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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
700,354 | KOU-6.7 | Kourovka Notebook Problem 6.7 | Suppose that $P$ is a finite 2-group. Does there
exist a characteristic subgroup $L(P)$ of $P$ such that $L(P)$ is
normal in $H$ for every finite group $H$ that satisfies the
following conditions: 1) $P$ is a Sylow 2-subgroup of $H$,
2) $H$ is ${\mathbb{S}_4 $-free, and 3) $C_H(O_2(H)) \leq O_2(H)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.7.
Proposed by: G. Glauberman
Published resolution:
Yes, there does (B. Stellmacher, *Israel J. Math.*, **94**
(1996), 367--379). | 2 | solved | G. Glauberman | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there does (B. Stellmacher, *Israel J. Math.*, **94**\n(1996), 367--379)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,355 | KOU-6.8 | Kourovka Notebook Problem 6.8 | Can a residually finite locally normal group be embedded in a
Cartesian product of finite groups in such a way that each
element of the group has at most finitely many central
projections? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.8.
Proposed by: Yu. M. Gorchakov
Published resolution:
Yes, it can (M. J. Tomkinson, *Bull. London Math. Soc.*, **13**
(1981),
133--137). | 2 | solved | Yu. M. Gorchakov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it can (M. J. Tomkinson, *Bull. London Math. Soc.*, **13**\n(1981),\n133--137)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,356 | KOU-6.9 | Kourovka Notebook Problem 6.9 | Is
the derived subgroup of any locally normal group decomposable into
a product of at most countable subgroups which commute
elementwise? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.9.
Proposed by: Yu. M. Gorchakov
Discussion and literature:
[Solved] Yes, it is (AI solution confirmed by M. Dixon: \ Achyuth Jayadevan, *Preprint*, 2026,
htt... | 2 | solved | Yu. M. Gorchakov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is (AI solution confirmed by M. Dixon: \\ Achyuth Jayadevan, *Preprint*, 2026,\n\nhttps://zenodo... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,357 | KOU-6.10 | Kourovka Notebook Problem 6.10 | Let $p$ be a prime and $n$ be an integer with $p > 2n + 1$. Let
$x, y$ be $p
$-elements in $GL_n({\mathbb{C})$. If the
subgroup $\left< x,y\right>$ is finite, then it is an abelian
$p
$-group (W. Feit, J. G. Thompson, *Pacif.
J. Math.*, **11**, no. 4 (1961), 1257--1262). What can be said
about $\left< x,y\right> $ in t... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.10.
Proposed by: J. D. Dixon | 2 | open | J. D. Dixon | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,358 | KOU-6.11 | Kourovka Notebook Problem 6.11 | Let ${\mathfrak{L}}$ be the class of locally compact groups with no
small subgroups (see D. Montgomery, L. Zippin, *Topological
transformation groups*, New York, 1955; V. M. Glushkov, *Uspekhi Matem. Nauk*, **12**, no. 2 (1957), 3--41 (Russian)).
Study extensions of groups in this class with the objective of
giving a d... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.11.
Proposed by: J. D. Dixon | 2 | open | J. D. Dixon | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,359 | KOU-6.12 | Kourovka Notebook Problem 6.12 | a) Is a metabelian group countable if it satisfies the weak
minimum condition for normal subgroups?
b) Is such a group minimax if it is
torsion-free? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.12.
Proposed by: D. I. Zaitsev
Published resolution:
Yes, it is, in both cases (D. I. Zaitsev, L. A. Kurdachenko,
A. V. Tushev, *Algebra and Logic*, **24** (1985), 412--436). | 2 | solved | D. I. Zaitsev | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is, in both cases (D. I. Zaitsev, L. A. Kurdachenko,\nA. V. Tushev, *Algebra and Logic*, **24** (1985), 412--436)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,360 | KOU-6.13 | Kourovka Notebook Problem 6.13 | Is it true that if a non-abelian Sylow 2-subgroup of a finite
group $G$ has a non-trivial abelian direct factor, then $G$ is
not simple? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.13.
Proposed by: A. S. Kondratiev
Published resolution:
Yes, it is, mod CFSG (V. V. Kabanov, A. S. Kondratiev, *Sylow
$2$-subgroups of finite groups (a survey)*, Inst. Math. Mech... | 2 | solved | A. S. Kondratiev | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is, mod CFSG (V. V. Kabanov, A. S. Kondratiev, *Sylow\n$2$-subgroups of finite groups (a survey)*, Inst. Math. Mech. UNC\nAN SSSR, Sverdlovsk, 1979 (Russ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,361 | KOU-6.14 | Kourovka Notebook Problem 6.14 | Are the following lattices locally finite: the lattice of
all locally finite varieties of groups? the lattice of all varieties
of groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.14.
Proposed by: A. V. Kuznetsov
Published resolution:
No, they are not (M. I. Anokhin, *Izv. Math.*, **63**,
no. 4 (1999),
649--665). | 2 | solved | A. V. Kuznetsov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, they are not (M. I. Anokhin, *Izv. Math.*, **63**,\nno. 4 (1999),\n649--665)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,362 | KOU-6.15 | Kourovka Notebook Problem 6.15 | A variety is said to be *pro-locally-finite* if it is not
locally finite while all of its proper subvarieties are locally
finite. An example --- the variety of abelian groups. How many
pro-locally-finite varieties of groups are there? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.15.
Proposed by: A. V. Kuznetsov
Published resolution:
There are continuum of
such varieties (P. A. Kozhevnikov, *On varieties of groups of large odd
exponent*, Dep. 1612-V00, VI... | 2 | solved | A. V. Kuznetsov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: There are continuum of\nsuch varieties (P. A. Kozhevnikov, *On varieties of groups of large odd\nexponent*, Dep. 1612-V00, VINITI, Moscow, 2000 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,363 | KOU-6.16 | Kourovka Notebook Problem 6.16 | A variety is called *sparse* if it has at most countably
many subvarieties. How many sparse varieties of groups are there? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.16.
Proposed by: A. V. Kuznetsov
Published resolution:
There are continuum of such varieties (P. A. Kozhevnikov, *On varieties of groups of large odd exponent*, Dep. 1612-V00,
VI... | 2 | solved | A. V. Kuznetsov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: There are continuum of such varieties (P. A. Kozhevnikov, *On varieties of groups of large odd exponent*, Dep. 1612-V00,\nVINITI, Moscow, 2000 (Russian); \\ S. V... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,364 | KOU-6.17 | Kourovka Notebook Problem 6.17 | Is every variety of groups generated by its finitely generated
groups that have soluble word problem? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.17.
Proposed by: A. V. Kuznetsov
Published resolution:
No (Yu. G. Kleiman, *Math. USSR--Izv.*, **22** (1984), 33--65). | 2 | solved | A. V. Kuznetsov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No (Yu. G. Kleiman, *Math. USSR--Izv.*, **22** (1984), 33--65)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,365 | KOU-6.18 | Kourovka Notebook Problem 6.18 | (Well-known problem). Suppose that a class ${\mathfrak{K}}$ of
2-generator groups generates the variety of all groups. Is a
non-cyclic free group residually in ${\mathfrak{K}}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.18.
Proposed by: V. M. Levchuk
Published resolution:
Not always (S. J. Pride, *Math. Z.*, **131** (1973), 245--248). | 2 | solved | V. M. Levchuk | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Not always (S. J. Pride, *Math. Z.*, **131** (1973), 245--248)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,366 | KOU-6.19 | Kourovka Notebook Problem 6.19 | Let $R$ be a nilpotent associative ring. Are the following two
statements for a subgroup $H$ of the adjoint group of $R$
always equivalent: \ 1) $H$ is a normal subgroup; \ 2) $H$ is an
ideal of the groupoid $R$ with respect to Lie multiplication? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.19.
Proposed by: V. M. Levchuk
Published resolution:
Not always. Let $R$ be the free nilpotent of index 3 associative
algebra over $\mathbb{F}_2$ on the free generators $x,y$. Le... | 2 | solved | V. M. Levchuk | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Not always. Let $R$ be the free nilpotent of index 3 associative\nalgebra over $\\mathbb{F}_2$ on the free generators $x,y$. Let $S$ be the\nsubalgebra generated... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,367 | KOU-6.20 | Kourovka Notebook Problem 6.20 | Does there exist a supersoluble group of odd order, all of
whose automorphisms are inner? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.20.
Proposed by: V. D. Mazurov
Published resolution:
Yes, there does (B. Hartley, D. J. S. Robinson, *Arch.
Math.*, **35** (1980), 67--74). | 2 | solved | V. D. Mazurov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there does (B. Hartley, D. J. S. Robinson, *Arch.\nMath.*, **35** (1980), 67--74)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,368 | KOU-6.21 | Kourovka Notebook Problem 6.21 | G. Higman proved that, for any prime number $p$, there exists a
natural number $\chi (p)$ such that the nilpotency class of any
finite group $G$ having an automorphism of order $p$ without
non-trivial fixed points does not exceed $\chi (p)$. At the same time
he showed that $\chi (p) \geq (p^2 - 1)/4$ for any such *Higm... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.21.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,369 | KOU-6.22 | Kourovka Notebook Problem 6.22 | Construct a braid that belongs to the derived subgroup of
the braid group but is not a commutator. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.22.
Proposed by: G. S. Makanin
Published resolution:
Let $x=\sigma _1, y=\sigma _2$ be the standard generators of the braid
group $B_3$; then
the braid $(xyxyx)^{12}(xy)^{-12}(xy... | 2 | solved | G. S. Makanin | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Let $x=\\sigma _1, y=\\sigma _2$ be the standard generators of the braid\ngroup $B_3$; then\nthe braid $(xyxyx)^{12}(xy)^{-12}(xyx)^{-12}$ belongs to the\nderive... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,370 | KOU-6.23 | Kourovka Notebook Problem 6.23 | A braid $K$ of the braid group ${\mathfrak{B}}_{n+1}$ is said to be *smooth* if removing any of the threads
in $K$ transforms $K$ into a braid that is equal to $1$ in ${\mathfrak{B}}_{n}$. It is known that smooth braids form a free subgroup.
Describe generators of this subgroup. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.23.
Proposed by: G. S. Makanin
Published resolution:
They are described in (D. L. Johnson, *Math. Proc. Camb.
Philos. Soc.*, **92** (1982), 425--427). | 2 | solved | G. S. Makanin | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: They are described in (D. L. Johnson, *Math. Proc. Camb.\nPhilos. Soc.*, **92** (1982), 425--427)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,371 | KOU-6.24 | Kourovka Notebook Problem 6.24 | The membership problem for the braid group on four strings. It is
known (T. A. Makanina, *Math. Notes*, **29**, no. 1
(1981), 16--17) that the membership problem is undecidable for
braid groups with more than four strings. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.24.
Proposed by: G. S. Makanin | 2 | open | G. S. Makanin | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,372 | KOU-6.25 | Kourovka Notebook Problem 6.25 | (Well-known problem). Find an algorithm for calculating the
rank of coefficient-free equations in a free group. The *rank* of an
equation is the maximal rank of the free subgroup generated by
a solution of this equation. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.25.
Proposed by: G. S. Makanin
Published resolution:
This was found (A. A. Razborov, *Math. USSR--Izv.*,
**25** (1984), 115--162). | 2 | solved | G. S. Makanin | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This was found (A. A. Razborov, *Math. USSR--Izv.*,\n**25** (1984), 115--162)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,373 | KOU-6.26 | Kourovka Notebook Problem 6.26 | Let $D$ be a normal set of involutions in a finite group $G$ and
let $\Gamma (D)$ be the graph with vertex set $D$ and edge set $\{
(a,b)\mid a,b \in D,\; ab=ba \ne 1\} $. Describe the finite groups
$G$ with non-connected graph $\Gamma (D)$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.26.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,374 | KOU-6.28 | Kourovka Notebook Problem 6.28 | Let $A$ be an elementary abelian 2-group which is a $ TI
$-subgroup of a finite group $G$. Investigate the structure of
$G$ under the hypothesis that the weak closure of $A$ in a Sylow
2-subgroup of $G$ is abelian. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.28.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,375 | KOU-6.29 | Kourovka Notebook Problem 6.29 | Suppose that a finite group $A$ is isomorphic to the group of all
topological automorphisms of a locally compact group $G$. Does
there always exist a discrete group whose automorphism group is
isomorphic to $A$? This is true if $A$ is cyclic; the condition of
local compactness of $G$ is essential (R. J. Wille, *Indag.
... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.29.
Proposed by: O. V. Mel'nikov | 2 | open | O. V. Mel'nikov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,376 | KOU-6.30 | Kourovka Notebook Problem 6.30 | Let $G$ be a residually finite Hopfian group, and let
$\widehat{G}$ be its profinite completion. Is $\widehat{G}$
necessarily Hopfian (in topological sense)? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.30.
Proposed by: O. V. Mel'nikov
Discussion and literature:
[Solved] No, not always (M. Brescia, E. Ingrosso, M. Trombetti, *Preprint*, 2026, https://arxiv.org... | 2 | solved | O. V. Mel'nikov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "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/2607.1919... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,377 | KOU-6.31 | Kourovka Notebook Problem 6.31 | b)
Let $G$ be a finitely generated residually finite group, $d(G)$
the minimal number of its generators, and $\delta (G)$ the minimal
number of topological generators of the profinite completion of
$G$. It is known that there exist groups
$G$ for which $d(G) > \delta
(G)$ (G. A. Noskov, *Math. Notes*, **33**, no. 4 (19... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 6.31.
Proposed by: O. V. Mel'nikov
Archive of Solved Problems entry:
Not always (G. A. Noskov, *Math. Notes*, **33** (1983),
249--254). | 2 | partially_solved | O. V. Mel'nikov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (Not always (G. A. Noskov, *Math. Notes*, **33** (1983),\n249--254).); 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 |
700,378 | KOU-6.32 | Kourovka Notebook Problem 6.32 | Let $F_n$ be the free profinite group of finite rank $n
> 1$. Is it true that for each normal subgroup $N$ of the free
profinite group of countable rank, there exists a normal
subgroup of $F_n$ isomorphic to $N$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.32.
Proposed by: O. V. Mel'nikov | 2 | open | O. V. Mel'nikov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,379 | KOU-6.34 | Kourovka Notebook Problem 6.34 | Let ${\mathfrak{o}}$ be an associative ring with
identity. A system of its ideals ${\mathfrak{A}}=\{ {\mathfrak{A}}_{ij}\mid
i,j \in {\mathbb{Z}\}$ is called a *carpet of ideals* if $
{\mathfrak{A}}_{ik}{\mathfrak{A}}_{kj} \subseteq {\mathfrak{A}}_{ij}$ for all
$i,j,k \in {\mathbb{Z}$. If ${\mathfrak{o}}$ is commutativ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.34.
Proposed by: Yu. I. Merzlyakov
Published resolution:
These are proved (V. M. Levchuk, *Sov. Math. Dokl.*, **42**, no. 1 (1991), 82--86; \ *Ukrain. Math. J.*, **44**,
no. 6 (1... | 2 | solved | Yu. I. Merzlyakov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: These are proved (V. M. Levchuk, *Sov. Math. Dokl.*, **42**, no. 1 (1991), 82--86; \\ *Ukrain. Math. J.*, **44**,\nno. 6 (1992), 710--718)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,380 | KOU-6.35 | Kourovka Notebook Problem 6.35 | (R. Bieri, R. Strebel). Let ${\mathfrak{o}}$ be an
associative ring with identity distinct from zero. A group $G$
is said to be *almost finitely presented
over* ${\mathfrak{o}}$ if it has a presentation $G=F/R$ where $F$ is
a finitely generated free group and the ${\mathfrak{o}}G $-module
$R/[R,R] \otimes _{{\mathbb{Z}... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.35.
Proposed by: Yu. I. Merzlyakov
Published resolution:
No, it is not true (M. Bestvina, N. Brady, *Invent. Math.*,
**129**, no. 3 (1997), 445--470). | 2 | solved | Yu. I. Merzlyakov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it is not true (M. Bestvina, N. Brady, *Invent. Math.*,\n**129**, no. 3 (1997), 445--470)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,381 | KOU-6.36 | Kourovka Notebook Problem 6.36 | (J. W. Grossman). The *nilpotent-completion diagram* of a group
$G$ is as follows: $G/\gamma _1G\leftarrow G/\gamma _2G\leftarrow \cdots
$, where $\gamma _iG$ is the $i$th term of the lower central series
and the arrows are natural homomorphisms. It is easy to see that
every nilpotent-completion diagram $G_1\leftarrow ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.36.
Proposed by: Yu. I. Merzlyakov
Published resolution:
Yes, such $
\gamma$-diagrams exist (N. S. Romanovskiĭ, *Sibirsk. Mat.
Zh.*, **26**, no. 4 (1985), 194--195 (Russian)). | 2 | solved | Yu. I. Merzlyakov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, such $\n\\gamma$-diagrams exist (N. S. Romanovskiĭ, *Sibirsk. Mat.\nZh.*, **26**, no. 4 (1985), 194--195 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,382 | KOU-6.37 | Kourovka Notebook Problem 6.37 | (H. Wielandt, O. H. Kegel). Is a finite group $G$ soluble if
it has soluble subgroups $A$, $B$, $C$ such that
$G=AB=AC=BC$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.37.
Proposed by: V. S. Monakhov
Published resolution:
Yes, it is (mod CFSG) (L. S. Kazarin, *Ukrain. Math. J.*, **43** (1991), 883--886). | 2 | solved | V. S. Monakhov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (mod CFSG) (L. S. Kazarin, *Ukrain. Math. J.*, **43** (1991), 883--886)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,383 | KOU-6.38 | Kourovka Notebook Problem 6.38 | b) Is it true that
an arbitrary subgroup of $GL_n(k)$
that intersects every conjugacy class is parabolic? How far is the
same statement true for subgroups of other groups of Lie type?
[Archived solved part(s) of 6.38:]
a)
Let $k$ be a (commutative) field. Find all irreducible subgroups
$G$ of $GL_n(k)$ having the prop... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 6.38.
Proposed by: P. M. Neumann
Archive of Solved Problems entry:
a) The conjecture is refuted
(S. A. Zyubin, *Algebra and Logic*, **45**, no. 5 (2006),
296-... | 2 | partially_solved | P. M. Neumann | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (a) The conjecture is refuted\n(S. A. Zyubin, *Algebra and Logic*, **45**, no. 5 (2006),\n296--305).); 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 |
700,384 | KOU-6.39 | Kourovka Notebook Problem 6.39 | A
class of groups ${\mathfrak{K}}$ is said to be *radical* if it is
closed under taking homomorphic images and normal subgroups and if
every group generated by its normal ${\mathfrak{K}} $-subgroups
also belongs to ${\mathfrak{K}}$. The question proposed relates to the
topic ``Radical classes and formulae of Narrow (fi... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.39.
Proposed by: B. I. Plotkin | 2 | open | B. I. Plotkin | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,385 | KOU-6.42 | Kourovka Notebook Problem 6.42 | Let $H$ be a strongly 3-embedded subgroup of a finite
group $G$. Suppose that $Z(H/O_{3'}(H))$ contains an element
of order 3. Does $Z(G/O_{3'}(G))$ necessarily contain an element
of order 3? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.42.
Proposed by: N. D. Podufalov
Published resolution:
Yes, it does (mod CFSG) (W. Xiao, *Sci. China (A)*, **33**
(1990), 1172--1181). | 2 | solved | N. D. Podufalov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does (mod CFSG) (W. Xiao, *Sci. China (A)*, **33**\n(1990), 1172--1181)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,386 | KOU-6.43 | Kourovka Notebook Problem 6.43 | Does the set of quasi-identities holding in the class of all finite
groups possess a basis in finitely many variables? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.43.
Proposed by: D. M. Smirnov
Published resolution:
No (A. K. Rumyantsev, *Algebra and Logic*, **19**
(1980), 297--311). | 2 | solved | D. M. Smirnov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No (A. K. Rumyantsev, *Algebra and Logic*, **19**\n(1980), 297--311)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,387 | KOU-6.44 | Kourovka Notebook Problem 6.44 | Construct a finitely generated infinite simple group requiring
more than two generators. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.44.
Proposed by: J. Wiegold
Published resolution:
This has been done (V. S. Guba, *Siberian Math. J.*, **27** (1986), 670--684). | 2 | solved | J. Wiegold | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This has been done (V. S. Guba, *Siberian Math. J.*, **27** (1986), 670--684)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,388 | KOU-6.45 | Kourovka Notebook Problem 6.45 | Construct a characteristic subgroup $N$ of a finitely generated
free group $F$ such that $F/N$ is infinite and simple. If no such
exists, it would follow that $d(S^2)=d(S)$ for every infinite
finitely generated simple group $S$. There is reason to believe
that this is false. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.45.
Proposed by: J. Wiegold
Discussion and literature:
[Solved] It is proved that for all $n\geq 2$, the free group $F_n$ admits continuum many pairwise non-is... | 2 | solved | J. Wiegold | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "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 proved that for all $n\\geq 2$, the free group $F_n$ admits continuum many pairwise non-isomorphic... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,389 | KOU-6.46 | Kourovka Notebook Problem 6.46 | If $G$ is $d $-generator group having no
non-trivial finite homomorphic images (in particular, if $G$ is an
infinite simple $d $-generator group) for some integer $d \geq
2$, must $G \times G$ be a $d $-generator group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.46.
Proposed by: John S. Wilson
Published resolution:
No, it need not (V. N. Obraztsov, *Proc. Roy. Soc.
Edinburgh*, **123** (1993), 839--855). | 2 | solved | John S. Wilson | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it need not (V. N. Obraztsov, *Proc. Roy. Soc.\nEdinburgh*, **123** (1993), 839--855)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,390 | KOU-6.47 | Kourovka Notebook Problem 6.47 | (C. D. H. Cooper). Let $G$ be a group, $v$ a group word in two
variables such that the operation $x \odot y=v(x,y)$ defines the
structure of a new group $G_v=\left< G,\odot \right>$ on the set
$G$. Does $G_v$ always lie in the variety generated by $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.47.
Proposed by: E. I. Khukhro | 2 | open | E. I. Khukhro | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,391 | KOU-6.48 | Kourovka Notebook Problem 6.48 | Is
every infinite binary finite $p
$-group non-simple?
Here $p$ is a prime number. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.48.
Proposed by: N. S. Chernikov | 2 | open | N. S. Chernikov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,392 | KOU-6.49 | Kourovka Notebook Problem 6.49 | Is the minimal condition for abelian normal subgroups inherited by subgroups of finite index?
This is true for the minimal condition for (all) abelian subgroups (J. S. Wilson, *Math. Z.*, **114** (1970), 19--21). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.49.
Proposed by: S. A. Chechin
Published resolution:
No, not always. Let $G=[(A\times B\times C\times D)\rtimes (\langle g\rangle\times \langle t\rangle )]\rtimes (Y\times \langl... | 2 | solved | S. A. Chechin | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always. Let $G=[(A\\times B\\times C\\times D)\\rtimes (\\langle g\\rangle\\times \\langle t\\rangle )]\\rtimes (Y\\times \\langle x\\rangle )$, where $A... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,393 | KOU-6.51 | Kourovka Notebook Problem 6.51 | Let ${\mathfrak{F}}$ be a local subformation of some formation ${\mathfrak{X}}$ of finite groups and let $\Omega$ be the set of all maximal
homogeneous ${\mathfrak{X}} $-screens of ${\mathfrak{F}}$. Find a way of
constructing the elements of $\Omega$ with the help of the maximal
inner local screen of ${\mathfrak{F}}$. ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.51.
Proposed by: L. A. Shemetkov | 2 | open | L. A. Shemetkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,394 | KOU-6.52 | Kourovka Notebook Problem 6.52 | Let $f$ be a local screen of a
formation which contains all finite nilpotent groups and let $A$ be a
group of automorphisms of a finite group $G$. Suppose that $A$ acts $
f $-stably on the socle of $G/\Phi (G)$. Is it true that $A$ acts $f
$-stably on $\Phi (G)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.52.
Proposed by: L. A. Shemetkov
Published resolution:
No, not always (A. N. Skiba, *Siber. Math. J.*, **34**
(1993), 953--958). | 2 | solved | L. A. Shemetkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always (A. N. Skiba, *Siber. Math. J.*, **34**\n(1993), 953--958)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,395 | KOU-6.53 | Kourovka Notebook Problem 6.53 | A group $G$ of the form
$G=F\leftthreetimes H$ is said to be a *Frobenius group with
kernel $F$ and complement* $H$ if $H \cap H^g=1$ for any $g \in
G\setminus H$ and $F\setminus \{ 1\} =G\setminus \bigcup\limits_{g\in
G} H^g$. What can be said about the kernel and the complement of a
Frobenius group? In particular, wh... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.53.
Proposed by: V. P. Shunkov
Published resolution:
Every group can be embedded into the kernel of a Frobenius group,
and every right-orderable group can be a complement in a Fr... | 2 | solved | V. P. Shunkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Every group can be embedded into the kernel of a Frobenius group,\nand every right-orderable group can be a complement in a Frobenius\ngroup (V. V. Bludov, *Sibe... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,396 | KOU-6.54 | Kourovka Notebook Problem 6.54 | Are there infinite finitely
generated Frobenius groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.54.
Proposed by: V. P. Shunkov
Published resolution:
Yes, such groups exist (A. I. Sozutov, *Siberian Math. J.*,
**35**, no. 4 (1994), 795--801). | 2 | solved | V. P. Shunkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, such groups exist (A. I. Sozutov, *Siberian Math. J.*,\n**35**, no. 4 (1994), 795--801)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,397 | KOU-6.55 | Kourovka Notebook Problem 6.55 | A group $G$ of the form $G=F\leftthreetimes H$ is
said to be a *Frobenius group with kernel $F$ and complement*
$H$ if $H \cap H^g=1$ for any $g \in G\setminus H$ and $F\setminus
\{ 1\} =G\setminus \bigcup\limits_{g\in G} H^g$. Do there exist
Frobenius $p
$-groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.55.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,398 | KOU-6.56 | Kourovka Notebook Problem 6.56 | Let $G=F\cdot \left< a\right>$ be a Frobenius group with the
complement $\left< a\right> $ of prime order.
a) Is $G$ locally finite if it is binary finite?
b) Is the kernel $F$ locally finite if the
groups $\left< a, a^g\right> $ are finite for all $g \in G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.56.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,399 | KOU-6.57 | Kourovka Notebook Problem 6.57 | A group $G$ is said to be (*conjugacy, $p $-conjugacy*)
*biprimitively finite* if, for any finite subgroup $H$,
any two elements of prime order (any two conjugate elements of
prime order, of prime order $p$) in $N_G(H)/H$ generate a finite
subgroup. Do the elements of finite order in a (conjugacy)
biprimitively finite ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.57.
Proposed by: V. P. Shunkov
Published resolution:
Not always (A. A. Cherep, *Algebra and Logic*, **29**
(1987), 311--313). | 2 | solved | V. P. Shunkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Not always (A. A. Cherep, *Algebra and Logic*, **29**\n(1987), 311--313)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,400 | KOU-6.58 | Kourovka Notebook Problem 6.58 | Are
a) the Al\"eshin $p $-groups and
b) the 2-generator Golod
$p $-groups
conjugacy biprimitively finite groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.58.
Proposed by: V. P. Shunkov
Published resolution:
a) Not always (A. V. Rozhkov, *Math.
USSR--Sb.*, **57** (1987), 437--448).
b) Not always (A. V. Timofeenko,
*Algebra and Log... | 2 | solved | V. P. Shunkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) Not always (A. V. Rozhkov, *Math.\nUSSR--Sb.*, **57** (1987), 437--448).\n\nb) Not always (A. V. Timofeenko,\n*Algebra and Logic*, **24** (1985), 129--139)."
... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,401 | KOU-6.59 | Kourovka Notebook Problem 6.59 | A group $G$ is said to be (*conjugacy, $p$-conjugacy*)
*biprimitively finite* if, for any finite subgroup $H$,
any two elements of prime order (any two conjugate elements of
prime order, of prime order $p$) in $N_G(H)/H$ generate a finite
subgroup. Prove that an arbitrary periodic (conjugacy)
biprimitively finite
group... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.59.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,402 | KOU-6.60 | Kourovka Notebook Problem 6.60 | Do
there exist infinite finitely generated simple periodic
(conjugacy) biprimitively finite groups which contain both
involutions and non-trivial elements of odd order? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.60.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,403 | KOU-6.61 | Kourovka Notebook Problem 6.61 | Is
every infinite periodic (conjugacy) biprimitively finite group
without involutions non-simple? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.61.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,404 | KOU-6.62 | Kourovka Notebook Problem 6.62 | Is
a (conjugacy, $p
$-conjugacy) biprimitively finite
group finite if it has a finite maximal subgroup ($p
$-subgroup)? There are affirmative answers for conjugacy
biprimitively finite $p
$-groups and for 2-conjugacy
biprimitively finite groups (V. P. Shunkov, *Algebra and
Logic*, **9**, no. 4 (1970), 291--297; **11**,... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 6th issue (1978), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 6.62.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,405 | KOU-6.63 | Kourovka Notebook Problem 6.63 | An infinite group $G$ is called a
*monster of the first kind* if it has elements of order $> 2$
and for any such an element $a$ and for any proper subgroup $H$ of
$G$, there is an element $g$ in $G\setminus H$, such that $\left<
a, a^g\right> =G$. Classify the monsters of the first kind all of
whose proper subgroups ar... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.63.
Proposed by: V. P. Shunkov
Published resolution:
The centre of such a group coincides with the set of elements of
order $\leq 2$ (V. P. Shunkov, *Algebra and Logic*, **7**
(1... | 2 | solved | V. P. Shunkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: The centre of such a group coincides with the set of elements of\norder $\\leq 2$ (V. P. Shunkov, *Algebra and Logic*, **7**\n(1968), no. 1 (1970), 66--69). An i... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,406 | KOU-6.64 | Kourovka Notebook Problem 6.64 | A group $G$ is called a *monster of the second kind* if
it has elements of order $> 2$ and if for any such element $a$
and any proper subgroup $H$ of $G$ there exists an infinite subset
${\mathfrak{M}}_{a,H}$ consisting of conjugates of $a$ by elements of
$G\setminus H$ such that $ \left< a,c\right> =G$ for all $ c\in
... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 6th issue (1978). Original problem number: 6.64.
Proposed by: V. P. Shunkov
Published resolution:
Yes, such monsters exist, in both cases (A. Yu. Olshanskii, *The
geometry of defining relations in groups*, Kluwer, Dordrecht... | 2 | solved | V. P. Shunkov | 1,978 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, such monsters exist, in both cases (A. Yu. Olshanskii, *The\ngeometry of defining relations in groups*, Kluwer, Dordrecht,\n1991)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,407 | KOU-7.1 | Kourovka Notebook Problem 7.1 | The free periodic groups $B(m,p)$ of prime exponent $p > 665$ are
known to possess many properties similar to those of absolutely
free groups (see S. I. Adian, *The Burnside Problem and
Identities in Groups*, Springer, Berlin, 1979). Is it true that
all normal subgroups of $B(m,p)$ are not free periodic groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.1.
Proposed by: S. I. Adian
Published resolution:
Yes, it is true for all sufficiently large $p$
(A. Yu. Olshanskii, in: *Groups, rings, Lie and Hopf
algebras, Int. Workshop, Can... | 2 | solved | S. I. Adian | 1,980 | 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 for all sufficiently large $p$\n(A. Yu. Olshanskii, in: *Groups, rings, Lie and Hopf\nalgebras, Int. Workshop, Canada, 2001*, Dordrecht, Kluwer, ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,408 | KOU-7.2 | Kourovka Notebook Problem 7.2 | Prove that the free periodic groups $B(m, n)$ of odd exponent
$n\geq 665$ with $ m \geq 2$ generators are non-amenable and
that random walks on these groups do not have the recurrence
property. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.2.
Proposed by: S. I. Adian
Published resolution:
Both assertions are proved (S. I. Adian, *Math. USSR--Izv.*, **21** (1982), 425--434). | 2 | solved | S. I. Adian | 1,980 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Both assertions are proved (S. I. Adian, *Math. USSR--Izv.*, **21** (1982), 425--434)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,409 | KOU-7.3 | Kourovka Notebook Problem 7.3 | Prove that the periodic product of odd exponent $n \geq 665$ of
non-trivial groups $F_1,\ldots ,F_k$ which do not contain
involutions cannot be generated by less than $k$ elements. This
would imply, on the basis of (S. I. Adian, *Sov. Math.
Doklady*, **19**, (1978), 910--913), the existence of $ k
$-generated simple gr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.3.
Proposed by: S. I. Adian | 2 | open | S. I. Adian | 1,980 | 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 |
700,410 | KOU-7.4 | Kourovka Notebook Problem 7.4 | Is a finitely generated group with quadratic growth almost
abelian? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.4.
Proposed by: V. V. Belyaev
Published resolution:
Yes, it is (M. Gromov, *Publ. Math. IHES*, **53** (1981), 53--73; \
J. A. Wolf, *Diff. Geometry*, **2** (1968), 421--446). | 2 | solved | V. V. Belyaev | 1,980 | 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 (M. Gromov, *Publ. Math. IHES*, **53** (1981), 53--73; \\\nJ. A. Wolf, *Diff. Geometry*, **2** (1968), 421--446)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,411 | KOU-7.5 | Kourovka Notebook Problem 7.5 | We
say that a group is *indecomposable* if any two of its
proper subgroups generate a proper subgroup. Describe the
indecomposable periodic metabelian groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.5.
Proposed by: V. V. Belyaev | 2 | open | V. V. Belyaev | 1,980 | 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 |
700,412 | KOU-7.6 | Kourovka Notebook Problem 7.6 | Describe the infinite simple locally finite groups
with a Chernikov Sylow 2-subgroup. In particular, are such groups
the Chevalley groups over locally finite fields of odd
characteristic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.6.
Proposed by: V. V. Belyaev, N. F. Sesekin
Published resolution:
No, as
every countably infinite locally finite $p$-group can be embedded as
a maximal $p$-subgroup of the simpl... | 2 | solved | V. V. Belyaev, N. F. Sesekin | 1,980 | 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, as\nevery countably infinite locally finite $p$-group can be embedded as\na maximal $p$-subgroup of the simple Hall's universal group $U$, the unique countab... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,413 | KOU-7.7 | Kourovka Notebook Problem 7.7 | (Well-known problem). Is the group $G=\left< a,b\mid a^9=1,\;\;
ab= b^2a^2\right>$ finite? This group contains $F(2, 9)$, the only
Fibonacci group for which it is not yet known whether it is finite
or infinite. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.7.
Proposed by: R. G. Burns
Published resolution:
No, it is infinite, since $F(2,9)$ is infinite (M. F. Newman,
*Arch. Math.*, **54**, no. 3 (1990), 209--212). | 2 | solved | R. G. Burns | 1,980 | 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 infinite, since $F(2,9)$ is infinite (M. F. Newman,\n*Arch. Math.*, **54**, no. 3 (1990), 209--212)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,414 | KOU-7.8 | Kourovka Notebook Problem 7.8 | Suppose that $H$ is a normal subgroup of a group $G$, where $H$
and $G$ are subdirect products of the same $n$ groups
$G_1,\ldots ,G_n$. Does the nilpotency class of $G/H$ increase
with $n$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.8.
Proposed by: Yu. M. Gorchakov
Published resolution:
Yes, it does (E. I. Khukhro, *Sibirsk. Mat. Zh.*, **23**,
no. 6 (1982), 178--180 (Russian)). | 2 | solved | Yu. M. Gorchakov | 1,980 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does (E. I. Khukhro, *Sibirsk. Mat. Zh.*, **23**,\nno. 6 (1982), 178--180 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,415 | KOU-7.12 | Kourovka Notebook Problem 7.12 | Find all groups with a Hall $2'$-subgroup. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.12.
Proposed by: R. L. Griess
Published resolution:
These are found mod CFSG (Z. Arad, M. B. Ward, *J.
Algebra*, **77** (1982), 234--246). | 2 | solved | R. L. Griess | 1,980 | 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: These are found mod CFSG (Z. Arad, M. B. Ward, *J.\nAlgebra*, **77** (1982), 234--246)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,416 | KOU-7.15 | Kourovka Notebook Problem 7.15 | Prove that if $F^{\displaystyle{*}} (G)$ is quasisimple and
$\alpha \in \mathrm{Aut} G$, $|\alpha |=2$, then $C_G(\alpha )$
contains an involution outside $Z(F^{\displaystyle{*}}(G))$,
except when $F ^{\displaystyle{*}}(G)$ has quaternion Sylow
2-subgroups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.15.
Proposed by: R. Griess
Discussion and literature:
[Solved] This is not always true (AI solution confirmed by A. Rizzoli: Achyuth Jayadevan, *Preprint*, 202... | 2 | solved | R. Griess | 1,980 | 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] This is not always true (AI solution confirmed by A. Rizzoli: Achyuth Jayadevan, *Preprint*, 2026,\nhttp... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,417 | KOU-7.16 | Kourovka Notebook Problem 7.16 | If $H$ is a proper subgroup of the finite group $G$, is there
always an element of prime-power order not conjugate to an
element of $H$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.16.
Proposed by: R. L. Griess
Published resolution:
Yes, always (mod CFSG) (B. Fein, W. M. Kantor,
M. Schacher, *J. Reine Angew. Math.*, **328** (1981), 39--57). | 2 | solved | R. L. Griess | 1,980 | 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, always (mod CFSG) (B. Fein, W. M. Kantor,\nM. Schacher, *J. Reine Angew. Math.*, **328** (1981), 39--57)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,418 | KOU-7.17 | Kourovka Notebook Problem 7.17 | Is
the number of maximal subgroups of the finite group $G$ at most
$|G| - 1$?
*Editors' remarks:* This is proved for
$G$ soluble (G. E. Wall, *J. Austral. Math. Soc.*, **2** (1961--62), 35--59) and for symmetric groups ${\mathbb{S} _n$ for
sufficiently large $n$ (M. Liebeck, A. Shalev, *J. Combin. Theory, Ser. A*, **7... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.17.
Proposed by: R. Griess
Published resolution:
No, not always (R. Guralnick, F. L\"ubeck, L. Scott, T. Sprowl; see (C. P. Bendel, B. D. Boe, C. M. Drupieski, D. K. Nakano, B. J... | 2 | solved | R. Griess | 1,980 | 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 (R. Guralnick, F. L\\\"ubeck, L. Scott, T. Sprowl; see (C. P. Bendel, B. D. Boe, C. M. Drupieski, D. K. Nakano, B. J. Parshall, C. Pillen, C. B. W... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,419 | KOU-7.19 | Kourovka Notebook Problem 7.19 | Construct an explicit example of a finitely
presented simple group with word problem not solvable by a
primitive recursive function. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.19.
Proposed by: F. B. Cannonito | 2 | open | F. B. Cannonito | 1,980 | 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 |
700,420 | KOU-7.21 | Kourovka Notebook Problem 7.21 | Is
there an algorithm which decides if an arbitrary finitely
presented solvable group is metabelian? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.21.
Proposed by: F. B. Cannonito | 2 | open | F. B. Cannonito | 1,980 | 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 |
700,421 | KOU-7.22 | Kourovka Notebook Problem 7.22 | Suppose that a finite group $G$ is realized as the automorphism group of some torsion-free abelian group. Is it true that for every infinite cardinal $\mathfrak{m}$ there exist $2^{\mathfrak{m}}$ non-isomorphic torsion-free abelian groups of cardinality $\mathfrak{m}$ whose automorphism groups are isomorphic to $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.22.
Proposed by: S. F. Kozhukhov
Published resolution:
Yes, this is true in the Zermelo--Frenkel system with axioms of choice and `weak diamond' (M. Dugas, R. G\"obel, *Proc. Lon... | 2 | solved | S. F. Kozhukhov | 1,980 | 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, this is true in the Zermelo--Frenkel system with axioms of choice and `weak diamond' (M. Dugas, R. G\\\"obel, *Proc. London Math. Soc. (3)*, **45**, no. 2 (... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,422 | KOU-7.23 | Kourovka Notebook Problem 7.23 | Does there exist an algorithm which decides, for a given list of
group identities, whether the elementary theory of the variety
given by this list is soluble, that is, by (A. P. Zamyatin, *Algebra and Logic*, **17**, no. 1 (1978), 13--17), whether this
variety is abelian? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.23.
Proposed by: A. V. Kuznetsov | 2 | open | A. V. Kuznetsov | 1,980 | 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 |
700,423 | KOU-7.24 | Kourovka Notebook Problem 7.24 | We say that a group is *sparse* if the variety generated
by it has at most countably many subvarieties. Does there exist
a finitely generated
sparse group that has undecidable word problem? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.24.
Proposed by: A. V. Kuznetsov
Published resolution:
Yes, such groups do exist; for example, a free group of ${\mathfrak{N}}_3{\mathfrak{A}}$. By (A. N. Krasil'nikov, *Math. US... | 2 | solved | A. V. Kuznetsov | 1,980 | 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 do exist; for example, a free group of ${\\mathfrak{N}}_3{\\mathfrak{A}}$. By (A. N. Krasil'nikov, *Math. USSR--Izv.*,\n**37** (1991), 539--553)... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,424 | KOU-7.25 | Kourovka Notebook Problem 7.25 | We associate with words in the alphabet $x$, $x^{-1}$,
$y$, $y^{-1}$, $z$, $z^{-1},\ldots $ operations which are
understood as functions in variables $x, y, z,\ldots $. We say
that a word $A$ *is expressible in words $B_1,\ldots ,B_n$ on
the group $G$* if $A$ can be constructed from the words
$B_1,\ldots ,B_n$ and vari... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.25.
Proposed by: A. V. Kuznetsov | 2 | open | A. V. Kuznetsov | 1,980 | 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 |
700,425 | KOU-7.26 | Kourovka Notebook Problem 7.26 | The *ordinal height* of a variety of groups is, by
definition, the supremum of order types (ordinals) of all
well-ordered by inclusion chains of its proper subvarieties. It is
clear that its cardinality is either finite or countably infinite,
or equal to $\omega _1$. Is every countable ordinal number the
ordinal height... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.26.
Proposed by: A. V. Kuznetsov | 2 | open | A. V. Kuznetsov | 1,980 | 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 |
700,426 | KOU-7.27 | Kourovka Notebook Problem 7.27 | Is it true that the group $SL_n(q)$ contains, for
sufficiently large $q$, a diagonal matrix which is not contained in
any proper irreducible subgroup of $SL_n(q)$ with the exception of
block-monomial ones? For $n=2, 3$ the answer is known to be
affirmative (V. M. Levchuk, *in: Some problems of the theory of
groups and ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.27.
Proposed by: V. M. Levchuk | 2 | open | V. M. Levchuk | 1,980 | 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 |
700,427 | KOU-7.28 | Kourovka Notebook Problem 7.28 | Let $G(K)$ be the Chevalley group over a commutative ring $K$
associated with the root system $\Phi$ as defined in (R. Steinberg,
*Lectures on Chevalley groups*, Yale Univ., New Haven, Conn.,
1967). This group is generated by the root subgroups $x_r(K)$, $r \in
\Phi $. We define an *elementary carpet of type $\Phi$ ove... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.28.
Proposed by: V. M. Levchuk | 2 | open | V. M. Levchuk | 1,980 | 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 |
700,428 | KOU-7.30 | Kourovka Notebook Problem 7.30 | Which finite simple groups can be generated by three involutions,
two of which commute? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.30.
Proposed by: V. D. Mazurov
Published resolution:
The answer is known mod CFSG. For the alternating groups and
groups of Lie type see (Ya. N. Nuzhin, *Algebra and Logic*,
**36... | 2 | solved | V. D. Mazurov | 1,980 | 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 answer is known mod CFSG. For the alternating groups and\ngroups of Lie type see (Ya. N. Nuzhin, *Algebra and Logic*,\n**36**, no. 4 (1997), 245--256). For s... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,429 | KOU-7.31 | Kourovka Notebook Problem 7.31 | Let $A$ be a group of automorphisms of a finite
non-abelian 2-group $G$ acting transitively on the set of involutions
of $G$. Is $A$ necessarily soluble?
Such groups $G$ were divided into several classes in
(F. Gross, *J. Algebra*, **40**, no. 2 (1976), 348--353).
For one of these classes a positive answer was given b... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.31.
Proposed by: V. D. Mazurov
Discussion and literature:
[Solved] Yes, it is (C. H. Li, Y. Z. Zhu, *Proc. London Math. Soc. (3)*, **129**, no. 5 (2024), Artic... | 2 | solved | V. D. Mazurov | 1,980 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is (C. H. Li, Y. Z. Zhu, *Proc. London Math. Soc. (3)*, **129**, no. 5 (2024), Article ID e70002... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,430 | KOU-7.33 | Kourovka Notebook Problem 7.33 | An
elementary $TI $-subgroup $V$ of a finite group $G$ is said to
be a *subgroup of non-root type* if $1 \ne N_V(V^g) \ne V$
for some $g \in G$. Describe the finite groups $G$ which contain a
2-subgroup $V$ of non-root type such that $[V, V^g]=1$ implies
that all involutions in $VV^g$ are conjugate to elements of $V$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.33.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,980 | 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 |
700,431 | KOU-7.34 | Kourovka Notebook Problem 7.34 | In
many of the sporadic finite simple groups the 2-ranks of the
centralizers of 3-elements are at most 2. Describe the finite
groups satisfying this condition. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.34.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,980 | 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 |
700,432 | KOU-7.35 | Kourovka Notebook Problem 7.35 | What varieties of groups ${\mathfrak{V}}$ have the following property:
the group $G/{\mathfrak{V}}(G)$ is residually finite for any residually
finite group $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.35.
Proposed by: O. V. Mel'nikov | 2 | open | O. V. Mel'nikov | 1,980 | 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 |
700,433 | KOU-7.36 | Kourovka Notebook Problem 7.36 | Is it true that every residually finite group in which every subgroup of finite index (including the group itself) is defined by a single defining relation is either free or isomorphic to the fundamental group of a compact surface? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.36.
Proposed by: O. V. Mel'nikov
Published resolution:
No; for example, let $H_n=\langle x,y\mid y^{-1}xy=x^n\rangle $, $n=2,3,\dots $; then every subgroup of finite index in $H_... | 2 | solved | O. V. Mel'nikov | 1,980 | 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; for example, let $H_n=\\langle x,y\\mid y^{-1}xy=x^n\\rangle $, $n=2,3,\\dots $; then every subgroup of finite index in $H_n$ is isomorphic to a group $H_m$ ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,434 | KOU-7.37 | Kourovka Notebook Problem 7.37 | We
say that a profinite group is *strictly complete* if each of
its subgroups of finite index is open. It is known (B. Hartley,
*Math. Z.*, **168**, no. 1 (1979), 71--76) that finitely
generated profinite groups having a finite series with
pronilpotent factors are strictly complete. Is a profinite group
strictly comple... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.37.
Proposed by: O. V. Mel'nikov
Published resolution:
a) Yes, it is (N. Nikolov,
D. Segal, *Ann. Math. (2)*, **165**, no. 1 (2007),
171--273).
b) Yes, it is (D. Segal, *Proc. L... | 2 | solved | O. V. Mel'nikov | 1,980 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) Yes, it is (N. Nikolov,\nD. Segal, *Ann. Math. (2)*, **165**, no. 1 (2007),\n171--273).\n\nb) Yes, it is (D. Segal, *Proc. London Math. Soc. (3)*, **81**\n(20... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,435 | KOU-7.38 | Kourovka Notebook Problem 7.38 | A *variety of profinite groups* is a
non-empty class of profinite groups closed under taking subgroups,
factor-groups, and Tikhonov products. A subvariety ${\mathfrak{V}}$ of the
variety ${\mathfrak{N}}$ of all pronilpotent groups is said to be *$($locally$)$ nilpotent* if all (finitely generated) groups in
${\mathfrak... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.38.
Proposed by: O. V. Mel'nikov
Discussion and literature:
[Solved] No, not always (AI solution confirmed by E. I. Khukhro: \ Achyuth Jayadevan, *Preprint*, 2... | 2 | partially_solved | O. V. Mel'nikov | 1,980 | 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] No, not always (AI solution confirmed by E. I. Khukhro: \\ Achyuth Jayadevan, *Preprint*, 2026... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,436 | KOU-7.39 | Kourovka Notebook Problem 7.39 | Let $G=\left< a, b\mid a^p=(ab)^3=b^2=(a^{\sigma}ba^{2/\sigma
}b)^2=1\right> ,$ where $p$ is a prime, $\sigma$ is an integer not
divisible by $p$. The group $PSL_2(p)$ is a factor-group of $G$ so
that there is a short exact sequence $1 \rightarrow N \rightarrow
G \rightarrow PSL_2(p) \rightarrow 1 .$ For each $p > 2$ t... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.39.
Proposed by: J. Mennicke | 2 | open | J. Mennicke | 1,980 | 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 |
700,437 | KOU-7.40 | Kourovka Notebook Problem 7.40 | Describe the (lattice of) subgroups of a given classical matrix group
over a ring which contain the subgroup consisting of all matrices in
that group with coefficients in some subring (see
Ju. I. Merzljakov, *J. Soviet Math.*, **1** (1973),
571--593). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.40.
Proposed by: Yu. I. Merzlyakov
Published resolution:
If the root system has single edges ($A_l$, $D_l$, $E_l$) and the larger ring is not quasi-algebraic over the smaller one... | 2 | solved | Yu. I. Merzlyakov | 1,980 | 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: If the root system has single edges ($A_l$, $D_l$, $E_l$) and the larger ring is not quasi-algebraic over the smaller one, then it is shown that the lattice in q... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,438 | KOU-7.41 | Kourovka Notebook Problem 7.41 | (John S. Wilson). Is every linear $ \overline{\! SI\vphantom{(}}
$-group an $ \overline{\! SN\vphantom{(}} $-group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.41.
Proposed by: Yu. I. Merzlyakov | 2 | open | Yu. I. Merzlyakov | 1,980 | 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 |
700,439 | KOU-7.42 | Kourovka Notebook Problem 7.42 | A group $U$ is called an *$F_q $-group*
(where $q \in \pi (U)$) if, for each finite subgroup $K$ of $U$
and for any two elements $a,b$ of order $q$ in $T=N_U(K)/K$, there
exists $c \in T$ such that the group $\left< a, b^c\right>$ is
finite. A group $U$ is called an {\it $F ^{\displaystyle{*}}
$-group} if each subgroup... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.42.
Proposed by: A. N. Ostylovski\u{\i
Published resolution:
No. A counterexample to both questions is given by an infinite
group all of whose subgroups are conjugate and have pr... | 2 | solved | A. N. Ostylovski\u{\i | 1,980 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No. A counterexample to both questions is given by an infinite\ngroup all of whose subgroups are conjugate and have prime order\n(A. Yu. Olshanskii, *Math. USSR-... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,440 | KOU-7.44 | Kourovka Notebook Problem 7.44 | Does a normal subgroup $H$ in a finite group $G$
possess a complement in $G$ if each Sylow subgroups of $H$ is
a direct factor in some Sylow subgroup of $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.44.
Proposed by: V. I. Sergiyenko
Published resolution:
Yes, it does, mod CFSG (W. Xiao, *J. Pure Appl. Algebra*,
**87**, no. 1 (1993), 97--98). | 2 | solved | V. I. Sergiyenko | 1,980 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does, mod CFSG (W. Xiao, *J. Pure Appl. Algebra*,\n**87**, no. 1 (1993), 97--98)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,441 | KOU-7.45 | Kourovka Notebook Problem 7.45 | Does the $Q
$-theory of the class of all finite groups
(in the sense of 2.40) coincide with the $Q
$-theory of a single finitely presented group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.45.
Proposed by: D. M. Smirnov | 2 | open | D. M. Smirnov | 1,980 | 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 |
700,442 | KOU-7.48 | Kourovka Notebook Problem 7.48 | (Well-known problem). Suppose that, in a finite
group $G$, each two elements of the same order are conjugate. Is
then $|G| \leq 6$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.48.
Proposed by: S. A. Syskin
Published resolution:
Yes, it is, mod CFSG (P. Fitzpatrick, *Proc. Roy. Irish Acad. Sect. A*, **85**, no. 1 (1985),
53--58); see also (W. Feit, G. S... | 2 | solved | S. A. Syskin | 1,980 | 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 (P. Fitzpatrick, *Proc. Roy. Irish Acad. Sect. A*, **85**, no. 1 (1985),\n53--58); see also (W. Feit, G. Seitz, *Illinois J.\nMath.*, **33**... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,443 | KOU-7.49 | Kourovka Notebook Problem 7.49 | Let $G$ be a finitely generated group, and $N$ a minimal normal
subgroup of $G$ which is an elementary abelian $p
$-group. Is it true that either $N$ is finite or the growth
function for the elements of $N$ with respect to a finite
generating set of $G$ is bounded below by an exponential function? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.49.
Proposed by: V. I. Trofimov | 2 | open | V. I. Trofimov | 1,980 | 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 |
700,444 | KOU-7.50 | Kourovka Notebook Problem 7.50 | Study the structure of the primitive permutation groups (finite
and infinite), in which the stabilizer of any three pairwise
distinct points is trivial. This problem is closely connected with
the problem of describing those group which have a Frobenius group
as one of maximal subgroups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.50.
Proposed by: A. N. Fomin | 2 | open | A. N. Fomin | 1,980 | 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 |
700,445 | KOU-7.51 | Kourovka Notebook Problem 7.51 | What are the primitive permutation groups (finite and infinite)
which have a regular sub-orbit, that is, in which a point
stabilizer acts faithfully and regularly on at least one of its
orbits? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.51.
Proposed by: A. N. Fomin | 2 | open | A. N. Fomin | 1,980 | 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 |
700,446 | KOU-7.52 | Kourovka Notebook Problem 7.52 | Describe the locally finite primitive permutation groups in which
the centre of any Sylow 2-subgroup contains involutions
stabilizing precisely one symbol. The case of finite groups was
completely determined by D. Holt in 1978. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.52.
Proposed by: A. N. Fomin | 2 | open | A. N. Fomin | 1,980 | 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 |
700,447 | KOU-7.53 | Kourovka Notebook Problem 7.53 | Let $p$ be a prime.
The law $ x\cdot x^{\varphi}\cdots x^{\varphi^{p-1}}=1$ from the
definition of a splitting automorphism (see Archive, 1.10) gives
rise to a variety of groups with operators $\left< \varphi\right>$
consisting of all groups that admit a splitting automorphism of
order $p$. Does the analogue of Kostrik... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 7th issue (1980). Original problem number: 7.53.
Proposed by: E. I. Khukhro
Published resolution:
Yes, it does (E. I. Khukhro, *Math. USSR--Sb.*, **58**
(1987), 119--126). | 2 | solved | E. I. Khukhro | 1,980 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does (E. I. Khukhro, *Math. USSR--Sb.*, **58**\n(1987), 119--126)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,448 | KOU-7.54 | Kourovka Notebook Problem 7.54 | Does there exist a group of infinite special rank which
can be represented as a product of two subgroups of finite special
rank? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.54.
Proposed by: N. S. Chernikov | 2 | open | N. S. Chernikov | 1,980 | 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 |
700,449 | KOU-7.55 | Kourovka Notebook Problem 7.55 | Is
it true that a group which is a product of two almost abelian
subgroups is almost soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.55.
Proposed by: N. S. Chernikov | 2 | open | N. S. Chernikov | 1,980 | 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 |
700,450 | KOU-7.56 | Kourovka Notebook Problem 7.56 | (B. Amberg). Does a group satisfy the minimum
(respectively, maximum) condition on subgroups if it is a product of
two subgroups satisfying the minimum (respectively, maximum)
condition on subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.56.
Proposed by: N. S. Chernikov | 2 | open | N. S. Chernikov | 1,980 | 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 |
700,451 | KOU-7.57 | Kourovka Notebook Problem 7.57 | a)
A set of generators of a finitely generated group $G$ consisting
of the least possible number $d(G)$ of elements is called a *basis of $G$*. Let $r_M(G)$ be the least number of relations
necessary to define $G$ in the basis $M$ and let $r(G)$ be the
minimum of the numbers $r_M(G)$ over all bases $M$ of $G$. It is
kn... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 7.57.
Proposed by: V. A. Churkin
Archive of Solved Problems entry:
b) Not always; c) not always (C. Hog, M. Lusztig, W. Metzler, in:
*Presentation classes, 3-... | 2 | partially_solved | V. A. Churkin | 1,980 | 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 (b) Not always; c) not always (C. Hog, M. Lusztig, W. Metzler, in:\n*Presentation classes, 3-manifolds and free products\n$($Lecture Notes in Math.*, **1167**), S... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,452 | KOU-7.58 | Kourovka Notebook Problem 7.58 | Suppose that $F$ is an absolutely free group, $R$ a normal
subgroup of $F$, and let ${\mathfrak{V}}$ be a variety of groups. It is
well-known (H. Neumann, *Varieties of Groups*, Springer, Berlin,
1967) that the group $F/{\mathfrak{V}}(R)$ is isomorphically
embeddable in the ${\mathfrak{V}} $-verbal wreath product of a ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 7th issue (1980), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 7.58.
Proposed by: G. G. Yabanzhi
Discussion and literature:
[Unverified AI claim] A criterion is found (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2... | 2 | open | G. G. Yabanzhi | 1,980 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] A criterion is found (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,453 | KOU-8.1 | Kourovka Notebook Problem 8.1 | Characterize all groups (or at least all soluble
groups) $G$ and fields $F$ such that every irreducible $ FG
$-module has finite dimension over the centre of its endomorphism
ring. For some partial answers see (B. A. F. Wehrfritz, *Glasgow Math. J.*, **24**, no. 1 (1983), 169--176). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.1.
Proposed by: B. A. F. Wehrfritz | 2 | open | B. A. F. Wehrfritz | 1,982 | 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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