finite soluble minimal non-${\mathfrak{F}} $-group $G$ is a
minimal non-${\mathfrak{N}}^{l(G)-1} $-group. Here ${\mathfrak{N}}$ is the
formation of all finite nilpotent groups, and $l(G)$ is th... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.72.
Proposed by: V. N. Semenchuk | 2 | open | V. N. Semenchuk | 1,992 | 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 |
700,891 | KOU-12.73 | Kourovka Notebook Problem 12.73 | A formation ${\mathfrak{H}}$ of finite groups is said to
have *length* $t$ if there is a chain of formations
$\varnothing ={\mathfrak{H}}_0 \subset {\mathfrak{H}}_1 \subset \cdots \subset
{\mathfrak{H}}_t={\mathfrak{H}}$ in which ${\mathfrak{H}}_{i-1}$ is a maximal
subformation of ${\mathfrak{H}}_i$. Is the lattice of ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.73.
Proposed by: A. N. Skiba | 2 | open | A. N. Skiba | 1,992 | 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 |
700,892 | KOU-12.74 | Kourovka Notebook Problem 12.74 | Let ${\mathfrak{F}}$ be a non-primary one-generator
composition formation of finite groups. Is it true that if ${\mathfrak{F}}={\mathfrak{M}}{\mathfrak{H}}$ and the formations ${\mathfrak{M}}$ and ${\mathfrak{H}}$
are non-trivial, then ${\mathfrak{M}}$ is a composition formation? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.74.
Proposed by: A. N. Skiba
Published resolution:
This is true if ${\mathfrak{F}}\ne {\mathfrak{H}}$ (A. N. Skiba, *Algebra of formations*, Belaruskaja Navuka, Minsk, 1997, p.... | 2 | solved | A. N. Skiba | 1,992 | 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 true if ${\\mathfrak{F}}\\ne {\\mathfrak{H}}$ (A. N. Skiba, *Algebra of formations*, Belaruskaja Navuka, Minsk, 1997, p. 144\n(Russian)). In general the ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,893 | KOU-12.75 | Kourovka Notebook Problem 12.75 | (B. Jonsson). Is the class $N$ of the lattices
of normal subgroups of groups a variety? It is proved
(C. Herrman,
W. Poguntke, *Algebra Univers.*,
**4**, no. 3 (1974), 280--286) that $N$ is an infinitely based
quasivariety. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.75.
Proposed by: D. M. Smirnov | 2 | open | D. M. Smirnov | 1,992 | 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 |
700,894 | KOU-12.76 | Kourovka Notebook Problem 12.76 | Is every group generated by a set of
3-transpositions locally finite? A *set of $3$-transpositions*
is, by definition, a normal set of involutions such that the orders
of their pairwise products are at most 3. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.76.
Proposed by: A. I. Sozutov
Published resolution:
Yes, it is
(H. Cuypers, J. I. Hall, in: *Groups, combinatorics and
geometry. Proc. L. M. S. Durham symp., July 5--15, 1990 ... | 2 | solved | A. I. Sozutov | 1,992 | 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\n(H. Cuypers, J. I. Hall, in: *Groups, combinatorics and\ngeometry. Proc. L. M. S. Durham symp., July 5--15, 1990 $($London\nMath. Soc. Lecture Note S... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,895 | KOU-12.77 | Kourovka Notebook Problem 12.77 | (Well-known problem). Does the order (if it is
greater than $p^2$) of a finite non-cyclic $p
$-group
divide the order of its automorphism group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.77.
Proposed by: A. I. Starostin
Published resolution:
No, not always (J. Gonz\'alez-S\'anchez, A. Jaikin-Zapirain, *Forum Math. Sigma*, **3**, Article ID e7, 11 p., electronic... | 2 | solved | A. I. Starostin | 1,992 | 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 (J. Gonz\\'alez-S\\'anchez, A. Jaikin-Zapirain, *Forum Math. Sigma*, **3**, Article ID e7, 11 p., electronic only (2015))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,896 | KOU-12.78 | Kourovka Notebook Problem 12.78 | (M. J. Curran). a) Does there exist a group of
order $p^6$ (here $p$ is a prime number), whose automorphism group
has also order $p^6$?
b) Is it true that for $p \equiv 1 (\mathrm{mod} 3)$, the smallest order of a $p
$-group that is
the automorphism group of a $p
$-group is $p^7$?
c) The same question for $p=3$ with ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.78.
Proposed by: A. I. Starostin
Published resolution:
a) No, there is no such a
group; b) yes, it is true; c) no, it is not true (S. Yu, G. Ban,
J. Zhang, *Algebra Colloq.*, *... | 2 | solved | A. I. Starostin | 1,992 | 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, there is no such a\ngroup; b) yes, it is true; c) no, it is not true (S. Yu, G. Ban,\nJ. Zhang, *Algebra Colloq.*, **3**, no. 2 (1996),\n97--106)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,897 | KOU-12.79 | Kourovka Notebook Problem 12.79 | Suppose that $a$ and $b$ are two elements of a
finite group $G$ such that the function
$\varphi
(g)=1^G(g) - 1_{\left< a\right>}^G(g) - 1_{\left< b\right>}^G(g)
- 1_{\left< ab\right>}^G(g) + 2$
is a character of $G$. Is it
true that $G=\left< a,b\right>$? The converse statement is true. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.79.
Proposed by: S. P. Strunkov
Published resolution:
No, not always: a counterexample is given by $G=A_4$, $a=b=(123)$. (S. V. Skresanov, *Algebra Logic*, **58**, no. 3 (2019)... | 2 | solved | S. P. Strunkov | 1,992 | 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: a counterexample is given by $G=A_4$, $a=b=(123)$. (S. V. Skresanov, *Algebra Logic*, **58**, no. 3 (2019),\n249--253)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,898 | KOU-12.80 | Kourovka Notebook Problem 12.80 | (K. W. Roggenkamp). a)
Is it true that the number of
$p
$-blocks of defect 0 of a finite group $G$ is equal
to the number of the conjugacy classes of elements $g \in G$ such
that the number of solutions of the equation $[x,y]=g$ in $G$ is
not divisible by $p$?
b) The same question in the case of $G$ being a simple gro... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.80.
Proposed by: S. P. Strunkov
Published resolution:
a) No it is not true; for example, if $p=2$ and $G={\mathbb{Z} _3\times
{\mathbb{S} _3$ (L. Barker, *Letter of May, 27, 19... | 2 | solved | S. P. Strunkov | 1,992 | 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 it is not true; for example, if $p=2$ and $G={\\mathbb{Z} _3\\times\n{\\mathbb{S} _3$ (L. Barker, *Letter of May, 27, 1996*).\n\nb) No, not always. For exa... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,899 | KOU-12.81 | Kourovka Notebook Problem 12.81 | What is the cardinality of the set of
subvarieties of the group variety ${\mathfrak{A}}_p^3$ (where ${\mathfrak{A}}_p$ is the variety of abelian groups of prime exponent $p$)? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.81.
Proposed by: V. I. Sushchanski\u{\i | 2 | open | V. I. Sushchanski\u{\i | 1,992 | 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 |
700,900 | KOU-12.82 | Kourovka Notebook Problem 12.82 | Find all pairs
$(n,r)$ such that the symmetric group ${\mathbb{S}_n$ contains a maximal
subgroup isomorphic to ${\mathbb{S}_r$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.82.
Proposed by: V. I. Sushchanski\u{\i
Published resolution:
They are found
(B. Newton, B. Benesh, *J. Algebra*, **304**, no. 2
(2006), 1108--1113). | 2 | solved | V. I. Sushchanski\u{\i | 1,992 | 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: They are found\n(B. Newton, B. Benesh, *J. Algebra*, **304**, no. 2\n(2006), 1108--1113)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,901 | KOU-12.84 | Kourovka Notebook Problem 12.84 | (Well-known problem). Is it true that if there
exist two non-isomorphic groups with the given set of orders
of the elements, then there are infinitely many groups with this
set of orders of the elements? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.84.
Proposed by: S. A. Syskin
Published resolution:
No, it is not
(V. D. Mazurov, *Algebra and Logic*, **33**, no. 1
(1994), 49--55). | 2 | solved | S. A. Syskin | 1,992 | 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. D. Mazurov, *Algebra and Logic*, **33**, no. 1\n(1994), 49--55)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,902 | KOU-12.85 | Kourovka Notebook Problem 12.85 | Does every variety which is generated by a (known)
finite simple group containing a soluble subgroup of derived length
$d$ contain a 2-generator soluble group of derived length $d$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.85.
Proposed by: S. A. Syskin | 2 | open | S. A. Syskin | 1,992 | 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 |
700,903 | KOU-12.86 | Kourovka Notebook Problem 12.86 | For each known finite simple group, find its
maximal 2-generator direct power. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.86.
Proposed by: S. A. Syskin | 2 | open | S. A. Syskin | 1,992 | 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 |
700,904 | KOU-12.87 | Kourovka Notebook Problem 12.87 | Let $\Gamma$ be a connected undirected graph
without loops or multiple edges and suppose that the automorphism
group $\mathrm{Aut} (\Gamma )$ acts transitively on the vertex set
of $\Gamma$. Is it true that at least one of the following
assertions holds?
1.
The stabilizer of a vertex
of $\Gamma$ in
$\mathrm{Aut} (\Ga... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.87.
Proposed by: V. I. Trofimov | 2 | open | V. I. Trofimov | 1,992 | 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 |
700,905 | KOU-12.88 | Kourovka Notebook Problem 12.88 | An undirected graph is called a *locally finite
Cayley graph of a group* $G$ if its vertex set can be identified
with the set of elements of $G$ in such a way that, for some
finite generating set $X=X^{-1}$ of $G$ not containing $1$, two
vertices $g$ and $h$ are adjacent if and only if $g^{-1}h \in X$.
Do there exist t... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.88.
Proposed by: V. I. Trofimov | 2 | open | V. I. Trofimov | 1,992 | 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 |
700,906 | KOU-12.89 | Kourovka Notebook Problem 12.89 | Describe the infinite connected graphs to which the
sequences of finite connected graphs with primitive automorphism
groups converge. For definitions, see (V. I. Trofimov, *Algebra and Logic*, **28**, no. 3 (1989), 220--237). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.89.
Proposed by: V. I. Trofimov | 2 | open | V. I. Trofimov | 1,992 | 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 |
700,907 | KOU-12.90 | Kourovka Notebook Problem 12.90 | Let $G$ be a finitely generated soluble minimax
group and let $H$ be a finitely generated residually finite group
which has precisely the same finite images as $G$. Must $H$ be a
minimax group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.90.
Proposed by: John S. Wilson
Published resolution:
Yes, it must
(A. Mann, D. Segal, *Proc.
London Math. Soc. (3)*, **61** (1990), 529--545; see also
A. V. Tushev, *Math. Not... | 2 | solved | John S. Wilson | 1,992 | 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 must\n(A. Mann, D. Segal, *Proc.\nLondon Math. Soc. (3)*, **61** (1990), 529--545; see also\nA. V. Tushev, *Math. Notes*, **56**, no. 5 (1994),\n1190--11... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,908 | KOU-12.91 | Kourovka Notebook Problem 12.91 | Every metabelian group
belonging to a Fitting class of (finite) supersoluble groups is
nilpotent. Does the following generalization also hold: Every
group belonging to a Fitting class of supersoluble groups has only
central minimal normal subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.91.
Proposed by: H. Heineken
Published resolution:
No, it does not
(H. Heineken, *Rend. Sem. Mat. Univ. Padova*, **98**
(1997), 241--251). | 2 | solved | H. Heineken | 1,992 | 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 does not\n(H. Heineken, *Rend. Sem. Mat. Univ. Padova*, **98**\n(1997), 241--251)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,909 | KOU-12.92 | Kourovka Notebook Problem 12.92 | For a field $K$ of characteristic 2, each finite
group $G=\{ g_1,\ldots ,g_n\}$ of odd order $n$ is determined up to
isomorphism by its group determinant (Formanek--Sibley, 1990) and
even by its *reduced norm* which is defined as the last
coefficient $s_m(x)$ of the minimal polynomial $\varphi (\lambda
;x)=\lambda ^m -... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.92.
Proposed by: H.-J. Hoehnke | 2 | open | H.-J. Hoehnke | 1,992 | 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 |
700,910 | KOU-12.93 | Kourovka Notebook Problem 12.93 | Let $N \rightarrowtail G \twoheadrightarrow Q$
be an extension of nilpotent groups, with $Q$ finitely generated,
which splits at every prime. Does the extension split? This is
known to be true if $N$ is finite or commutative. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.93.
Proposed by: P. Hilton
Published resolution:
No, not always (K. Lorensen,
*Math. Proc. Cambridge Philos. Soc.*, **123**, no. 2
(1998), 213--215). | 2 | solved | P. Hilton | 1,992 | 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 (K. Lorensen,\n*Math. Proc. Cambridge Philos. Soc.*, **123**, no. 2\n(1998), 213--215)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,911 | KOU-12.94 | Kourovka Notebook Problem 12.94 | Let $G$ be a finitely generated
pro-$p
$-group not involving the wreath product $C_p\wr
{\mathbb{Z}_p$ as a closed section (where $C_p$ is a cyclic group of
order $p$ and ${\mathbb{Z}_p$ is the group of $p
$-adic
integers). Does it follow that $G$ is $p
$-adic
analytic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.94.
Proposed by: A. Shalev
Published resolution:
No, it does not (A. Jaikin-Zapirain, B. Klopsch, *J. London Math. Soc. (2)*, **76**, no. 2 (2007), 365--383). | 2 | solved | A. Shalev | 1,992 | 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 does not (A. Jaikin-Zapirain, B. Klopsch, *J. London Math. Soc. (2)*, **76**, no. 2 (2007), 365--383)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,912 | KOU-12.95 | Kourovka Notebook Problem 12.95 | Let $G$ be a finitely generated
pro-$p
$-group and let $g_1,\ldots ,g_n \in G$. Let $H$
be an open subgroup of $G$, and suppose there exists a non-trivial
word $w=w(X_1,\ldots ,X_n)$ such that $w(a_1,\ldots ,a_n)=1$
whenever $a_1 \in g_1H,\ldots ,a_n \in g_nH$ (that is, $G$
satisfies a *coset identity*). Does it follow... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.95.
Proposed by: A. Shalev | 2 | open | A. Shalev | 1,992 | 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 |
700,913 | KOU-12.96 | Kourovka Notebook Problem 12.96 | Find a
non-empty Fitting class ${\mathfrak{F}}$ and a non-soluble finite group
$G$ such that $G$ has no ${\mathfrak{F}} $-injectors. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.96.
Proposed by: L. A. Shemetkov
Published resolution:
Found by E. Salomon
(Mainz Univ., unpublished); his example is presented in \S 7.1 of
(A. Ballester-Bolinches, L. M. Ezqu... | 2 | solved | L. A. Shemetkov | 1,992 | 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: Found by E. Salomon\n(Mainz Univ., unpublished); his example is presented in \\S 7.1 of\n(A. Ballester-Bolinches, L. M. Ezquerro, *Classes of finite\ngroups*, Sp... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,914 | KOU-12.97 | Kourovka Notebook Problem 12.97 | Let ${\mathfrak{F}}$ be the formation of all finite
groups all of whose composition factors are isomorphic to some
fixed simple non-abelian group $T$. Prove that ${\mathfrak{F}}$ is
indecomposable into a product of two non-trivial subformations. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.97.
Proposed by: L. A. Shemetkov
Published resolution:
It is proved (O. V. Mel'nikov,
*Problems in Algebra*, **9**, Gomel', 1996, 42--47 (Russian)). | 2 | solved | L. A. Shemetkov | 1,992 | 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 (O. V. Mel'nikov,\n*Problems in Algebra*, **9**, Gomel', 1996, 42--47 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,915 | KOU-12.98 | Kourovka Notebook Problem 12.98 | Let $F$ be a free group of finite rank, $R$ its
recursively defined normal subgroup. Is it true that
a) the word problem for
$F/R$ is soluble if and only if it is soluble for $F/[R,R]$?
b) the conjugacy problem for
$F/R$ is soluble if and only if it is soluble for $F/[R,R]$?
c) the conjugacy problem
for $F/[R,R]$ is... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.98.
Proposed by: V. E. Shpil'rain
Published resolution:
a) Yes, it is; b)
No, it is not; c) No, it is not (M. I. Anokhin, *Math.
Notes*, **61**, no. 1--2 (1997), 3--8). | 2 | solved | V. E. Shpil'rain | 1,992 | 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; b)\nNo, it is not; c) No, it is not (M. I. Anokhin, *Math.\nNotes*, **61**, no. 1--2 (1997), 3--8)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,916 | KOU-12.100 | Kourovka Notebook Problem 12.100 | Is every periodic group with a regular
automorphism of order 4 locally finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.100.
Proposed by: P. V. Shumyatsky | 2 | open | P. V. Shumyatsky | 1,992 | 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 |
700,917 | KOU-12.101 | Kourovka Notebook Problem 12.101 | We call a group $G$ containing an involution $i$ a
*$T_0 $-group* if
1) the order of the product of any two
involutions conjugate
to $i$ is finite;
2) all 2-subgroups of $G$ are either cyclic
or generalized
quaternion;
3) the centralizer $C$ of the involution
$i$ in $G$ is infinite, distinct from $G$, and has finite... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.101.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,992 | 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 |
700,918 | KOU-12.102 | Kourovka Notebook Problem 12.102 | Is
every proper factor-group of a group of Golod (see 9.76) residually
finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.102.
Proposed by: V. P. Shunkov
Published resolution:
No, not every (L. Hammoudi, *Algebra Colloq.*, **5** (1998),
371--376). | 2 | solved | V. P. Shunkov | 1,992 | 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 (L. Hammoudi, *Algebra Colloq.*, **5** (1998),\n371--376)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,919 | KOU-13.1 | Kourovka Notebook Problem 13.1 | Let $U(K)$ denote the group
of units of a ring $K$. Let $G$ be a finite group, ${\mathbb{Z} G$
the integral group ring of $G$, and ${\mathbb{Z} _pG$ the group ring
of $G$ over the residues modulo a prime number $p$.
Describe the homomorphism
from $U({\mathbb{Z} G)$ into $U({\mathbb{Z} _pG)$ induced by reducing the
coef... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.1.
Proposed by: R. Zh. Aleev | 2 | open | R. Zh. Aleev | 1,995 | 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 |
700,920 | KOU-13.2 | Kourovka Notebook Problem 13.2 | Does there exist a finitely
based variety of groups
${\mathfrak{V}}$ such that the word problem is solvable
in $F_n({\mathfrak{V}})$ for every positive integer $n$,
but is unsolvable in
$F_{\infty}({\mathfrak{V}})$ (with respect to a free
generating system)? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.2.
Proposed by: M. I. Anokhin | 2 | open | M. I. Anokhin | 1,995 | 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 |
700,921 | KOU-13.3 | Kourovka Notebook Problem 13.3 | Let ${\mathfrak{M}}$ be an
arbitrary variety of groups. Is it true that every infinitely
generated projective group in
${\mathfrak{M}}$ is an ${\mathfrak{M}} $-free product of countably generated projective groups in
${\mathfrak{M}}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.3.
Proposed by: V. A. Artamonov | 2 | open | V. A. Artamonov | 1,995 | 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 |
700,922 | KOU-13.4 | Kourovka Notebook Problem 13.4 | Let $G$ be a group with a
normal (pro-)
2-subgroup $N$ such that $G/N$ is
isomorphic to $GL_n(2)$, inducing its natural module on $N/\Phi
(N)$, the Frattini factor-group of $N$. For $n=3$ determine $G$
such that $N$ is as large as possible. (For $n>3$ it can be proved
that already the Frattini subgroup $\Phi (N)$ of $N... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.4.
Proposed by: B. Baumann | 2 | open | B. Baumann | 1,995 | 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 |
700,923 | KOU-13.5 | Kourovka Notebook Problem 13.5 | Groups $A$ and $B$ are said to
be *locally equivalent* if for every finitely generated
subgroup $X\leq A$ there is a subgroup $Y\leq B$ isomorphic to
$X$, and conversely, for every finitely generated subgroup
$Y\leq B$ there is a subgroup $X\leq A$, isomorphic to
$Y$. We call a group $G$ *categorical* if $G$ is isomorp... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.5.
Proposed by: V. V. Belyaev | 2 | open | V. V. Belyaev | 1,995 | 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 |
700,924 | KOU-13.6 | Kourovka Notebook Problem 13.6 | (B. Hartley). Is it true that
a locally finite group containing an element with Chernikov
centralizer is almost soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.6.
Proposed by: V. V. Belyaev | 2 | open | V. V. Belyaev | 1,995 | 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 |
700,925 | KOU-13.7 | Kourovka Notebook Problem 13.7 | (B. Hartley). Is it true that
a simple locally finite group containing a finite subgroup with
finite centralizer is linear? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.7.
Proposed by: V. V. Belyaev | 2 | open | V. V. Belyaev | 1,995 | 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 |
700,926 | KOU-13.8 | Kourovka Notebook Problem 13.8 | (B. Hartley). Is it true that a locally soluble
periodic group has a finite normal series with locally nilpotent
factors if it contains an element
a)
with finite centralizer?
b) with Chernikov centralizer? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.8.
Proposed by: V. V. Belyaev
Discussion and literature:
[Solved] a) Yes, it is true (E. Khukhro,
*Bull. London Math. Soc.*, **58**, no. 1 (2026), Article ID... | 2 | partially_solved | V. V. Belyaev | 1,995 | 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": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] a) Yes, it is true (E. Khukhro,\n*Bull. London Math. Soc.*, **58**, no. 1 (2026), Article ID e... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,927 | KOU-13.9 | Kourovka Notebook Problem 13.9 | Is it true that a locally
soluble periodic group containing a finite nilpotent subgroup with
Chernikov centralizer has a finite normal series with locally
nilpotent factors? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.9.
Proposed by: V. V. Belyaev,
B. Hartley | 2 | open | V. V. Belyaev,
B. Hartley | 1,995 | 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 |
700,928 | KOU-13.10 | Kourovka Notebook Problem 13.10 | Is there a function
$f\! : {\mathbb{N} \rightarrow {\mathbb{N}$ such that, for every soluble
group $G$ of derived length $k$ generated by a set $A$, the
validity of the identity
$x^4=1$ on each subgroup generated by at most $f(k)$
elements of $A$ implies that $G$ is a group of exponent 4? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.10.
Proposed by: V. V. Bludov
Published resolution:
No, there is not (G. S. Deryabina, A. N. Krasil'nikov, *Siberian Math. J.*, **44**, no. 1 (2003), 58--60). | 2 | solved | V. V. Bludov | 1,995 | 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, there is not (G. S. Deryabina, A. N. Krasil'nikov, *Siberian Math. J.*, **44**, no. 1 (2003), 58--60)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,929 | KOU-13.11 | Kourovka Notebook Problem 13.11 | Is a torsion-free
group almost polycyclic if it has a finite set of generators
$a_1,\ldots , a_n$ such that every element of the group has a
unique presentation in the form $a_1^{k_1}\cdots a_n^{k_n}$,
where $k_1,\ldots , k_n\in {\mathbb{Z}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.11.
Proposed by: V. V. Bludov
Published resolution:
No, not always
(A. Muranov, *Trans. Amer. Math. Soc.*, **359** (2007),
3609--3645). | 2 | solved | V. V. Bludov | 1,995 | 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(A. Muranov, *Trans. Amer. Math. Soc.*, **359** (2007),\n3609--3645)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,930 | KOU-13.12 | Kourovka Notebook Problem 13.12 | Is the group of all automorphisms of an
arbitrary hyperbolic group finitely presented?
*Comment of 2001:* This is proved for the
group of automorphisms of a torsion-free hyperbolic group that
cannot be decomposed into a free product (Z. Sela, *Geom.
Funct. Anal.*, **7**, no. 3 (1997) 561--593). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.12.
Proposed by: O. V. Bogopolski | 2 | open | O. V. Bogopolski | 1,995 | 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 |
700,931 | KOU-13.13 | Kourovka Notebook Problem 13.13 | Let $G$ and $H$ be finite $p
$-groups
with isomorphic Burnside rings. Is the nilpotency class of
$H$ bounded by some function of the class of $G$? There is an
example where $G$ is of class 2 and $H$ is of class 3. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.13.
Proposed by: R. Brandl | 2 | open | R. Brandl | 1,995 | 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 |
700,932 | KOU-13.14 | Kourovka Notebook Problem 13.14 | Is the lattice of quasivarieties of nilpotent
torsion-free groups of nilpotency class $\leq 2$
distributive? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.14.
Proposed by: A. I. Budkin | 2 | open | A. I. Budkin | 1,995 | 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 |
700,933 | KOU-13.15 | Kourovka Notebook Problem 13.15 | Does a free non-abelian nilpotent group of class
3 possess an independent basis of quasiidentities in the class of
torsion-free groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.15.
Proposed by: A. I. Budkin | 2 | open | A. I. Budkin | 1,995 | 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 |
700,934 | KOU-13.16 | Kourovka Notebook Problem 13.16 | Is every locally nilpotent group with minimum
condition on centralizers hypercentral? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.16.
Proposed by: F. O Wagner
Published resolution:
Yes, it is (V. V. Bludov,
*Algebra and Logic*, **37**, no. 3 (1998), 151--156). | 2 | solved | F. O Wagner | 1,995 | 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. V. Bludov,\n*Algebra and Logic*, **37**, no. 3 (1998), 151--156)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,935 | KOU-13.17 | Kourovka Notebook Problem 13.17 | A representation of a group $G$ on a vector
space $V$ is called a *nil-representation* if for every
$v$ in $V$ and $g$ in $G$ there exists $n=n(v,g)$ such that $v
(g-1)^n = 0$. Is it true that in zero characteristic every
irreducible nil-representation is trivial? (In prime
characteristic it is not true.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.17.
Proposed by: S. Vovsi | 2 | open | S. Vovsi | 1,995 | 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 |
700,936 | KOU-13.18 | Kourovka Notebook Problem 13.18 | Let $F$ be a finitely generated non-Abelian free group and let $G$
be the Cartesian (unrestricted) product of countable infinity of
copies of $F$. Must the Abelianization $G/G'$ of $G$ be
torsion-free? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.18.
Proposed by: A. M. Gaglione, D. Spellman
Published resolution:
No, it may contain elements of order $2$ \ (O. Kharlampovich,
A. Myasnikov, in: *Knots, braids, and mapping c... | 2 | solved | A. M. Gaglione, D. Spellman | 1,995 | 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 may contain elements of order $2$ \\ (O. Kharlampovich,\nA. Myasnikov, in: *Knots, braids, and mapping class\ngroups --- papers dedicated to Joan S. Birma... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,937 | KOU-13.19 | Kourovka Notebook Problem 13.19 | Suppose that both a group $Q$ and its normal
subgroup
$H$ are subgroups of the direct product $G_1\times \cdots \times
G_n$ such that for each $i$ the projections of both $Q$ and $H$
onto $G_i$ coincide with $G_i$. If $Q/H$ is a $p
$-group, is $Q/H$ a regular $p
$-group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.19.
Proposed by: Yu. M. Gorchakov | 2 | open | Yu. M. Gorchakov | 1,995 | 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 |
700,938 | KOU-13.20 | Kourovka Notebook Problem 13.20 | Is it true that the generating series of the
growth function of every finitely generated group with one
defining relation represents an algebraic (or even a rational)
function? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.20.
Proposed by: R. I. Grigorchuk | 2 | open | R. I. Grigorchuk | 1,995 | 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 |
700,939 | KOU-13.21 | Kourovka Notebook Problem 13.21 | b) What is the minimal possible rate of
growth of the function $
\pi (n)=\max\limits_{\delta (g)\leq n} |g|$ for the class of
groups indicated in part ``a'' of this problem (see Archive)? It is
known (R. I. Grigorchuk, *Math. USSR--Izv.*, **25**
(1985), 259--300) that there exist $p
$-groups with
$\pi (n)\leq n^{\lambd... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 13.21.
Proposed by: R. I. Grigorchuk
Archive of Solved Problems entry:
Yes, there is (A. V. Rozhkov, *Dokl. Math.*, **58**,
no. 2 (1998), 234--237). | 2 | partially_solved | R. I. Grigorchuk | 1,995 | 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 is (A. V. Rozhkov, *Dokl. Math.*, **58**,\nno. 2 (1998), 234--237).); 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 |
700,940 | KOU-13.22 | Kourovka Notebook Problem 13.22 | Let the group $G=AB$ be the product of two polycyclic subgroups
$A$ and $B$, and assume that $G$ has an ascending normal series
with locally nilpotent factors (i. e. $G$ is radical). Is it true
that $G$ is polycyclic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.22.
Proposed by: F. de Giovanni | 2 | open | F. de Giovanni | 1,995 | 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 |
700,941 | KOU-13.23 | Kourovka Notebook Problem 13.23 | Let the group $G$ have a finite normal
series with infinite cyclic factors (containing of course $G$ and
$\{1\}$). Is it true that $G$ has a non-trivial outer
automorphism? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.23.
Proposed by: F. de Giovanni
Discussion and literature:
[Solved] No, not always (M. Brescia, E. Ingrosso, M. Trombetti, *Preprint*, 2026, https://arxiv.or... | 2 | solved | F. de Giovanni | 1,995 | 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 (M. Brescia, E. Ingrosso, M. Trombetti, *Preprint*, 2026, https://arxiv.org/abs/2607.1177... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,942 | KOU-13.24 | Kourovka Notebook Problem 13.24 | Let $G$ be a non-discrete topological group with
only finitely many ultrafilters that converge to the identity. Is
it true that
$G$ contains a countable open subgroup of exponent 2?
*Comment of 2005:* This is proved for $G$
countable (E. G. Zelenyuk, *Mat. Studii*, **7**
(1997), 139--144). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.24.
Proposed by: E. G. Zelenyuk | 2 | open | E. G. Zelenyuk | 1,995 | 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 |
700,943 | KOU-13.25 | Kourovka Notebook Problem 13.25 | Let $(G,\tau )$ be a topological group with
finite semigroup $\tau (G)$ of ultrafilters converging to the
identity (I. V. Protasov, *Siberian Math. J.*, **34**,
no. 5 (1993), 938--951). Is it true that
$\tau (G)$ is a semigroup of idempotents?
*Comment of 2005:* This is proved for $G$
countable
(E. G. Zelenyuk, *Mat. ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.25.
Proposed by: E. G. Zelenyuk | 2 | open | E. G. Zelenyuk | 1,995 | 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 |
700,944 | KOU-13.26 | Kourovka Notebook Problem 13.26 | Is it true that a countable topological group of
exponent 2 with unique free ultrafilter converging to the identity
has a basis of neighborhoods of the identity consisting of
subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.26.
Proposed by: E. G. Zelenyuk,
I. V. Protasov
Published resolution:
Assuming Martin's Axiom, the answer is ``No'' (Ye. Zelenyuk, *Adv. Math.*, **229**, no. 4 (2012), 2415--24... | 2 | solved | E. G. Zelenyuk,
I. V. Protasov | 1,995 | 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: Assuming Martin's Axiom, the answer is ``No'' (Ye. Zelenyuk, *Adv. Math.*, **229**, no. 4 (2012), 2415--2426)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,945 | KOU-13.27 | Kourovka Notebook Problem 13.27 | (B. Amberg). Suppose that $G=AB=AC=BC$ for a
group $G$ and its subgroups
$A,B,C$.
a) Is $G$ a Chernikov group if
$
A,B,C$ are Chernikov groups?
b) Is $G$ almost polycyclic if
$A,B,C$ are almost polycyclic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.27.
Proposed by: L. S. Kazarin | 2 | open | L. S. Kazarin | 1,995 | 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 |
700,946 | KOU-13.28 | Kourovka Notebook Problem 13.28 | (D. M. Evans). A permutation group on an
infinite set is *cofinitary* if its non-identity elements
fix only finitely many points. Is it true that a closed cofinitary
permutation group is locally compact (in the topology of
pointwise convergence)? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.28.
Proposed by: P. J. Cameron
Published resolution:
No, not always (G. Hjorth, *J. Algebra*, **200**, no. 2
(1998), 439--448). | 2 | solved | P. J. Cameron | 1,995 | 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 (G. Hjorth, *J. Algebra*, **200**, no. 2\n(1998), 439--448)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,947 | KOU-13.29 | Kourovka Notebook Problem 13.29 | Given an infinite set $\Omega$, define an
algebra $A$ (the *reduced incidence algebra of finite
subsets*) as follows. Let $V_n$ be the set of functions from the
set of $n $-element subsets of $\Omega$ to the rationals $\mathbb{Q}$.
Now let $A=\bigoplus V_n$, with multiplication as follows: for
$f\in V_n$, $g\in V_m$, a... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.29.
Proposed by: P. J. Cameron
Published resolution:
Conjecture is proved (M. Pouzet, *Theor. Inf. App.*, **42**, no. 1 (2008), 83--103). | 2 | solved | P. J. Cameron | 1,995 | 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: Conjecture is proved (M. Pouzet, *Theor. Inf. App.*, **42**, no. 1 (2008), 83--103)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,948 | KOU-13.30 | Kourovka Notebook Problem 13.30 | A group
$G$ is called a *$B$-group* if every primitive
permutation group which contains the regular representation of
$G$ is doubly transitive. Are there any countable $B $-groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.30.
Proposed by: P. J. Cameron | 2 | open | P. J. Cameron | 1,995 | 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 |
700,949 | KOU-13.31 | Kourovka Notebook Problem 13.31 | Let
$G$ be a permutation group on a set
$\Omega$. A sequence of points of $\Omega$ is a *base* for
$G$ if its pointwise stabilizer in $G$ is the identity. *The
greedy algorithm* for a base chooses each point in the sequence
from an orbit of maximum size of the stabilizer of its
predecessors. Is it true that there is a ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.31.
Proposed by: P. J. Cameron | 2 | open | P. J. Cameron | 1,995 | 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 |
700,950 | KOU-13.32 | Kourovka Notebook Problem 13.32 | (Well-known problem). On a group, a partial
order that is directed upwards and has linearly ordered cone of
positive elements is said to be *semilinear* if this order
is invariant under right multiplication by the group elements. Can
every semilinear order of a group be extended to a right order of
the group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.32.
Proposed by: V. M. Kopytov | 2 | open | V. M. Kopytov | 1,995 | 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 |
700,951 | KOU-13.33 | Kourovka Notebook Problem 13.33 | (F. Gross).
Is a normal subgroup of a finite $D_{\pi} $-group (see Archive, 3.62)
always a $D_{\pi} $-group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.33.
Proposed by: V. D. Mazurov
Published resolution:
Yes, it is mod CFSG
(E. P. Vdovin, D. O. Revin, *in: Ischia Group Theory 2004 $($Contemp.
Math.*, **402**), Amer. Math. Soc... | 2 | solved | V. D. Mazurov | 1,995 | 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\n(E. P. Vdovin, D. O. Revin, *in: Ischia Group Theory 2004 $($Contemp.\nMath.*, **402**), Amer. Math. Soc., 2006, 229--263)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,952 | KOU-13.34 | Kourovka Notebook Problem 13.34 | (I. D. Macdonald). If the identity $[x,
y]^n=1$ holds on a group, is the derived subgroup of the group
periodic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.34.
Proposed by: V. D. Mazurov
Published resolution:
No, not always (G. S. Deryabina, P. A. Kozhevnikov, *Commun. Algebra*, **27**, no. 9 (1999), 4525--4530; \
S. I. Adyan, *Do... | 2 | solved | V. D. Mazurov | 1,995 | 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 (G. S. Deryabina, P. A. Kozhevnikov, *Commun. Algebra*, **27**, no. 9 (1999), 4525--4530; \\\nS. I. Adyan, *Dokl.\nMath.*, **62**, no. 2 (2000), 1... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,953 | KOU-13.35 | Kourovka Notebook Problem 13.35 | Does every non-soluble pro-$p
$-group
of cohomological dimension
2 contain a free non-abelian
pro-$p
$-subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.35.
Proposed by: O. V. Mel'nikov | 2 | open | O. V. Mel'nikov | 1,995 | 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 |
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