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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
701,154 | KOU-15.66 | Kourovka Notebook Problem 15.66 | For a class $\mathfrak{X}$ of groups let $g_{\mathfrak{X}}(n)$ be
the number of groups of order $n$ in the class $\mathfrak{X}$ (up
to isomorphism). Many years ago I formulated the following
problem: find good upper bounds for the quotient $g_{\mathfrak{V}}(n)/g_{\mathfrak{U}}(n)$, where $\mathfrak{V}$ is a variety tha... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.66.
Proposed by: P. M. Neumann | 2 | open | P. M. Neumann | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,155 | KOU-15.67 | Kourovka Notebook Problem 15.67 | Which adjoint Chevalley groups (of
normal type) over the integers are generated by three involutions
two of which commute?
The groups $SL_n({\mathbb{Z})$, $n>13$, satisfy
this condition (M. C. Tamburini, P. Zucca, *J. Algebra*,
**195**, no. 2 (1997), 650--661). The groups
$PSL_n({\mathbb{Z})$ satisfy it if and only if... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.67.
Proposed by: Ya. N. Nuzhin | 2 | open | Ya. N. Nuzhin | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,156 | KOU-15.68 | Kourovka Notebook Problem 15.68 | Does there exist an infinite
finitely generated 2-group (of finite exponent) all of whose
proper subgroups are locally finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.68.
Proposed by: A. Yu. Olshanskii | 2 | open | A. Yu. Olshanskii | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,157 | KOU-15.69 | Kourovka Notebook Problem 15.69 | Is it true that every hyperbolic group has a free normal subgroup
with the factor-group of finite exponent? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.69.
Proposed by: A. Yu. Olshanskii | 2 | open | A. Yu. Olshanskii | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,158 | KOU-15.70 | Kourovka Notebook Problem 15.70 | Do there exist groups of arbitrarily large cardinality that satisfy
the minimum condition for subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.70.
Proposed by: A. Yu. Olshanskii
Discussion and literature:
[Solved] A positive answer is consistent with the ZFC axioms of set theory, and the consistency... | 2 | solved | A. Yu. Olshanskii | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] A positive answer is consistent with the ZFC axioms of set theory, and the consistency of a negative ans... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,159 | KOU-15.71 | Kourovka Notebook Problem 15.71 | (B. Huppert). Let $G$ be a finite solvable group, and let $\rho(G)$
denote the set of prime divisors of the degrees of irreducible
characters of $G$. Is it true that there always exists an
irreducible character of $G$ whose degree is divisible by at least
$|\rho(G)|/2$ different primes? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.71.
Proposed by: P. P. P\'alfy | 2 | open | P. P. P\'alfy | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,160 | KOU-15.72 | Kourovka Notebook Problem 15.72 | For a fixed prime $p$ does there exist a sequence of groups $P_n$ of
order $p^n$ such that the number of conjugacy classes $k(P_n)$
satisfies $\lim_{n\to\infty} \log k(P_n) / \sqrt{n} = 0$?
Note that J. M. Riedl (*J. Algebra*, **218** (1999), 190--215) constructed $p $-groups for which the above
limit is $2 \log p$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.72.
Proposed by: P. P. P\'alfy | 2 | open | P. P. P\'alfy | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,161 | KOU-15.73 | Kourovka Notebook Problem 15.73 | Is it true that for every finite lattice $L$ there
exists a finite group $G$ and a subgroup $H \le G$ such that the
interval $Int(H;G)$ in the subgroup lattice of $G$ is isomorphic to
$L$? (Probably not.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.73.
Proposed by: P. P. P\'alfy | 2 | open | P. P. P\'alfy | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,162 | KOU-15.74 | Kourovka Notebook Problem 15.74 | For every prime $p$ find a finite $p $-group of nilpotence class
$p$ such that its lattice of normal subgroups cannot be embedded
into the subgroup lattice of any abelian group. (Solved for $p=2$,
$3$.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.74.
Proposed by: P. P. P\'alfy | 2 | open | P. P. P\'alfy | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,163 | KOU-15.75 | Kourovka Notebook Problem 15.75 | b) Consider the sequence $u_1 = [x,y],\ldots$, $ u_{n+1} =
[[u_n, x],[u_n,y]]$. Is it true that an arbitrary finite group is
soluble if and only if it satisfies one of these identities $u_n =
1$?
[Archived solved part(s) of 15.75:]
a) Does there exist a sequence of identities in two variables
$u_1=1$, $u_2=1,\ldots $ ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 15.75.
Proposed by: B. I. Plotkin
Archive of Solved Problems entry:
Yes, such sequences exist (T. Bandman, F. Grunewald,
G.-M. Greuel, B. Kunyavskii, G. Pfis... | 2 | partially_solved | B. I. Plotkin | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (Yes, such sequences exist (T. Bandman, F. Grunewald,\nG.-M. Greuel, B. Kunyavskii, G. Pfister, E. Plotkin, *Compositio Math.*, **142** (2006), 734--764; \\\n\nJ.... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,164 | KOU-15.76 | Kourovka Notebook Problem 15.76 | b)
If $\Theta$ is a variety of groups, then let $\Theta ^0$ denote the
category of all free groups of finite rank in $\Theta$. It is proved
(G. Mashevitzky, B. Plotkin, E. Plotkin *J. Algebra*, **282** (2004), 490--512) that if $\Theta$ is the variety of all groups,
then every automorphism of the category $\Theta ^0$ i... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 15.76.
Proposed by: B. I. Plotkin
Main section comment:
[Unverified AI claim] Yes, it is true (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,... | 2 | partially_solved | B. I. Plotkin | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (Yes, it is, even for\nthe variety of nilpotent groups of any class $n$ (A. Tsurkov,\n*Int. J. Algebra Comput.*, **17** (2007), 1273--1281).); remaining part(s) i... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,165 | KOU-15.77 | Kourovka Notebook Problem 15.77 | Elements $a,b$ of a group $G$ are said to be *symmetric with
respect to an element* $ g\in G$ if $a=gb^{-1}g$. Let $G$ be an
infinite abelian group, $\alpha$ a cardinal, $\alpha <|G|$. Is it
true that for any $n$-colouring $\chi : G\rightarrow\{ 0, 1,\ldots
,n-1\}$ there exists a monochrome subset of cardinality $\alph... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.77.
Proposed by: I. V. Protasov
Discussion and literature:
[Solved] Yes, it is true (AI solution confirmed by I. V. Protasov: \ Achyuth Jayadevan, *Preprint*... | 2 | solved | I. V. Protasov | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 true (AI solution confirmed by I. V. Protasov: \\ Achyuth Jayadevan, *Preprint*, 2026,\nhttps... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,166 | KOU-15.78 | Kourovka Notebook Problem 15.78 | (R. I. Grigorchuk). Is it true that for any
$n$-colouring of a free group of any rank there exists a
monochrome subset that is symmetric with respect to some element
of the group? This is true for $n=2$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.78.
Proposed by: I. V. Protasov | 2 | open | I. V. Protasov | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,167 | KOU-15.79 | Kourovka Notebook Problem 15.79 | Does there exist a
Hausdorff group topology on ${\mathbb{Z}$ such that the sequence
$ \{ 2^n+3^n \}$ converges to zero? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 15th issue (2002). Original problem number: 15.79.
Proposed by: I. V. Protasov
Published resolution:
Yes, there does,
as follows from Theorem 3 in (I. Z. Ruzsa, \emph{Proc. Conf. Number theory (Budapest, 1987). Vol. \mathrm... | 2 | solved | I. V. Protasov | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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,\nas follows from Theorem 3 in (I. Z. Ruzsa, \\emph{Proc. Conf. Number theory (Budapest, 1987). Vol. \\mathrm{I}: Elementary and analytic}, *Coll... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,168 | KOU-15.80 | Kourovka Notebook Problem 15.80 | A sequence $\{ F_n\}$
of pairwise disjoint finite subsets of a topological group is
called *expansive* if for every open subset $U$ there is a
number $m$ such that $ F_n\cap U\ne \varnothing$ for all $n>m$.
Suppose that a group $G$ can be partitioned into countably many
dense subsets. Is it true that in $ G$ there exis... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.80.
Proposed by: I. V. Protasov
Discussion and literature:
[Solved] Not necessarily. One example is $G = \prod_{\mathbb{N}}\mathbb{R}$ under the box topology... | 2 | solved | I. V. Protasov | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] Not necessarily. One example is $G = \\prod_{\\mathbb{N}}\\mathbb{R}$ under the box topology. It is well... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,169 | KOU-15.81 | Kourovka Notebook Problem 15.81 | Let $G$ be a finite non-supersoluble group. Is it true that $G$
has a non-cyclic Sylow subgroup $P$ such that some maximal
subgroup of $P$ has no proper complement in $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 15th issue (2002). Original problem number: 15.81.
Proposed by: A. N. Skiba
Published resolution:
Yes, it is (Wang, Yanming, Wei, Huaquan, *Sci. China Ser. A*,
**47** (2004), no. 1, 96--103). | 2 | solved | A. N. Skiba | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (Wang, Yanming, Wei, Huaquan, *Sci. China Ser. A*,\n**47** (2004), no. 1, 96--103)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,170 | KOU-15.82 | Kourovka Notebook Problem 15.82 | Suppose that a periodic group $G$ contains a strongly isolated
$2$-subgroup $U$. Is it true that either
$G$ is locally finite, or $U$ is a normal subgroup of $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.82.
Proposed by: A. I. Sozutov, N. M. Suchkov | 2 | open | A. I. Sozutov, N. M. Suchkov | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,171 | KOU-15.83 | Kourovka Notebook Problem 15.83 | (Yu. I. Merzlyakov). Does there
exist a rational number $\alpha $ such that $|\alpha |<2$ and the
matrices
{\small $\left(
\begin{array}{cc}1&\alpha \\0&1\end{array}\right)$} and {\small
$\left(
\begin{array}{cc}1&0\\\alpha &1\end{array}\right)$} generate a free
group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.83.
Proposed by: Yu. V. Sosnovski\u{\i | 2 | open | Yu. V. Sosnovski\u{\i | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,172 | KOU-15.84 | Kourovka Notebook Problem 15.84 | Yu. I. Merzlyakov (*Sov. Math. Dokl.*, **19**
(1978), 64--68) proved that
if the complex numbers $\alpha ,\beta ,\gamma $ are each at
least 3 in absolute value,
then the matrices
{\small $\left(
\begin{array}{cc}1&\alpha \\0&1\end{array}\right)$}, \
{\small $\left(
\begin{array}{cc}1&0 \\\beta&1\end{array}\right)$}, \ ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.84.
Proposed by: Yu. V. Sosnovski\u{\i | 2 | open | Yu. V. Sosnovski\u{\i | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,173 | KOU-15.85 | Kourovka Notebook Problem 15.85 | A torsion-free group all of whose subgroups
are subnormal is nilpotent (H. Smith, *Arch. Math.*, **76**, no. 1 (2001), 1--6). Is a torsion-free group with the
normalizer condition
a) hyperabelian?
b) hypercentral? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.85.
Proposed by: Yu. V. Sosnovski\u{\i | 2 | open | Yu. V. Sosnovski\u{\i | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,174 | KOU-15.86 | Kourovka Notebook Problem 15.86 | A group $G$ is called *discriminating* if for any finite set
of nontrivial elements of the direct square $G\times G$ there is a
homomorphism $G\times G\rightarrow G$ which does not annihilate
any of them (G. Baumslag, A. G. Myasnikov,
V. N. Remeslennikov). A group $G$ is called *squarelike*
if $G$ is universally equiva... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 15th issue (2002). Original problem number: 15.86.
Proposed by: D. Spellman
Published resolution:
Yes, it must
(O. Belegradek, *J. Group Theory*, **7**, no. 4 (2004),
521--532; \ B. Fine, A. M. Gaglione, D. Spellman, *Archi... | 2 | solved | D. Spellman | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it must\n(O. Belegradek, *J. Group Theory*, **7**, no. 4 (2004),\n521--532; \\ B. Fine, A. M. Gaglione, D. Spellman, *Archiv\nMath.*, **83**, no. 2 (2004), ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,175 | KOU-15.87 | Kourovka Notebook Problem 15.87 | Suppose that a $2$-group $G$ admits a regular periodic locally
cyclic group of automorphisms that transitively permutes the set
of involutions of $G$.
Is $G$ locally finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.87.
Proposed by: N. M. Suchkov | 2 | open | N. M. Suchkov | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,176 | KOU-15.88 | Kourovka Notebook Problem 15.88 | Let ${\mathfrak{A}}$ and ${\mathfrak{N}}_c$ denote the varieties of abelian groups and nilpotent groups of
class $\leq c$, respectively. Let $F_r= F_r({\mathfrak{A}}{\mathfrak{N}}_c)$ be
a free group of rank $r$ in $ {\mathfrak{A}}{\mathfrak{N}}_c$. An element of
$ F_r$ is called *primitive* if it can
be included in a ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.88.
Proposed by: E. I. Timoshenko | 2 | open | E. I. Timoshenko | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,177 | KOU-15.89 | Kourovka Notebook Problem 15.89 | Let $\Gamma$ be an infinite undirected connected vertex-symmetric
graph of finite valency without loops or multiple edges. Is it true
that every complex number is an eigenvalue of the adjacency matrix of
$\Gamma$ under its natural action as a linear operator on the complex
vector space of all complex-valued functions o... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.89.
Proposed by: V. I. Trofimov
Discussion and literature:
[Solved] No, not always (AI solution confirmed by V. I. Trofimov: \ Achyuth Jayadevan, *Preprint*,... | 2 | solved | V. I. Trofimov | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (AI solution confirmed by V. I. Trofimov: \\ Achyuth Jayadevan, *Preprint*, 2026, https:/... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,178 | KOU-15.90 | Kourovka Notebook Problem 15.90 | Let $\Gamma$ be an infinite directed graph, and $\overline{ \Gamma}$ the
underlying
undirected graph. Suppose that the graph $\Gamma$ admits a vertex-transitive
group of automorphisms, and the graph $ \overline{ \Gamma}$ is connected and
of finite valency. Does there exist a positive integer $k$ (possibly
depending
on ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.90.
Proposed by: V. I. Trofimov | 2 | open | V. I. Trofimov | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,179 | KOU-15.91 | Kourovka Notebook Problem 15.91 | Is it true that any irreducible faithful
representation of a linear group $G$ of finite rank over a finitely
generated field of characteristic zero is induced from an irreducible
representation of a finitely generated dense subgroup of the group
$G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.91.
Proposed by: A. V. Tushev | 2 | open | A. V. Tushev | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,180 | KOU-15.92 | Kourovka Notebook Problem 15.92 | A group $\left< x,y\mid x^l=y^m=(xy)^n\right>$ is called the *triangle group* with parameters $(l, m, n)$. It is proved in
(A. M. Brunner, R. G. Burns, J. Wiegold, *Math. Scientist*,
**4** (1979), 93--98) that the triangle group $ (2, 3, 30)$ has
uncountably many non-isomorphic homomorphic
images that are residually fi... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.92.
Proposed by: J. Wiegold
Discussion and literature:
[Unverified AI claim] Yes, it is true (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,
... | 2 | open | J. Wiegold | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] Yes, it is true (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,\n... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,181 | KOU-15.93 | Kourovka Notebook Problem 15.93 | Let $G$ be a pro-$p$ group, $p>2$, and $\varphi$ an
automorphism of $G$ of order 2. Suppose that the centralizer
$C_G(\varphi )$ is abelian. Is it true that $G$ satisfies a pro-$p$
identity?
An affirmative answer would generalize a result
of A. N. Zubkov (*Siberian Math. J.*, **28**, no. 5
(1987), 742--747) saying tha... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.93.
Proposed by: A. Jaikin-Zapirain | 2 | open | A. Jaikin-Zapirain | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,182 | KOU-15.94 | Kourovka Notebook Problem 15.94 | Define the *weight* of a group $G$ to be the minimum number of
generators of $G$ as a normal subgroup of itself.
Let $G=G_1*\cdots *G_n$ be a free product of $n$ nontrivial groups.
a) Is it true that the
weight of $G$ is at
least $n/2$?
b) Is it true if the $G_i$ are cyclic?
c) Is it true for $n=3$ (without assuming... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.94.
Proposed by: J. Howie | 2 | open | J. Howie | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,183 | KOU-15.95 | Kourovka Notebook Problem 15.95 | (A. Mann, Ch. Praeger). Suppose
that all fixed-point-free elements of a transitive permutation
group $G$ have prime order $p$. If $G$ is a finite $p $-group,
must $G$ have exponent $p$?
This is true for $p=2, 3$; it is also known
that the
exponent of $G$ is bounded in terms of $p$ (A. Hassani,
M. Khayaty, E. I. Khukhr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.95.
Proposed by: E. I. Khukhro | 2 | open | E. I. Khukhro | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,184 | KOU-15.96 | Kourovka Notebook Problem 15.96 | An automorphism $\varphi $ of a group $G$ is
called a *splitting automorphism of order
$n$* if $\varphi ^n=1$ and $xx^{\varphi}x^{\varphi
^{2}}\cdots x^{\varphi ^{n-1}}=1$ for any $x\in G$.
a) Is it true that the derived length of a
$d$-generated nilpotent $p $-group admitting a splitting
automorphism of order $p^n$ i... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.96.
Proposed by: E. I. Khukhro | 2 | open | E. I. Khukhro | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,185 | KOU-15.97 | Kourovka Notebook Problem 15.97 | Let $p$ be a prime. A group $G$ satisfies the
*$p $-minimal condition* if there are no infinite
descending chains $G_1>G_2>\ldots $ of subgroups of $G$ such that
each difference $G_i\setminus G_{i+1}$ contains a $p $-element
(S. N. Chernikov). Suppose that a locally finite group $G$
satisfies the $p $-minimal condition... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.97.
Proposed by: N. S. Chernikov | 2 | open | N. S. Chernikov | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,186 | KOU-15.98 | Kourovka Notebook Problem 15.98 | Let $\mathfrak{F}$ be a saturated formation, and $G$
a finite soluble minimal non-$\mathfrak{F}$-group such that
$G^{\mathfrak{F}}$ is a Sylow $p $-subgroup of $G$. Is it true
that $G^{\mathfrak{F}}$ is isomorphic to a factor group of the Sylow
$p $-subgroup of $PSU(3, p^{2n})$? This is true for the
formation of finite... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.98.
Proposed by: L. A. Shemetkov | 2 | open | L. A. Shemetkov | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,187 | KOU-15.99 | Kourovka Notebook Problem 15.99 | Let $f(n)$ be the number of isomorphism classes of finite groups
of order $n$. Is it true that the equation $f(n)=k$ has a solution
for any positive integer $k$? The answer is affirmative for all $k
\leq 1000$ (G. M. Wei, *Southeast Asian Bull. Math.*, **22**, no. 1 (1998), 93--102). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.99.
Proposed by: W. J. Shi | 2 | open | W. J. Shi | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,188 | KOU-15.100 | Kourovka Notebook Problem 15.100 | Is a periodic group locally finite if it has
a non-cyclic subgroup of order 4 that coincides with its centralizer? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.100.
Proposed by: A. K. Shl\"epkin | 2 | open | A. K. Shl\"epkin | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,189 | KOU-15.101 | Kourovka Notebook Problem 15.101 | Is a periodic group locally finite if it has an involution whose
centralizer is a locally finite group with Sylow 2-subgroup of order
2? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.101.
Proposed by: A. K. Shl\"epkin | 2 | open | A. K. Shl\"epkin | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,190 | KOU-15.102 | Kourovka Notebook Problem 15.102 | (W. Magnus). An
element $r$ of a free group $F_n$ is called a *normal root* of
an element $u\in F_n$ if $u$ belongs to the normal closure of $r$
in $F_n$.
Can an element $u$ that lies outside the commutator
subgroup $[F_n, F_n]$ have infinitely many non-conjugate normal
roots? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.102.
Proposed by: V. E. Shpilrain | 2 | open | V. E. Shpilrain | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,191 | KOU-15.103 | Kourovka Notebook Problem 15.103 | (Well-known problem). Is the group $\mathrm{Out}(F_3)$ of outer automorphisms of a free group of rank $3$ linear? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.103.
Proposed by: V. E. Shpilrain | 2 | open | V. E. Shpilrain | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,192 | KOU-15.104 | Kourovka Notebook Problem 15.104 | Let $n$ be a positive integer and let
$w$ be a
group word in the variables $x_1, x_2,\ldots $.
Suppose that a residually finite group $G$ satisfies the identity $
w^n=1$.
Does it follow that the verbal subgroup $w(G)$ is
locally finite?
This is the Restricted Burnside Problem if
$w=x_1$. A positive answer was obtained... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 15.104.
Proposed by: P. V. Shumyatsky | 2 | open | P. V. Shumyatsky | 2,002 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,193 | KOU-16.1 | Kourovka Notebook Problem 16.1 | Let $G$ be a finite non-abelian group, and
$Z(G)$ its centre. One can associate a graph $\Gamma_G$ with $G$
as follows: take $G\backslash Z(G)$ as vertices of $\Gamma_G$ and
join two vertices $x$ and $y$ if $xy\not=yx$. Let $H$ be a finite
non-abelian group such that $\Gamma_G \cong \Gamma_H$.
b) If $H$ is nilpotent,
... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 16.1.
Proposed by: A. Abdollahi, S. Akbari, H. R. Maimani
Archive of Solved Problems entry:
a) Yes, it is mod CFSG
(Ch. Khan', G. Ch\`en', S. Go, *
Siberian ... | 2 | partially_solved | A. Abdollahi, S. Akbari, H. R. Maimani | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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) Yes, it is mod CFSG\n(Ch. Khan', G. Ch\\`en', S. Go, *\nSiberian Math. J.*, **49**, no. 6 (2008), 1138--1146, for sporadic simple groups; \\ A. Abdollahi, H. ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,194 | KOU-16.2 | Kourovka Notebook Problem 16.2 | A group $G$ is *subgroup-separable* if for
any subgroup $H\leq G$ and element $x\in G\setminus H$ there is a
homomorphism to a finite group $f: G\rightarrow F$ such that
$f(x)\not\in f(H)$. Is it true that a finitely generated solvable
group is locally subgroup-separable if and only if it does not
contain a solvable Ba... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.2.
Proposed by: R. C. Alperin
Published resolution:
No, it is not (J. O. Button, *Ricerche di Matematica*, **61**, no. 1 (2012), 139--145). | 2 | solved | R. C. Alperin | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (J. O. Button, *Ricerche di Matematica*, **61**, no. 1 (2012), 139--145)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,195 | KOU-16.3 | Kourovka Notebook Problem 16.3 | Is it true that if $G$ is a finite group with
all conjugacy classes of distinct sizes, then $G\cong {\mathbb{S}_3$?
This is true if $G$ is solvable (J. Zhang,
*J. Algebra*, **170** (1994), 608--624); it
is also known that $F(G)$ is nontrivial (Z. Arad,
M. Muzychuk, A. Oliver, *J. Algebra*, **280** (2004), 537--576). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.3.
Proposed by: Z. Arad | 2 | open | Z. Arad | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,196 | KOU-16.4 | Kourovka Notebook Problem 16.4 | Let $G$ be a finite group with $C,D$ two nontrivial
conjugacy classes such that $CD$ is also a conjugacy class. Can
$G$ be a non-abelian simple group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.4.
Proposed by: Z. Arad | 2 | open | Z. Arad | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,197 | KOU-16.5 | Kourovka Notebook Problem 16.5 | A group is said to be *perfect* if it
coincides with its derived subgroup. Does there exist a perfect
locally finite $p$-group
a) all of whose proper subgroups are
hypercentral?
b) all of whose proper subgroups are solvable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.5.
Proposed by: A. O. Asar | 2 | open | A. O. Asar | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,198 | KOU-16.6 | Kourovka Notebook Problem 16.6 | Can a perfect locally finite $p $-group be
generated by a subset of bounded exponent
a) if all of its proper subgroups are
hypercentral?
b) if all of its proper subgroups are
solvable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.6.
Proposed by: A. O. Asar | 2 | open | A. O. Asar | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,199 | KOU-16.7 | Kourovka Notebook Problem 16.7 | Is it true that the membership problem is
undecidable for any semidirect product $F_n \leftthreetimes F_n$
of non-abelian free groups $F_n$?
An affirmative answer would imply an answer
to 6.24. Recall that the membership problem is decidable for
$F_n$ (M. Hall, 1949) and undecidable for $F_n \times F_n$
(K. A. Mikhail... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.7.
Proposed by: V. G. Bardakov | 2 | open | V. G. Bardakov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,200 | KOU-16.8 | Kourovka Notebook Problem 16.8 | The *width* $w(G')$ of the derived
subgroup $G'$ of a finite non-abelian group $G$ is the smallest
positive integer $m$ such that every element of $G'$ is a product
of $\leq m$ commutators. Is it true that the maximum value of the
ratio $w(G')/|G|$ is $1/6$ (attained at the symmetric group ${\mathbb{S}_3$)? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.8.
Proposed by: V. G. Bardakov
Published resolution:
Yes, it is (T. Bonner,
*J. Algebra*, **320** (2008), 3165--3171). | 2 | solved | V. G. Bardakov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (T. Bonner,\n*J. Algebra*, **320** (2008), 3165--3171)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,201 | KOU-16.9 | Kourovka Notebook Problem 16.9 | An element $g$ of a free group $F_n$ on the
free generators $x_1,\ldots,x_n$ is called a *palindrome*
with respect to these generators if the reduced word
representing $g$ is the same when read from left to right or from
right to left. The *palindromic length* of an element $w \in
F_n$ is the smallest number of palindr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.9.
Proposed by: V. G. Bardakov, V. A. Tolstykh, V. E. Shpilrain
Discussion and literature:
[Unverified AI claim] Yes, there is (unverified AI solutions: Achy... | 2 | open | V. G. Bardakov, V. A. Tolstykh, V. E. Shpilrain | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] Yes, there is (unverified AI solutions: Achyuth Jayadevan, *Preprint of 11 Septem... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,202 | KOU-16.10 | Kourovka Notebook Problem 16.10 | Is there an algorithm for finding the primitive
length of a given element of $F_n$? The definition of a primitive
element is given in 14.84; the primitive length is defined
similarly to the palindromic length in 16.9. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.10.
Proposed by: V. G. Bardakov, V. A. Tolstykh,
V. E. Shpilrain | 2 | open | V. G. Bardakov, V. A. Tolstykh,
V. E. Shpilrain | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,203 | KOU-16.11 | Kourovka Notebook Problem 16.11 | Let $G$ be a finite $p$-group. Does there
always exist a finite $p$-group $H$ such that $\Phi(H)\cong
[G,G]$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.11.
Proposed by: Ya. G. Berkovich
Discussion and literature:
[Solved] No, not always (V. Ionin, A. Semidetnov, *Preprint*, 2026, https://arxiv.org/pdf/2608.2... | 2 | solved | Ya. G. Berkovich | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (V. Ionin, A. Semidetnov, *Preprint*, 2026, https://arxiv.org/pdf/2608.29219)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,204 | KOU-16.12 | Kourovka Notebook Problem 16.12 | Given a finite $p $-group $G$, we define a
*$\Phi$-extension* of $G$ as any finite $p $-group $H$
containing a normal subgroup $N$ of order $p$ such that $H/N\cong
G$ and $N\leq\Phi(H)$. Is it true that for every finite
$p $-group $G$ there exists an infinite sequence
$G=G_1, G_2,\ldots $ such that $G_{i+1}$ is a $\Phi... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.12.
Proposed by: Ya. G. Berkovich
Published resolution:
Yes, it is
(S. F. Kamornikov, *Izv. Gomel' Univ.*, **5** (2008),
200--201 (Russian)). | 2 | solved | Ya. G. Berkovich | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is\n(S. F. Kamornikov, *Izv. Gomel' Univ.*, **5** (2008),\n200--201 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,205 | KOU-16.13 | Kourovka Notebook Problem 16.13 | Does there exist a finite $p$-group $G$
all of whose maximal subgroups $H$ are special, that is, satisfy
$Z(H)=[H,H]=\Phi(H)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.13.
Proposed by: Ya. G. Berkovich
Published resolution:
Yes, for all primes there are groups of arbitrarily large size with this property (J. Cossey, *Bull. Austral. Math. Soc.... | 2 | solved | Ya. G. Berkovich | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, for all primes there are groups of arbitrarily large size with this property (J. Cossey, *Bull. Austral. Math. Soc.*, **89** (2014), 415--419); an example f... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,206 | KOU-16.14 | Kourovka Notebook Problem 16.14 | Let $G$ be a finite $2$-group such that
$\Omega_1(G)\leq Z(G)$. Is it true that the rank of $G/G^2$ is at
most double the rank of $Z(G)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.14.
Proposed by: Ya. G. Berkovich | 2 | open | Ya. G. Berkovich | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,207 | KOU-16.15 | Kourovka Notebook Problem 16.15 | An element $g$ of a group $G$ is an *Engel
element* if for every $h\in G$ there exists $k$ such that
$[h,g,\ldots ,g]=1$, where $g$ occurs $k$ times; if there is such
$k$ independent of $h$, then $g$ is said to be *boundedly
Engel*.
b) Does the set of boundedly
Engel elements form a subgroup in a torsion-free group?
... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 16.15.
Proposed by: V. V. Bludov
Archive of Solved Problems entry:
a) No, not always (A. I. Sozutov,
*Siberian Math. J.*, **60**, no. 6 (2019), 1099--1100). | 2 | partially_solved | V. V. Bludov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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) No, not always (A. I. Sozutov,\n*Siberian Math. J.*, **60**, no. 6 (2019), 1099--1100).); 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 |
701,208 | KOU-16.16 | Kourovka Notebook Problem 16.16 | {a)} Does the set of (not necessarily
boundedly) Engel elements of a group without elements of order 2
form a subgroup?
b) The same question for torsion-free groups.
c) The same question for right-ordered groups.
d) The same question for linearly ordered
groups.
There are examples of 2-groups where a product of two... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.16.
Proposed by: V. V. Bludov | 2 | open | V. V. Bludov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,209 | KOU-16.17 | Kourovka Notebook Problem 16.17 | Is it true that a non-abelian simple group cannot contain Engel
elements other than the identity element? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.17.
Proposed by: V. V. Bludov
Published resolution:
No, it can: since an
involution is an Engel element in any 2-group, counterexamples are
provided by non-abelian simple 2-gro... | 2 | solved | V. V. Bludov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 can: since an\ninvolution is an Engel element in any 2-group, counterexamples are\nprovided by non-abelian simple 2-groups; see Archive, 4.74a).\n(V. V. B... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,210 | KOU-16.18 | Kourovka Notebook Problem 16.18 | Does there exist a linearly orderable soluble
group of derived length exactly $n$ that has a single proper
normal relatively convex subgroup
a) for $n=3$?
b) for $n > 4$?
Such groups do exist for $ n=2$ and for
$n=4$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.18.
Proposed by: V. V. Bludov | 2 | open | V. V. Bludov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,211 | KOU-16.19 | Kourovka Notebook Problem 16.19 | Is the variety of lattice-ordered groups
generated by nilpotent groups finitely based? (Here the variety is
considered in the signature of group and lattice operations.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.19.
Proposed by: V. V. Bludov | 2 | open | V. V. Bludov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,212 | KOU-16.20 | Kourovka Notebook Problem 16.20 | Let $M$ be a quasivariety of groups. The *dominion*
$\mathrm{dom}^M_A(H)$ of a subgroup $H$ of a group $A$ (in
$M$) is the set of all elements $a\in A$ such that for any two
homomorphisms $f,g:A\rightarrow B\in M$, if $f,g$ coincide on $H$,
then $f(a)=g(a)$. Suppose that the set $\{ \mathrm{dom}^N_A(H)\mid N
\mbox{ is ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.20.
Proposed by: A. I. Budkin
Discussion and literature:
[Unverified AI claim] Yes, it can (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,
h... | 2 | open | A. I. Budkin | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] Yes, it can (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,\n\nht... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,213 | KOU-16.21 | Kourovka Notebook Problem 16.21 | Given a non-central matrix $\alpha \in
SL_n(F)$ over a field $F$ for $n>2$, is it true that every
non-central matrix in
$SL_n(F)$ is a product of $n$ matrices, each similar to $\alpha
$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.21.
Proposed by: L. Vaserstein | 2 | open | L. Vaserstein | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,214 | KOU-16.22 | Kourovka Notebook Problem 16.22 | (Well-known problem.) Let
$E_n(A)$ be the subgroup of $GL_n(A)$ generated by elementary
matrices. Is $SL_2(A) = E_2(A)$ when $A= {\mathbb{Z}[x, 1/x]$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.22.
Proposed by: L. Vaserstein | 2 | open | L. Vaserstein | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,215 | KOU-16.23 | Kourovka Notebook Problem 16.23 | Is there, for some $n>2$ and a ring $A$ with
$1$, a matrix in $E_n(A)$ that is nonscalar modulo any proper
ideal and is not a commutator? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.23.
Proposed by: L. Vaserstein | 2 | open | L. Vaserstein | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,216 | KOU-16.24 | Kourovka Notebook Problem 16.24 | The *spectrum* of a finite group is the
set of orders of its elements. Does there exist a finite group
$G$ whose spectrum coincides with the spectrum of a finite simple
exceptional group $L$ of Lie type, but $G$ is not isomorphic to
$L$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.24.
Proposed by: A. V. Vasil'ev
Published resolution:
Yes, for example, for $L={}^3D_4(2)$ (V. D. Mazurov,
*Algebra Logic*, **52**, no. 5 (2013), 400--403). Further examples a... | 2 | solved | A. V. Vasil'ev | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, for example, for $L={}^3D_4(2)$ (V. D. Mazurov,\n\n*Algebra Logic*, **52**, no. 5 (2013), 400--403). Further examples are given in\nM. A. Grechkoseeva, M. A... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,217 | KOU-16.25 | Kourovka Notebook Problem 16.25 | Do there exist three pairwise non-isomorphic
finite non-abelian simple groups with the same spectrum? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.25.
Proposed by: A. V. Vasil'ev
Published resolution:
No (A. A. Buturlakin, *Sibirsk. Electron. Math. Reports*, **7** (2010), 111--114). | 2 | solved | A. V. Vasil'ev | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. A. Buturlakin, *Sibirsk. Electron. Math. Reports*, **7** (2010), 111--114)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,218 | KOU-16.26 | Kourovka Notebook Problem 16.26 | We say that the prime graphs of finite groups
$G$ and $H$ coincide if the sets of primes dividing their orders
are the same, $\pi (G)=\pi (H)$, and for any distinct $p,q\in \pi
(G)$ there is an element of order $pq$ in $G$ if and only if there
is such an element in $H$. Does there exist a positive integer $k$
such that... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.26.
Proposed by: A. V. Vasil'ev | 2 | open | A. V. Vasil'ev | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,219 | KOU-16.27 | Kourovka Notebook Problem 16.27 | Suppose that a finite group $G$ has the
same spectrum as an alternating group. Is it true that $G$ has at
most one non-abelian composition factor?
For any finite simple group other than
alternating
the answer to the corresponding question is
affirmative (mod CFSG). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.27.
Proposed by: A. V. Vasil'ev and
V. D. Mazurov
Published resolution:
Yes, it is (I. B. Gorshkov, *Algebra and Logic*, **52**, no. 1 (2013), 41--45). | 2 | solved | A. V. Vasil'ev and
V. D. Mazurov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
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"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 (I. B. Gorshkov, *Algebra and Logic*, **52**, no. 1 (2013), 41--45)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,220 | KOU-16.28 | Kourovka Notebook Problem 16.28 | Let $G$ be a connected linear reductive
algebraic group over a field of positive characteristic, $X$ a
closed subset of $G$, and let $X^k=\{x_1\cdots x_k\mid x_i\in
X\}$.
a) Is it true that there always exists a
positive integer $c=c(X)>1$ such that $X^c$ is closed?
b) If $X$ is a conjugacy class of $G$ such that
$X^... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.28.
Proposed by: E. P. Vdovin
Discussion and literature:
[Solved] No, not always (AI solution confirmed by E. P. Vdovin: \ Achyuth Jayadevan, *Preprint of 12... | 2 | partially_solved | E. P. Vdovin | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. P. Vdovin: \\ Achyuth Jayadevan, *Preprint of 12 S... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,221 | KOU-16.29 | Kourovka Notebook Problem 16.29 | Which finite simple groups of Lie type $G$ have the following property:
for every semisimple abelian subgroup $A$ and proper subgroup $H$ of $G$ there exists $x\in G$ such that $A^x\cap H=1$?
This property holds when $A$ is a cyclic subgroup
(J. Siemons, A. Zalesskii, *J. Algebra*, **256** (2002), 611--625), as well a... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.29.
Proposed by: E. P. Vdovin | 2 | open | E. P. Vdovin | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,222 | KOU-16.30 | Kourovka Notebook Problem 16.30 | Suppose that $A$ and $B$ are subgroups of a
group $G$ and $G = AB$. Will $G$ have composition (principal)
series if $A$ and $B$ have composition (respectively, principal)
series? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.30.
Proposed by: V. A. Vedernikov
Discussion and literature:
[Unverified AI claim] No, not always, both for composition and principal series (unverified AI s... | 2 | open | V. A. Vedernikov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] No, not always, both for composition and principal series (unverified AI solution... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,223 | KOU-16.31 | Kourovka Notebook Problem 16.31 | Suppose that a group $G$ has a composition
series and let ${\mathfrak{F}}(G)$ be the formation generated by $G$.
Is the set of all subformations of ${\mathfrak{F}}(G)$ finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.31.
Proposed by: V. A. Vedernikov
Published resolution:
No, not always, even for $G$ finite (V. P. Burichenko, *J. Algebra*, **372**
(2012), 428--458). | 2 | solved | V. A. Vedernikov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, even for $G$ finite (V. P. Burichenko, *J. Algebra*, **372**\n(2012), 428--458)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,224 | KOU-16.32 | Kourovka Notebook Problem 16.32 | Suppose that a group $G$ has a composition
series and let $\mathrm{Fit}(G)$ be the Fitting class generated by
$G$. Is the set of all Fitting subclasses of $\mathrm{Fit}(G)$
finite? Cf. 14.31. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.32.
Proposed by: V. A. Vedernikov | 2 | open | V. A. Vedernikov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,225 | KOU-16.33 | Kourovka Notebook Problem 16.33 | Suppose that a finite $p $-group $G$ has
an abelian subgroup $A$ of order $p^n$. Does $G$ contain an
abelian subgroup $B$ of order $p^n$ that is normal in $\langle
B^G\rangle$
a) if $p=3$?
b) if $p=2$?
c) If $p=3$ and $A$ is elementary abelian, does
$G$ contain an elementary abelian subgroup $B$ of order $3^n$ that
... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.33.
Proposed by: G. Glauberman | 2 | open | G. Glauberman | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,226 | KOU-16.34 | Kourovka Notebook Problem 16.34 | Suppose that $G$ is a finitely generated
group acting faithfully on a regular rooted tree by finite-state
automorphisms. Is the conjugacy problem decidable for $G$?
See the definitions in (R. I. Grigorchuk,
V. V. Nekrashevich,
V. I. Sushchanskiĭ,
*Proc. Steklov Inst. Math.*, **2000**, no. 4 (231),
128--203). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.34.
Proposed by: R. I. Grigorchuk,
V. V. Nekrashevich, V. I. Sushchanski\u{\i
Discussion and literature:
[Solved] No, not always (Z. \v{S}uni\'{c}, E. Ventur... | 2 | solved | R. I. Grigorchuk,
V. V. Nekrashevich, V. I. Sushchanski\u{\i | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (Z. \\v{S}uni\\'{c}, E. Ventura, *J. Algebra*, **364** (2012), 148--154)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,227 | KOU-16.35 | Kourovka Notebook Problem 16.35 | Is every finitely presented soluble group
nilpotent-by-nilpotent-by-finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.35.
Proposed by: J. R. J. Groves
Published resolution:
No: for a prime $p$ the
group $G=\langle a,b,c,d\mid b^a =b^p,\; c^a=c^p,\; d^a=d,\;
c^b=c, d^c=dd^b,\; [d,d^b]=1,\; d^p=... | 2 | solved | J. R. J. Groves | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 a prime $p$ the\ngroup $G=\\langle a,b,c,d\\mid b^a =b^p,\\; c^a=c^p,\\; d^a=d,\\;\nc^b=c, d^c=dd^b,\\; [d,d^b]=1,\\; d^p=1\\rangle $ is soluble\nof deri... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,228 | KOU-16.36 | Kourovka Notebook Problem 16.36 | (Well-known problem).
We call a finite group *rational* if all of its
ordinary characters are rational-valued. Is every Sylow 2-subgroup
of a rational group also a rational group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.36.
Proposed by: M. R. Darafsheh
Published resolution:
No, not always
(I. M. Isaacs, G. Navarro, *Math. Z.*, **272** (2012), 937--945). | 2 | solved | M. R. Darafsheh | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always\n(I. M. Isaacs, G. Navarro, *Math. Z.*, **272** (2012), 937--945)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,229 | KOU-16.37 | Kourovka Notebook Problem 16.37 | Let $G$ be a solvable rational finite group with an
extra-special Sylow 2-subgroup. Is it true that either $G$ is a
2-nilpotent group, or there is a normal subgroup $E$ of $G$ such
that $G/E$ is an extension of a normal 3-subgroup by an elementary
abelian 2-group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.37.
Proposed by: M. R. Darafsheh
Published resolution:
Yes, it is (M. R. Darafsheh, H. Sharifi, *Extracta Math.*,
**22**, no. 1 (2007), 83--91). | 2 | solved | M. R. Darafsheh | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. R. Darafsheh, H. Sharifi, *Extracta Math.*,\n**22**, no. 1 (2007), 83--91)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,230 | KOU-16.38 | Kourovka Notebook Problem 16.38 | Let $G$ be a soluble group, and let $A$ and $B$ be periodic
subgroups of $G$. Is it true that any subgroup of $G$ contained in
the set $AB=\{ ab\;|\; a\in A,\; b\in B\}$ is periodic? This is
known to be true if $AB$ is a subgroup of $G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.38.
Proposed by: F. de Giovanni
Discussion and literature:
[Solved] Yes, it is true (AI solution confirmed by B. Amberg: \ M. Kinyon, J. Urban, *Preprint*, 2... | 2 | solved | F. de Giovanni | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 true (AI solution confirmed by B. Amberg: \\ M. Kinyon, J. Urban, *Preprint*, 2026,\nhttps://... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,231 | KOU-16.39 | Kourovka Notebook Problem 16.39 | (J. E. Humphreys, D. N. Verma).
Let $G$ be a semisimple algebraic group over an algebraically
closed field $k$ of characteristic $p>0$. Let ${\mathfrak{g}}$ be the
Lie algebra of $G$ and let $u=u({\mathfrak{g}})$ be the restricted
enveloping algebra of ${\mathfrak{g}}$. By a theorem of Curtis every
irreducible restrict... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.39.
Proposed by: S. Donkin | 2 | open | S. Donkin | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,232 | KOU-16.40 | Kourovka Notebook Problem 16.40 | Let $\Delta$ be a subgroup of the automorphism
group of a free pro-$p$ group of finite rank $F$ such that
$\Delta$ is isomorphic (as a profinite group) to the group $Z_p$
of $ p $-adic integers. Is the subgroup of fixed points of
$\Delta$ in $F$ finitely generated (as a profinite group)? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.40.
Proposed by: P. A. Zalesski\u{\i | 2 | open | P. A. Zalesski\u{\i | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,233 | KOU-16.41 | Kourovka Notebook Problem 16.41 | Let $F$ be a free pro-$p$ group of finite rank
$n>1$. Does $\mathrm{Aut} F$ possess an open subgroup of finite
cohomological dimension? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.41.
Proposed by: P. A. Zalesski\u{\i | 2 | open | P. A. Zalesski\u{\i | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,234 | KOU-16.42 | Kourovka Notebook Problem 16.42 | Is a topological Abelian group $(G,\tau)$
compact if every group topology $\tau '\subseteq\tau$ on $G$ is
complete? (The answer is yes if every continuous homomorphic image
of $(G,\tau)$ is complete.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.42.
Proposed by: E. G. Zelenyuk
Published resolution:
Yes, it is (T. Banakh, *Topology Appl.*, **271** (2020), Article ID 106983, 17 p.). | 2 | solved | E. G. Zelenyuk | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (T. Banakh, *Topology Appl.*, **271** (2020), Article ID 106983, 17 p.)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,235 | KOU-16.43 | Kourovka Notebook Problem 16.43 | Is there a
partition of the group $\bigoplus_{\omega_1}({\mathbb{Z}}/3{\mathbb{Z}})$ into three subsets whose complements do not contain cosets
modulo infinite subgroups? (There is a partition into two such
subsets.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.43.
Proposed by: E. G. Zelenyuk
Published resolution:
Yes, moreover, every
infinite abelian group with finitely many involutions can be
partitioned into infinitely many subsets... | 2 | solved | E. G. Zelenyuk | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, moreover, every\ninfinite abelian group with finitely many involutions can be\npartitioned into infinitely many subsets such that every coset\nmodulo an inf... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,236 | KOU-16.44 | Kourovka Notebook Problem 16.44 | Is there in ZFC a countable non-discrete
topological group not containing discrete subsets with a single
accumulation point? (Such a group is known to exist under Martin's
Axiom.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.44.
Proposed by: E. G. Zelenyuk | 2 | open | E. G. Zelenyuk | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,237 | KOU-16.45 | Kourovka Notebook Problem 16.45 | Let $G$ be a permutation group on a set $\Omega$. A sequence of
points of $\Omega$ is a *base* for $G$ if its pointwise
stabilizer in $G$ is the identity; it is *minimal* if no
point may be removed. Let $b(G)$ be the maximum, over all
permutation representations of the finite group $G$, of the
maximum size of a minimal... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.45.
Proposed by: P. J. Cameron
Discussion and literature:
[Solved] No, not always (AI solution confirmed by P. J. Cameron: \ A. Kriebel, *Preprint*, 2026, ht... | 2 | solved | P. J. Cameron | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (AI solution confirmed by P. J. Cameron: \\ A. Kriebel, *Preprint*, 2026, https://zenodo.... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,238 | KOU-16.46 | Kourovka Notebook Problem 16.46 | Among the finitely-presented groups that act arc-transitively on
the (infinite) 3-valent tree with finite vertex-stabilizer are the
two groups $G_3 = \left< h, a, P, Q \mid h^3\right.$, $ a^2$, $
P^2$, $ Q^2$, $ [P,Q]$, $ [h,P]$, $ (hQ)^2$, $\left. a^{-1}PaQ
\right>$ \ and \ $G_4 = \left< h, a, p, q, r \mid
h^3\right.$... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.46.
Proposed by: M. Conder | 2 | open | M. Conder | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,239 | KOU-16.47 | Kourovka Notebook Problem 16.47 | (P. Conrad).
Is it true that every torsion-free
abelian group admits an Archimedean lattice ordering? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.47.
Proposed by: V. M. Kopytov, N. Ya. Medvedev | 2 | open | V. M. Kopytov, N. Ya. Medvedev | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,240 | KOU-16.48 | Kourovka Notebook Problem 16.48 | A group $H$ is said to have *generalized
torsion* if there exists an element $h\ne 1$ such that
$h^{x_1}h^{x_2}\cdots h^{x_n}= 1$ for some $n$ and some $x_i\in
H$. Is every group without generalized torsion right-orderable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.48.
Proposed by: V. M. Kopytov, N. Ya. Medvedev
Discussion and literature:
[Solved] No, not every (T. W. Cai, A. Clay, *Preprint*, 2025, https://arxiv.org/ab... | 2 | solved | V. M. Kopytov, N. Ya. Medvedev | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not every (T. W. Cai, A. Clay, *Preprint*, 2025, https://arxiv.org/abs/2504.08084)"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,241 | KOU-16.49 | Kourovka Notebook Problem 16.49 | Is it true that a free product of groups
without generalized torsion is a group without generalized
torsion? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.49.
Proposed by: V. M. Kopytov,
N. Ya. Medvedev
Published resolution:
Yes, it is true; moreover, the generalized torsion in a free product of torsion-free groups is conjugate t... | 2 | solved | V. M. Kopytov,
N. Ya. Medvedev | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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; moreover, the generalized torsion in a free product of torsion-free groups is conjugate to a generalized torsion of one of its factor groups (T.... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,242 | KOU-16.50 | Kourovka Notebook Problem 16.50 | Do there exist simple finitely generated
right-orderable groups?
There exist finitely generated right-orderable
groups coinciding with the derived subgroup (G. M. Bergman). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.50.
Proposed by: V. M. Kopytov, N. Ya. Medvedev
Published resolution:
Yes, such groups do exist (J. Hyde, Y. Lodha, *Invent. Math.*, **218** (2019), 83--112). | 2 | solved | V. M. Kopytov, N. Ya. Medvedev | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (J. Hyde, Y. Lodha, *Invent. Math.*, **218** (2019), 83--112)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,243 | KOU-16.51 | Kourovka Notebook Problem 16.51 | Do there exist groups that can be
right-ordered in infinitely countably many ways?
There exist groups that can be fully ordered in
infinitely countably many ways (R. Buttsworth). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.51.
Proposed by: V. M. Kopytov, N. Ya. Medvedev
Discussion and literature:
[Solved] No, there are no such groups (A. Clay, *Preprint*, 2009, http://arxiv.org... | 2 | solved | V. M. Kopytov, N. Ya. Medvedev | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, there are no such groups (A. Clay, *Preprint*, 2009, http://arxiv.org/abs/0909.0273; \\ A. Navas, C.... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,244 | KOU-16.52 | Kourovka Notebook Problem 16.52 | Is every finitely presented elementary amenable group
solvable-by-finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.52.
Proposed by: P. Linnell, T. Schick
Published resolution:
No, not every. I. Belegradek and Y. Cornulier have pointed out that the
groups in (C. H. Houghton, *Arch. Math. (Ba... | 2 | solved | P. Linnell, T. Schick | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not every. I. Belegradek and Y. Cornulier have pointed out that the\ngroups in (C. H. Houghton, *Arch. Math. (Basel)*, **31**, no. 3 (1978/79), 254--258) are... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,245 | KOU-16.53 | Kourovka Notebook Problem 16.53 | Let $d(G)$ denote the smallest cardinality of a
generating set of the group $G$. Suppose that
$G=\left<A,B\right>$, where $A$ and $B$ are two $d$-generated
finite groups of coprime orders. Is it true that $d(G)\leq d+1$?
See Archive 12.71 and (A. Lucchini, *J. Algebra*, **245** (2001), 552--561).
*Comment of 2021*:
t... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.53.
Proposed by: A. Lucchini
Discussion and literature:
[Solved] No, not always (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/w... | 2 | solved | A. Lucchini | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/wp-content/up... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,246 | KOU-16.54 | Kourovka Notebook Problem 16.54 | We say that a group $G$ acts *freely*
on a group $V$ if $vg\ne v$ for any nontrivial elements $g\in
G,\; v\in V$. Is it true that a group $G$ that can act
freely on a non-trivial abelian group is embeddable in the
multiplicative group of some skew-field? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.54.
Proposed by: V. D. Mazurov
Published resolution:
No, it is not. For example, the group $2.A_5.2$ with a quaternion Sylow 2-subgroup can act freely on an elementary abelian ... | 2 | solved | V. D. Mazurov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. For example, the group $2.A_5.2$ with a quaternion Sylow 2-subgroup can act freely on an elementary abelian group of order $7^4$ but is not embedd... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,247 | KOU-16.55 | Kourovka Notebook Problem 16.55 | (Well-known
problem). Let $V$ be a faithful absolutely irreducible module for
a finite group $G$. Is it true that $\dim H^1(G,V)\leq 2$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.55.
Proposed by: V. D. Mazurov
Published resolution:
No, it is not
(L. L. Scott, *J. Algebra*, **260** (2003), 416--425; see
also J. N. Bray, R. A. Wilson, *J. Group Theory*, *... | 2 | solved | V. D. Mazurov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it is not\n(L. L. Scott, *J. Algebra*, **260** (2003), 416--425; see\nalso J. N. Bray, R. A. Wilson, *J. Group Theory*, **11**\n(2008), 669--673)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,248 | KOU-16.56 | Kourovka Notebook Problem 16.56 | The *spectrum* $\omega(G)$ of a group
$G$ is the set of orders of elements of $G$. Suppose that
$\omega(G)=\{1,2,3,4,5,6\}$. Is $G$ locally finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.56.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,249 | KOU-16.57 | Kourovka Notebook Problem 16.57 | Suppose that $\omega(G)=\omega(L_2(7))=\{1,2,3,4,7\}$. Is $G\cong
L_2(7)$? This is true for finite $G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.57.
Proposed by: V. D. Mazurov
Published resolution:
Yes, it is
(D. V. Lytkina, A. A. Kuznetsov, *Siberian Electr. Math.
Rep.*, **4** (2007), 136--140; http://semr.math.nsc.ru)... | 2 | solved | V. D. Mazurov | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is\n(D. V. Lytkina, A. A. Kuznetsov, *Siberian Electr. Math.\nRep.*, **4** (2007), 136--140; http://semr.math.nsc.ru)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,250 | KOU-16.58 | Kourovka Notebook Problem 16.58 | Is $SU_2({\mathbb{C})$ the only group
that has just one irreducible complex representation of dimension
$n$ for each $n = 1, 2,\ldots $?
(If $R[n]$ is the $n$-dimensional irreducible
complex representation of $SU_2({\mathbb{C})$, then $R[2]$ is the natural
two-dimensional representation, and $R[2]\otimes R[n] = R[n-1]... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.58.
Proposed by: J. McKay
Published resolution:
No, it is not. It is known that ${\mathbb{C}}$ has infinitely many (discrete) automorphisms. For $\varphi\in\mathrm{Aut}({\mathb... | 2 | solved | J. McKay | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. It is known that ${\\mathbb{C}}$ has infinitely many (discrete) automorphisms. For $\\varphi\\in\\mathrm{Aut}({\\mathbb{C}})$ and a matrix $x$, le... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,251 | KOU-16.59 | Kourovka Notebook Problem 16.59 | Given a finite group $K$, does there exist a
finite group $G$ such that $K\cong \mathrm{Out }G=\mathrm{Aut } G/\mathrm{Inn }G$? (It is known that an infinite group $G$ exists with this
property.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.59.
Proposed by: D. MacHale
Published resolution:
Yes, it does (Y. Cornulier, https://mathoverflow.net/questions/372480/is-every-finite-group-the-outer-automorphism-group-of-a-... | 2 | solved | D. MacHale | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (Y. Cornulier, https://mathoverflow.net/questions/372480/is-every-finite-group-the-outer-automorphism-group-of-a-finite-group/372563)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,252 | KOU-16.60 | Kourovka Notebook Problem 16.60 | If $G$ is a finite group, let $T(G)$ be
the sum of the degrees of the irreducible complex representations
of $G$, $T(G)=\sum\limits_{i=1}^{k(G)}d_i$, where $G$ has $k(G)$
conjugacy classes. If $\alpha \in \mathrm{Aut } G$, let $S_{\alpha
}=\{g\in G\mid \alpha (g)=g^{-1}\}$. Is it true that $T(G)\geq
|S_{\alpha }|$ for ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.60.
Proposed by: D. MacHale | 2 | open | D. MacHale | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,253 | KOU-16.61 | Kourovka Notebook Problem 16.61 | A subgroup $H$ of a group $G$ is *fully invariant* if
$\vartheta (H)\leq H$ for every endomorphism $\vartheta$ of $G$.
Let $G$ be a finite group such that $G$ has a fully invariant
subgroup of order $d$ for every $d$ dividing $|G|$. Must $G$ be
cyclic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.61.
Proposed by: D. MacHale
Published resolution:
No: take $G=C_p\times
C_{p^2}$, where $C_p,C_{p^2}$ are cyclic $p$-groups of orders
$p,\;p^2$ (A. Abdollahi, *Letter of 4 Marc... | 2 | solved | D. MacHale | 2,006 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: take $G=C_p\\times\nC_{p^2}$, where $C_p,C_{p^2}$ are cyclic $p$-groups of orders\n$p,\\;p^2$ (A. Abdollahi, *Letter of 4 March 2009*)."
} | 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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