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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
701,254 | KOU-16.62 | Kourovka Notebook Problem 16.62 | Let $G$ be a group such that every $\alpha
\in \mathrm{Aut } G$ fixes every conjugacy class of $G$ (setwise).
Must $\mathrm{Aut } G =\mathrm{Inn }G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.62.
Proposed by: D. MacHale
Published resolution:
Not necessarily: the paper
(H. Heineken, *Arch. Math. $($Basel$)$*, **33**
(1979/80), no. 6, 497--503), in particular, produce... | 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: Not necessarily: the paper\n(H. Heineken, *Arch. Math. $($Basel$)$*, **33**\n(1979/80), no. 6, 497--503), in particular, produces finite\n$p$-groups all of whose... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,255 | KOU-16.63 | Kourovka Notebook Problem 16.63 | Is there a non-trivial finite $p$-group $G$
of odd order such that $|\mathrm{Aut } G| =|G|$?
See also Archive 12.77. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.63.
Proposed by: D. MacHale
Discussion and literature:
[Unverified AI claim] Yes, there is (unverified AI solution: A. Kriebel, *Preprint*, 2026,
https://ko... | 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 | {
"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 solution: A. Kriebel, *Preprint*, 2026,\n\nhttps://k... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,256 | KOU-16.64 | Kourovka Notebook Problem 16.64 | A non-abelian variety in which all finite
groups are abelian is called *pseudo-abelian*. A group variety
is called a *$t$-variety* if for all groups in this variety
the relation of being a normal subgroup is transitive. By
(O. Macedo\'nska, A. Storozhev, *Commun. Algebra*, **25**,
no. 5 (1997), 1589--1593) each non-abe... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.64.
Proposed by: O. Macedo\'nska | 2 | open | O. Macedo\'nska | 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 |
701,257 | KOU-16.65 | Kourovka Notebook Problem 16.65 | Does there exist a finitely presented residually torsion-free
nilpotent group with a free presentation $G=F/R$ such that the
group $F/[F,R]$ is not residually nilpotent? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.65.
Proposed by: R. Mikhailov,
I. B. S. Passi
Published resolution:
Yes, there does
(V. V. Bludov, *J. Group Theory*, **12**, no. 4 (2009),
579--590). | 2 | solved | R. Mikhailov,
I. B. S. Passi | 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, there does\n(V. V. Bludov, *J. Group Theory*, **12**, no. 4 (2009),\n579--590)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,258 | KOU-16.66 | Kourovka Notebook Problem 16.66 | For a group $G$, let $D_n(G)$ denote the $n$-th
dimension subgroup of $G$, and $\zeta_n(G)$ the $n$-th term of its
upper central series. For a given integer $n\geq 1$, let
$f(n)=\max\{ m\mid \exists\;\text{a nilpotent group}\;G\;\text{of
class}\;n\;\text{with}\;D_m(G)\neq 1\}$ and $g(n)=\max\{ m\ |\
\exists\ \text{a ni... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.66.
Proposed by: R. Mikhailov, I. B. S. Passi | 2 | open | R. Mikhailov, I. B. S. Passi | 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 |
701,259 | KOU-16.67 | Kourovka Notebook Problem 16.67 | *Conjecture:* Given any integer $k$, there exists an integer
$n_0=n_0(k)$ such that if $n\geq n_0$ then the symmetric group of
degree $n$ has at least $k$ different ordinary irreducible
characters of equal degrees. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.67.
Proposed by: A. Moret\'o
Published resolution:
This is proved
(D. Craven, *Proc. London Math. Soc.*, **96** (2008),
26--50). | 2 | solved | A. Moret\'o | 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: This is proved\n(D. Craven, *Proc. London Math. Soc.*, **96** (2008),\n26--50)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,260 | KOU-16.68 | Kourovka Notebook Problem 16.68 | Let $W(x, y)$ be a non-trivial reduced group
word, and $G$ one of the groups $PSL(2,{\mathbb{R})$, \ $PSL(2,
{\mathbb{C})$, or $SO(3, {\mathbb{R})$. Are all the maps $W:G \times
G\rightarrow G$ surjective?
For $SL(2, {\mathbb{R})$, \ $SL(2, {\mathbb{C})$, \
$GL(2, {\mathbb{C})$, and for the group $S^3$ of quaternions ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.68.
Proposed by: J. Mycielski
Discussion and literature:
[Solved] These maps are not surjective for $PSL(2,{\mathbb{R})$ and $SO(3, {\mathbb{R})$, and surjec... | 2 | solved | J. Mycielski | 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] These maps are not surjective for $PSL(2,{\\mathbb{R})$ and $SO(3, {\\mathbb{R})$, and surjective for $P... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,261 | KOU-16.69 | Kourovka Notebook Problem 16.69 | Let $W$ be as above.
a) Must the range of the function $\mathrm{Tr}(W(x, y))$ for $x, y\in GL(2, {\mathbb{R})$ include the interval
$[-2, +\infty)$?
b) For $x, y$ being non-zero quaternions,
must the range of the function $\mathrm{Re}(W(x, y))$ include the interval
$[-5/27, 1]$?
(For some comments see *ibid*.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.69.
Proposed by: J. Mycielski
Discussion and literature:
[Solved] a) No, it need not (AI solution confirmed by A. Thom: \ Nilradical, *Preprint*, 2026,
http... | 2 | solved | J. Mycielski | 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] a) No, it need not (AI solution confirmed by A. Thom: \\ Nilradical, *Preprint*, 2026,\n\nhttps://kourov... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,262 | KOU-16.70 | Kourovka Notebook Problem 16.70 | Suppose that a finitely generated group $G$ acts
freely on an $A$-tree, where $A$ is an ordered abelian group. Is
it true that $G$ acts freely on a ${\mathbb{Z}^n$-tree for some
$n$?
*Comment of 2013:*
It was proved that every finitely presented group acting freely on an $A$-tree acts freely on some $\mathbb{R}^n$-tre... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.70.
Proposed by: A. G. Myasnikov,
V. N. Remeslennikov, O. G. Kharlampovich | 2 | open | A. G. Myasnikov,
V. N. Remeslennikov, O. G. Kharlampovich | 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 |
701,263 | KOU-16.71 | Kourovka Notebook Problem 16.71 | Is the elementary theory of a torsion-free
hyperbolic group decidable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.71.
Proposed by: A. G. Myasnikov, O. G. Kharlampovich
Published resolution:
Yes, it is (O. Kharlampovich, A. Myasnikov, https://arxiv.org/pdf/1303.0760.pdf). | 2 | solved | A. G. Myasnikov, O. G. Kharlampovich | 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 (O. Kharlampovich, A. Myasnikov, https://arxiv.org/pdf/1303.0760.pdf)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,264 | KOU-16.72 | Kourovka Notebook Problem 16.72 | Does there exist an exponential-time
algorithm for obtaining a JSJ-decomposition of a finitely
generated fully residually free group?
Some algorithm was found in (O. Kharlampovich,
A. Myasnikov, in: *Contemp. Math. AMS $($Algorithms,
Languages, Logic*), **378** (2005), 87--212). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.72.
Proposed by: A. G. Myasnikov,
O. G. Kharlampovich | 2 | open | A. G. Myasnikov,
O. G. Kharlampovich | 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 |
701,265 | KOU-16.73 | Kourovka Notebook Problem 16.73 | b) Let $G$ be a group generated by a finite set
$S$, and let $l(g)$ denote the word length function of $g\in G$
with respect to $S$. The group $G$ is said to be *contracting* if there exist a faithful action of $G$ on the set
$X^*$ of finite words over a finite alphabet $X$ and constants
$0<\lambda <1$ and $C>0$ such t... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 16.73.
Proposed by: V. V. Nekrashevich
Archive of Solved Problems entry:
a) No, it cannot
(V. V. Nekrashevich, *Groups Geom. Dynam.*, **4**, no. 4 (2010), 84... | 2 | partially_solved | V. V. Nekrashevich | 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": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (a) No, it cannot\n(V. V. Nekrashevich, *Groups Geom. Dynam.*, **4**, no. 4 (2010), 847--862).); 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 |
701,266 | KOU-16.74 | Kourovka Notebook Problem 16.74 | b) Is it true that all groups
generated by automata of polynomial growth in the sense of
S. Sidki (*Geom. Dedicata*, **108** (2004), 193--204) are
amenable?
[Archived solved part(s) of 16.74:]
a)
Let $G=\left<\alpha,\beta\right>$ be the group generated by the
following two permutations of ${\mathbb{Z}$: \ $\alpha (n)=... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 16.74.
Proposed by: V. V. Nekrashevich
Archive of Solved Problems entry:
a) Yes, it is
(G. Amir, O. Angel, B. Vir\'ag, *J. Eur. Math. Soc.*, **15**, no. 3 (2... | 2 | partially_solved | V. V. Nekrashevich | 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\n(G. Amir, O. Angel, B. Vir\\'ag, *J. Eur. Math. Soc.*, **15**, no. 3 (2013), 705--730).); 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 |
701,267 | KOU-16.75 | Kourovka Notebook Problem 16.75 | Can a non-abelian one-relator group be the
group of all automorphisms of some group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.75.
Proposed by: M. V. Neshchadim
Published resolution:
Yes, it can: in
(D. J. Collins, *Proc. London Math. Soc. (3)*, **36**
(1978), no. 3, 480--493) it was proved that for in... | 2 | solved | M. V. Neshchadim | 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 can: in\n(D. J. Collins, *Proc. London Math. Soc. (3)*, **36**\n(1978), no. 3, 480--493) it was proved that for integers $r,s$\nsuch that $(r,s)= 1$, $|r... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,268 | KOU-16.76 | Kourovka Notebook Problem 16.76 | We call a group $G$ *strictly real* if
each of its non-trivial elements is conjugate to its inverse by
some involution in $G$. In which groups of Lie type over a field
of characteristic 2 the maximal unipotent subgroups are
strictly real? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.76.
Proposed by: Ya. N. Nuzhin | 2 | open | Ya. N. Nuzhin | 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 |
701,269 | KOU-16.77 | Kourovka Notebook Problem 16.77 | It is known that in every Noetherian group the
nilpotent radical coincides with the collection of all Engel
elements (R. Baer, *Math. Ann.*, **133** (1957), 256--270;
B. I. Plotkin, *Izv. Vysš. Učebn. Zaved. Mat.*, **1958**, no. 1(2), 130--135 (Russian)). It would be nice to find a
similar characterization of the solva... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.77.
Proposed by: B. I. Plotkin | 2 | open | B. I. Plotkin | 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 |
701,270 | KOU-16.78 | Kourovka Notebook Problem 16.78 | Do there exist linear non-abelian simple
groups without involutions? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.78.
Proposed by: B. Poizat | 2 | open | B. Poizat | 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 |
701,271 | KOU-16.79 | Kourovka Notebook Problem 16.79 | Is it true that in any finitely generated
$AT$-group over a sequence of cyclic groups of uniformly bounded
orders all Sylow subgroups are locally finite? For the definition
of an *$AT $-group* see (A. V. Rozhkov, *Math. Notes*,
**40** (1986), 827--836) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.79.
Proposed by: A. V. Rozhkov
Published resolution:
No, it is not true (A. V. Rozhkov, in *Group theory and its applications*, Proc. XXII School-Conf. on Group Theory, Kuban' ... | 2 | solved | A. V. Rozhkov | 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 true (A. V. Rozhkov, in *Group theory and its applications*, Proc. XXII School-Conf. on Group Theory, Kuban' Univ., Krasnodar, 2018, 126--131 (Russ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,272 | KOU-16.80 | Kourovka Notebook Problem 16.80 | Suppose that a group $G$ is obtained from the
free product of torsion-free groups $A_1, \ldots , A_n$ by
imposing $m$ additional relations, where $m<n$. Is it true that
the free product of some $n-m$ of the $A_i$ embeds into $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.80.
Proposed by: N. S. Romanovski\u{\i | 2 | open | N. S. Romanovski\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 |
701,273 | KOU-16.82 | Kourovka Notebook Problem 16.82 | Let $\mathcal{X}$ be a non-empty class of finite groups closed under
taking homomorphic images, subgroups, and direct products. With
every group $G\in\mathcal{X}$ we associate some set $\tau(G)$ of
subgroups of $G$. We say that $\tau$ is a *subgroup functor
on* $\mathcal{X}$ if:
1) $G\in\tau(G)$ for all $G\in\mathcal{... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.82.
Proposed by: A. N. Skiba
Published resolution:
No such classes exist
(S. F. Kamornikov, *Siberian Math. J.*, **51**, no. 5
(2010), 824--829). | 2 | solved | A. N. Skiba | 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 such classes exist\n(S. F. Kamornikov, *Siberian Math. J.*, **51**, no. 5\n(2010), 824--829)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,274 | KOU-16.83 | Kourovka Notebook Problem 16.83 | b) Let $E_n$ be a free locally nilpotent $n$-Engel
group on countably many generators, and let $\pi(E_n)$ be the set
of prime divisors of the orders of elements of the periodic part
of $E_n$. It is known that $2, 3, 5\in \pi(E_4)$.
Is it true that $\pi(E_n)=\pi(E_{n+1}) $
for all sufficiently large $ n$?
[Archived so... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 16.83.
Proposed by: Yu. V. Sosnovski\u{\i
Archive of Solved Problems entry:
a) Yes, $n=6$, since the 2-generator free nilpotent
6-Engel group has an element ... | 2 | partially_solved | Yu. V. Sosnovski\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 | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (a) Yes, $n=6$, since the 2-generator free nilpotent\n6-Engel group has an element of order 7\n(W. Nickel, *J. Austral. Math. Soc., Ser. A*, **67**, no. 2 (1999),... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,275 | KOU-16.84 | Kourovka Notebook Problem 16.84 | Can the braid group ${\mathfrak{B}}_n$, $n\geq 4$,
act faithfully on a regular rooted tree by finite-state
automorphisms? Such action is known for ${\mathfrak{B}}_3$.
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: 16.84.
Proposed by: V. I. Sushchanski\u{\i | 2 | open | 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",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,276 | KOU-16.85 | Kourovka Notebook Problem 16.85 | Suppose that groups $G, H$ act faithfully on a
regular rooted tree by finite-state automorphisms. Can their free
product $G*H$ act faithfully on a regular rooted tree by finite
state automorphisms? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.85.
Proposed by: V. I. Sushchanski\u{\i
Published resolution:
Yes, it can (M. Fedorova, A. Oliynyk, *Algebra Discrete Math.*, **23**, no. 2 (2017), 230--236). | 2 | solved | 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",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it can (M. Fedorova, A. Oliynyk, *Algebra Discrete Math.*, **23**, no. 2 (2017), 230--236)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,277 | KOU-16.86 | Kourovka Notebook Problem 16.86 | Does the group of all finite-state
automorphisms of a regular rooted tree possess an irreducible
system of generators? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.86.
Proposed by: V. I. Sushchanski\u{\i
Published resolution:
Yes, it does (Ya. Lavrenyuk, *Geometriae Dedicata*, **183**, no. 1 (2016), 59--67). | 2 | solved | 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",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (Ya. Lavrenyuk, *Geometriae Dedicata*, **183**, no. 1 (2016), 59--67)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,278 | KOU-16.87 | Kourovka Notebook Problem 16.87 | Let ${\mathfrak{M}}$ be a variety of groups and
let $G_r$ be a free $r$-generator group in $\mathfrak{M}$. A subset
$S\subseteq G_r$ is called a *test set* if every
endomorphism of $G_r$ identical on
$S$ is an automorphism. The minimum of the cardinalities of test
sets is called the test rank of $G_r$. Suppose that the... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.87.
Proposed by: E. I. Timoshenko
Discussion and literature:
[Solved] a), b) No, it is not true in both cases (AI solution confirmed by E. I. Timoshenko: \ M... | 2 | solved | E. I. Timoshenko | 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] a), b) No, it is not true in both cases (AI solution confirmed by E. I. Timoshenko: \\ M. Kinyon, J. Urb... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,279 | KOU-16.88 | Kourovka Notebook Problem 16.88 | (G. M. Bergman). The *width* of a group $G$
with respect to a generating set $X$ means the supremum over all
$g\in G$ of the least length of a group word in elements of $X$
expressing $g$. A group $G$ has *finite width* if the width
of $G$ with respect to every
generating set is
finite. Does there exist a countably inf... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.88.
Proposed by: V. Tolstykh | 2 | open | V. Tolstykh | 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 |
701,280 | KOU-16.89 | Kourovka Notebook Problem 16.89 | (G. M. Bergman). Is it true that the automorphism
group of an infinitely generated free group $F$ has finite width?
The answer is affirmative if $F$ is countably generated. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.89.
Proposed by: V. Tolstykh | 2 | open | V. Tolstykh | 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 |
701,281 | KOU-16.90 | Kourovka Notebook Problem 16.90 | Is it true that the automorphism group
$\text{Aut} F$ of an infinitely generated free group $F$ is
a) the normal closure of a single element?
b) the normal closure of some involution in $\text{Aut} F$?
For a free group $F_n$ of
finite rank $n$ \ M. Bridson and K. Vogtmann
have recently shown
that $\text{Aut} F_n$ is... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.90.
Proposed by: V. Tolstykh | 2 | open | V. Tolstykh | 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 |
701,282 | KOU-16.91 | Kourovka Notebook Problem 16.91 | Let $F$ be an infinitely generated free group.
Is there an $IA$-automorphism of $F$ whose normal closure in
$\text{Aut} F$ is the group of all $IA$-automorphisms of $F?$ | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.91.
Proposed by: V. Tolstykh | 2 | open | V. Tolstykh | 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 |
701,283 | KOU-16.92 | Kourovka Notebook Problem 16.92 | Let $F$ be an infinitely generated free group.
Is $\text{Aut} F$ equal to its derived subgroup? This is true if
$F$ is countably generated (R. Bryant,
V. A. Roman'kov, *J. Algebra*, **209** (1998), 713--723). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.92.
Proposed by: V. Tolstykh | 2 | open | V. Tolstykh | 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 |
701,284 | KOU-16.93 | Kourovka Notebook Problem 16.93 | Let $F_n$ be a free group of finite rank $n \geq
2$. Is the group $\text{Inn} F_n$ of inner automorphisms of $F_n$
a first-order definable subgroup of $\text{Aut} F_n$?
It is known that the set of inner automorphisms induced by
powers of primitive elements is definable in $\text{Aut} F_n$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.93.
Proposed by: V. Tolstykh | 2 | open | V. Tolstykh | 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 |
701,285 | KOU-16.94 | Kourovka Notebook Problem 16.94 | If $G=[G,G]$, then the *commutator width*
of the group $G$ is its width relative to the set of commutators.
Let $V$ be an infinite-dimensional vector space over a division
ring. It is known that the commutator width of $GL(V)$ is finite.
Is it true that the commutator width of $GL(V)$ is one? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.94.
Proposed by: V. Tolstykh | 2 | open | V. Tolstykh | 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 |
701,286 | KOU-16.95 | Kourovka Notebook Problem 16.95 | *Conjecture:* If $F$ is a field and
$A$ is in $GL(n,F)$, then there is a permutation matrix $P$ such
that $AP$ is cyclic, that is, the minimal polynomial of $AP$ is
also its characteristic polynomial. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.95.
Proposed by: J. G. Thompson | 2 | open | J. G. Thompson | 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 |
701,287 | KOU-16.96 | Kourovka Notebook Problem 16.96 | Let $G$ be a locally finite $n$-Engel
$p $-group where $p$ a prime greater than $n$. Is $G$ then a
Fitting group?
(Examples of N. Gupta and F. Levin show that the
condition $p>n$ is necessary in general.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.96.
Proposed by: G. Traustason | 2 | open | G. Traustason | 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 |
701,288 | KOU-16.97 | Kourovka Notebook Problem 16.97 | Let $G$ be a torsion-free group with all
subgroups subnormal of defect at most $n$. Must $G$ then be
nilpotent of class at most $n$? (This is known to be true for $n <
5$). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.97.
Proposed by: G. Traustason | 2 | open | G. Traustason | 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 |
701,289 | KOU-16.98 | Kourovka Notebook Problem 16.98 | Suppose that $G$ is a solvable finite group
and $A$ is a group of automorphisms of $G$ of relatively prime
order. Is there a bound for the Fitting height $h(G)$ of $G$ in
terms of $A$ and $h(C_G(A))$, or even in terms of the length
$l(A)$ of the longest chain of nested subgroups of $A$ and
$h(C_G(A))$?
When $A$ is sol... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.98.
Proposed by: A. Turull | 2 | open | A. Turull | 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 |
701,290 | KOU-16.99 | Kourovka Notebook Problem 16.99 | Suppose that $G$ is a finite solvable group, $A
\leq \mathrm{Aut} G$, \ $C_G(A)=1$, the orders of $G$ and $A$ are
coprime, and let $l(A)$ be the length of the longest chain of
nested subgroups of $A$. Is the Fitting height of $G$ bounded
above by $l(A)$?
For $A$ solvable the question coincides
with 5.30. It is proved ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.99.
Proposed by: A. Turull | 2 | open | A. Turull | 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 |
701,291 | KOU-16.100 | Kourovka Notebook Problem 16.100 | Is there an (infinite) 2-generator
simple group $G$ such that $\mathrm{Aut} F_2$ is transitive on the
set of normal subgroups $N$ of the free group $F_2$ such that
$F_2/N\cong G$? Cf. 6.45. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.100.
Proposed by: J. Wiegold | 2 | open | J. Wiegold | 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 |
701,292 | KOU-16.101 | Kourovka Notebook Problem 16.101 | Do there exist uncountably many infinite 2-groups that are
quotients of the group $\left<x, y \mid x^2 = y^4 = (xy)^8 =
1\right>$? There certainly exists one, namely the subgroup of
finite index in Grigorchuk's first group generated by $b$ and
$ad$; see (R. I. Grigorchuk, *Functional Anal. Appl.*,
**14** (1980), 41--43... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.101.
Proposed by: J. Wiegold
Published resolution:
Yes, there do
(R. I. Grigorchuk, *Algebra Discrete Math.*, **2009**, no. 4 (2009), 78--96 (subm. 25 November 2009); \
A. Mina... | 2 | solved | J. Wiegold | 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, there do\n(R. I. Grigorchuk, *Algebra Discrete Math.*, **2009**, no. 4 (2009), 78--96 (subm. 25 November 2009); \\\nA. Minasyan, A. Yu. Olshanskii, D. Sonki... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,293 | KOU-16.102 | Kourovka Notebook Problem 16.102 | We say that a group $G$ is *rational*
if any two elements $x,y\in
G$ satisfying $\left< x \right>=\left< y \right>$ are conjugate.
Is it true that for any $d$ there exist only finitely many finite
rational $d$-generated $2$-groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.102.
Proposed by: A. Jaikin-Zapirain | 2 | open | A. Jaikin-Zapirain | 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 |
701,294 | KOU-16.103 | Kourovka Notebook Problem 16.103 | Is there a rank analogue of the
Leedham-Green--McKay--Shepherd
theorem on $p $-groups of maximal class? More precisely, suppose
that $P$ is a 2-generator finite $p $-group whose lower central
quotients $\gamma _i(P)/\gamma _{i+1}(P)$ are cyclic for all
$i\geq 2$. Is it true that $P$ contains a normal subgroup $N$ of
ni... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.103.
Proposed by: E. I. Khukhro
Published resolution:
No, moreover, there are no functions $d(p)$ and
$r(p)$ such that a group with these properties would necessarily
have a no... | 2 | solved | E. I. Khukhro | 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, moreover, there are no functions $d(p)$ and\n$r(p)$ such that a group with these properties would necessarily\nhave a normal subgroup of derived length ${}\\... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,295 | KOU-16.104 | Kourovka Notebook Problem 16.104 | If \( G \) is a finite group, then every element \( a \) of the
rational group algebra \( {\mathbb{Q}}[G] \) has a unique Jordan
decomposition \( a =a _{s}+a _{n} \), where \( a _{n}\in {\mathbb{Q}}[G] \) is nilpotent, \( a _{s}\in {\mathbb{Q}}[G] \) is
semisimple over \( \mathbb{Q} \), and \( a _{s}a _{n}=a _{n}a _{s}... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.104.
Proposed by: A. W. Hales, I. B. S. Passi
Published resolution:
Yes, it is true (A. W. Hales, I. B. S. Passi, L. E. Wilson, *J. Algebra*, **316**, no. 1 (2007), 109--132; *... | 2 | solved | A. W. Hales, I. B. S. Passi | 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 (A. W. Hales, I. B. S. Passi, L. E. Wilson, *J. Algebra*, **316**, no. 1 (2007), 109--132; **371** (2012), 665--666)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,296 | KOU-16.105 | Kourovka Notebook Problem 16.105 | Is it true that a locally graded group which
is a product of two almost polycyclic subgroups (equivalently, of
two almost soluble subgroups with the maximal condition) is almost
polycyclic?
By (N. S. Chernikov, *Ukrain. Math. J.*, **32** (1980) 476--479) a locally graded
group which is a product of two Chernikov subgr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.105.
Proposed by: N. S. Chernikov | 2 | open | N. S. Chernikov | 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 |
701,297 | KOU-16.106 | Kourovka Notebook Problem 16.106 | Let $\pi _e(G)$ denote the set of orders of
elements of a group $G$, and $h(\Gamma )$ the number of
non-isomorphic finite groups $G$ with $\pi _e(G)=\Gamma$. Do there
exist two finite groups $G_1, G_2$
such that $\pi _e(G_1)=\pi _e(G_2)$, \ $h(\pi _e(G_1)) < \infty$,
and neither of the two groups $G_1, G_2$ is isomorph... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.106.
Proposed by: W. J. Shi
Published resolution:
Yes, there do: for example, $G_1=L_{15}(2^{60}).3$ and
$G_2=L_{15}(2^{60}).5$ (M. A. Grechkoseeva, *Algebra and Logic*, **47**... | 2 | solved | W. J. Shi | 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, there do: for example, $G_1=L_{15}(2^{60}).3$ and\n$G_2=L_{15}(2^{60}).5$ (M. A. Grechkoseeva, *Algebra and Logic*, **47**, no. 4\n(2008), 229--241)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,298 | KOU-16.107 | Kourovka Notebook Problem 16.107 | Is
it true that almost every alternating group
${A}_n$ is uniquely determined in the class of finite groups by
its set of element orders, i. e. that $h(\pi _e({ A}_n))=1$ for
all large enough $n$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.107.
Proposed by: W. J. Shi
Published resolution:
Yes, it is (I. B. Gorshkov, *Algebra and Logic*, **52**, no. 1 (2013), 41--45). | 2 | solved | W. J. Shi | 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 (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 |
701,299 | KOU-16.108 | Kourovka Notebook Problem 16.108 | Do braid groups $B_n,\; n>4$, have
non-elementary hyperbolic factor groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.108.
Proposed by: V. E. Shpilrain | 2 | open | 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 |
701,300 | KOU-16.109 | Kourovka Notebook Problem 16.109 | Is there a polynomial time algorithm for
solving the word problem in the group $\mathrm{Aut} F_n$ (with
respect to some particular finite presentation), where $F_n$ is
the free group of rank $n \geq 2$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 16th issue (2006). Original problem number: 16.109.
Proposed by: V. E. Shpilrain
Published resolution:
Yes, there is
(S. Schleimer, *Comment. Math. Helv.*, **83** (2008),
741--765). | 2 | solved | 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": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there is\n(S. Schleimer, *Comment. Math. Helv.*, **83** (2008),\n741--765)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,301 | KOU-16.110 | Kourovka Notebook Problem 16.110 | (I. Kapovich, P. Schupp). Is there an
algorithm which, when given two elements $u, v$ of a free group
$F_n$, decides whether or not the cyclic length of $\phi(u)$
equals the cyclic length of $\phi(v)$ for every automorphism
$\phi$ of $F_n$?
*Comment of 2009:* The answer was shown
to be positive for $n=2$ in (D. Lee, *... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.110.
Proposed by: V. E. Shpilrain | 2 | open | 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 |
701,302 | KOU-16.111 | Kourovka Notebook Problem 16.111 | Must an infinite simple periodic group with a
dihedral Sylow 2-subgroup be isomorphic to $L_2(P)$ for a
locally finite field $P$ of odd characteristic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 16.111.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 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 |
701,303 | KOU-17.1 | Kourovka Notebook Problem 17.1 | (I. M. Isaacs). Does there exist a finite group
partitioned by subgroups of equal order not all of which are
abelian? (Cf. 15.26.) It is known that such a group must be of
prime exponent (I. M. Isaacs, *Pacific J. Math.*, **49**
(1973), 109--116). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.1.
Proposed by: A. Abdollahi | 2 | open | A. Abdollahi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,304 | KOU-17.2 | Kourovka Notebook Problem 17.2 | (P. Schmid). Does there exist a finite non-abelian $p$-group $G$ such that $H^1({G}/{\Phi(G)}, Z(\Phi(G) ) )=0$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.2.
Proposed by: A. Abdollahi
Published resolution:
Yes, it does (A. Abdollahi, *J. Algebra*, **342** (2011), 154--160). | 2 | solved | A. Abdollahi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does (A. Abdollahi, *J. Algebra*, **342** (2011), 154--160)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,305 | KOU-17.3 | Kourovka Notebook Problem 17.3 | Let $G$ be a group in which every $4$-element
subset contains two elements generating a nilpotent subgroup. Is
it true that every $2$-generated subgroup of $G$ is nilpotent? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.3.
Proposed by: A. Abdollahi
Published resolution:
No, not always (A. I. Sozutov,
*Siberian Math. J.*, **60**, no. 6 (2019), 1099--1100). | 2 | solved | A. Abdollahi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always (A. I. Sozutov,\n*Siberian Math. J.*, **60**, no. 6 (2019), 1099--1100)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,306 | KOU-17.4 | Kourovka Notebook Problem 17.4 | Let $x$ be a right $4$-Engel element of a group
$G$.
a) Is it true that the normal closure $\langle
x \rangle ^G$ of $x$ in $G$ is nilpotent if $G$ is locally
nilpotent?
b) If the answer to a) is affirmative, is there
a bound on the nilpotency class of $\langle x \rangle ^G$?
c) Is it true that $\langle x \rangle ^G... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.4.
Proposed by: A. Abdollahi | 2 | open | A. Abdollahi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,307 | KOU-17.5 | Kourovka Notebook Problem 17.5 | Is the nilpotency class of a nilpotent group
generated by $d$ left $3$-Engel elements bounded in terms of $d$?
A group generated by two left $3$-Engel elements is nilpotent of
class at most $4$ (*J. Pure Appl. Algebra*, **188** (2004),
1--6.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.5.
Proposed by: A. Abdollahi | 2 | open | A. Abdollahi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,308 | KOU-17.6 | Kourovka Notebook Problem 17.6 | a) Is there a function $f:\mathbb{N}\times\mathbb{N}\rightarrow
\mathbb{N}$ such that every nilpotent group generated by $d$ left
$n$-Engel elements is nilpotent of class at most $f(n,d)$?
b) Is there a function
$g:\mathbb{N}\times\mathbb{N}\rightarrow \mathbb{N}$ such that
every nilpotent group generated by $d$ right... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.6.
Proposed by: A. Abdollahi | 2 | open | A. Abdollahi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,309 | KOU-17.7 | Kourovka Notebook Problem 17.7 | a) Is there a group in which the set of right Engel elements does
not form a subgroup?
b) Is there a group in which the set of
bounded right Engel elements does not form a subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.7.
Proposed by: A. Abdollahi | 2 | open | A. Abdollahi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,310 | KOU-17.8 | Kourovka Notebook Problem 17.8 | a) Is there a group containing a right Engel element which is not
a left Engel element?
b) Is there a group containing a bounded right
Engel element which is not a left Engel element? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.8.
Proposed by: A. Abdollahi | 2 | open | A. Abdollahi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,311 | KOU-17.9 | Kourovka Notebook Problem 17.9 | Is there a group containing a left
Engel element whose inverse is not a left Engel element? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.9.
Proposed by: A. Abdollahi | 2 | open | A. Abdollahi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,312 | KOU-17.10 | Kourovka Notebook Problem 17.10 | Is there a group containing a right
$4$-Engel element which does not belong to the Hirsch--Plotkin
radical? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.10.
Proposed by: A. Abdollahi | 2 | open | A. Abdollahi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,313 | KOU-17.11 | Kourovka Notebook Problem 17.11 | Is there a group containing a left
$3$-Engel element which does not belong to the Hirsch--Plotkin
radical? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.11.
Proposed by: A. Abdollahi | 2 | open | A. Abdollahi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,314 | KOU-17.12 | Kourovka Notebook Problem 17.12 | Are there functions
$e,c:\mathbb{N}\rightarrow \mathbb{N}$ such that if in a nilpotent
group $G$ a normal subgroup $H$ consists of right $n$-Engel
elements of $G$, then $H^{e(n)}\leq \zeta_{c(n)}(G)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.12.
Proposed by: A. Abdollahi
Published resolution:
Yes, there are
(P. G. Crosby, G. Traustason, *J. Algebra*, **324** (2010), 875--883). | 2 | solved | A. Abdollahi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 are\n(P. G. Crosby, G. Traustason, *J. Algebra*, **324** (2010), 875--883)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,315 | KOU-17.13 | Kourovka Notebook Problem 17.13 | Let $G$ be a totally imprimitive $p$-group of
finitary permutations on an infinite set. Suppose that the support
of any cycle in the cyclic decomposition of every element of $G$
is a block for $G$. Does $G$ necessarily contain a minimal
non-$FC$-subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.13.
Proposed by: A. O. Asar | 2 | open | A. O. Asar | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,316 | KOU-17.14 | Kourovka Notebook Problem 17.14 | Can the braid group $B_n$ for $n \geq 4$ be embedded into the
automorphism group $Aut (F_{n-2})$ of a free group $F_{n-2}$ of
rank $n-2$?
Artin's theorem implies that $B_n$ can be
embedded into $Aut (F_n)$, and an embedding of $B_n$, $n \geq 3$,
into $Aut (F_{n-1})$ was constructed in (B. Perron,
J. P. Vannier, *Math.... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.14.
Proposed by: V. G. Bardakov
Published resolution:
No, it cannot for $n=4$ (V. G. Bardakov, P. Bellingeri, in *Knot theory and its applications. ICTS program knot theory and... | 2 | solved | V. G. Bardakov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 cannot for $n=4$ (V. G. Bardakov, P. Bellingeri, in *Knot theory and its applications. ICTS program knot theory and its applications (KTH-2013), IISER Moh... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,317 | KOU-17.15 | Kourovka Notebook Problem 17.15 | Construct an algorithm which, for a given polynomial $f \in
{\mathbb{Z}}[x_1, x_2, \ldots, x_n]$, finds (explicitly, in terms of
generators) the maximal subgroup $G_f$ of the group $Aut ({\mathbb{Z}}[x_1, x_2, \ldots, x_n])$ that leaves $f$ fixed. This question
is related to description of the solution set of the Dioph... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.15.
Proposed by: V. G. Bardakov | 2 | open | V. G. Bardakov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,318 | KOU-17.16 | Kourovka Notebook Problem 17.16 | Let $A$ be an Artin group of finite type, that is, the
corresponding Coxeter group is finite. It is known that the group
$A$ is linear (S. Bigelow, *J. Amer. Math. Soc.*, **14**
(2001), 471--486; \ D. Krammer, *Ann. Math.*, **155**
(2002), 131--156; \ F. Digne, *J. Algebra*, **268**
(2003), 39--57; \ A. M. Cohen, D. B.... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.16.
Proposed by: V. G. Bardakov | 2 | open | V. G. Bardakov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,319 | KOU-17.17 | Kourovka Notebook Problem 17.17 | If a finitely generated group $G$ has
$n<\infty$ maximal subgroups, must $G$ be finite? In particular,
what if $n=3$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.17.
Proposed by: G. M. Bergman
Published resolution:
No, it need not. An example of an infinite group with 3 maximal subgroups is given by a 2-generated 2-LERF group in \S 7 of... | 2 | solved | G. M. Bergman | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it need not. An example of an infinite group with 3 maximal subgroups is given by a 2-generated 2-LERF group in \\S 7 of (M. Ershov, A. Jaikin-Zapirain, *J. ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,320 | KOU-17.18 | Kourovka Notebook Problem 17.18 | Let $\mathbf{A}$ be the class of compact groups
$A$ with the property that whenever two compact groups $B$ and $C$
contain $A$, they can be embedded in a common compact group $D$ by
embeddings agreeing on $A$. I showed (*Manuscr. Math.*, **58** (1987) 253--281) that all members of $\mathbf{A}$ are (not
necessarily conn... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.18.
Proposed by: G. M. Bergman | 2 | open | G. M. Bergman | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,321 | KOU-17.19 | Kourovka Notebook Problem 17.19 | If $F$ is a free group of finite rank, $R$ a
retract of $F$, and $H$ a subgroup of $F$ of finite rank, must
$H\cap R$ be a retract of $H$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.19.
Proposed by: G. M. Bergman
Published resolution:
No, it need not (I. Snopce, S. Tanushevski, P. Zalesskii, *Int. Math. Res. Notices*, **2022**, no. 11 (2022), 8280--8294). | 2 | solved | G. M. Bergman | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it need not (I. Snopce, S. Tanushevski, P. Zalesskii, *Int. Math. Res. Notices*, **2022**, no. 11 (2022), 8280--8294)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,322 | KOU-17.20 | Kourovka Notebook Problem 17.20 | If $M$ is a real manifold with nonempty
boundary, and $G$ the group of self-homeomorphisms of $M$
which fix the boundary pointwise, is $G$ right-orderable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.20.
Proposed by: G. M. Bergman
Published resolution:
No, not always (J. Hyde,
*Ann. of Math. (2)*, **190**, no. 2 (2019), 657--661). | 2 | solved | G. M. Bergman | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always (J. Hyde,\n*Ann. of Math. (2)*, **190**, no. 2 (2019), 657--661)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,323 | KOU-17.21 | Kourovka Notebook Problem 17.21 | a) If $A, B, C$ are torsion-free abelian
groups with $A \cong A\oplus B\oplus C$, must $A\cong A\oplus B$?
b) What if, furthermore, $B\cong C$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.21.
Proposed by: G. M. Bergman
Published resolution:
No, not necessarily, even in case b). See
(A. L. S. Corner, in *Proc. Colloq. Abelian
Groups (Tihany, 1963)*, Budapest, 196... | 2 | solved | G. M. Bergman | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not necessarily, even in case b). See\n(A. L. S. Corner, in *Proc. Colloq. Abelian\nGroups (Tihany, 1963)*, Budapest, 1964, 43--48); I also have an example w... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,324 | KOU-17.22 | Kourovka Notebook Problem 17.22 | Suppose $A$ is a group which belongs to a
variety $\mathbf V$ of groups, and which is embeddable in the full
symmetric group $S$ on an infinite set. Must the coproduct in
$\mathbf V$ of two copies of $A$ also be embeddable in $S$?
(N. G. de Bruijn proved that this is true if $\mathbf V$ is the
variety of all groups.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.22.
Proposed by: G. M. Bergman
Discussion and literature:
[Unverified AI claim] No, it need not (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026,... | 2 | open | G. M. Bergman | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, it need not (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026,\n\n... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,325 | KOU-17.23 | Kourovka Notebook Problem 17.23 | Suppose the full symmetric group $S$ on a
countably infinite set is generated by the union of two subgroups
$G$ and $H$. Must $S$ be finitely generated over one of these
subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.23.
Proposed by: G. M. Bergman | 2 | open | G. M. Bergman | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,326 | KOU-17.24 | Kourovka Notebook Problem 17.24 | (A. Blass, J. Irwin, G. Schlitt). Does
$\mathbb{Z}^\omega$ have a subgroup whose dual is free of
uncountable rank? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.24.
Proposed by: G. M. Bergman
Published resolution:
Yes, it does
(G. M. Bergman, *Portugaliae Math.*, **69** (2012) 69--84). | 2 | solved | G. M. Bergman | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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\n(G. M. Bergman, *Portugaliae Math.*, **69** (2012) 69--84)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,327 | KOU-17.25 | Kourovka Notebook Problem 17.25 | (S. P. Farbman). Let $X$ be the set of
complex numbers $\alpha$ such that the group generated by the two
$2\times 2$ matrices $I+\alpha e_{12}$ and $I+ e_{21}$ is not free
on those generators.
a) Does $X$ contain all rational numbers in the
interval $(-4,4)$?
b) Does $X$ contain any rational number in the
interval $[... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.25.
Proposed by: G. M. Bergman | 2 | open | G. M. Bergman | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,328 | KOU-17.26 | Kourovka Notebook Problem 17.26 | Are the classes of right-orderable and right-ordered groups closed
under taking solutions of equations
$w(a_1,\dots,a_k,x_1,\dots,x_n)=1$? (Here the closures are under
group embeddings and order-preserving embeddings, respectively.)
This is true for equations with a single constant, when $k=1$
(*J. Group Theory*, **11*... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.26.
Proposed by: V. V. Bludov, A. M. W. Glass | 2 | open | V. V. Bludov, A. M. W. Glass | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,329 | KOU-17.27 | Kourovka Notebook Problem 17.27 | Can the free product of two ordered groups with order-isomorphic
amalgamated subgroups be lattice orderable but not orderable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.27.
Proposed by: V. V. Bludov, A. M. W. Glass
Discussion and literature:
[Unverified AI claim] Yes, it can (unverified AI solution: Achyuth Jayadevan, *Prepr... | 2 | open | V. V. Bludov, A. M. W. Glass | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: Achyuth Jayadevan, *Preprint*, 2026,\n\nhttp... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,330 | KOU-17.28 | Kourovka Notebook Problem 17.28 | Is there a soluble right-orderable group with insoluble word problem? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.28.
Proposed by: V. V. Bludov, A. M. W. Glass
Published resolution:
Yes, there is (A. Darbinyan, *J. Symbolic Logic*, **85**, no. 4 (2020), 1588--1598). | 2 | solved | V. V. Bludov, A. M. W. Glass | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there is (A. Darbinyan, *J. Symbolic Logic*, **85**, no. 4 (2020), 1588--1598)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,331 | KOU-17.29 | Kourovka Notebook Problem 17.29 | Construct a finitely presented orderable group
with insoluble word problem. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.29.
Proposed by: V. V. Bludov, A. M. W. Glass
Published resolution:
Such a group is constructed (V. V. Bludov, A. M. W. Glass, *Bull. London Math. Soc.*, **44** (2012), 85--98)... | 2 | solved | V. V. Bludov, A. M. W. Glass | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: Such a group is constructed (V. V. Bludov, A. M. W. Glass, *Bull. London Math. Soc.*, **44** (2012), 85--98)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,332 | KOU-17.30 | Kourovka Notebook Problem 17.30 | Is there a constant $l$ such that every finitely presented soluble group
has a subgroup of finite index of nilpotent length at most $l$?
Cf. 16.35 in Archive. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.30.
Proposed by: V. V. Bludov, J. R. J. Groves | 2 | open | V. V. Bludov, J. R. J. Groves | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,333 | KOU-17.31 | Kourovka Notebook Problem 17.31 | Can a soluble right-orderable group have finite
quotient by the derived subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.31.
Proposed by: V. V. Bludov, A. H. Rhemtulla | 2 | open | V. V. Bludov, A. H. Rhemtulla | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,334 | KOU-17.32 | Kourovka Notebook Problem 17.32 | Is the following analogue of the Cayley--Hamilton theorem
true for the free group $F_n$ of rank $n$:
If $w\in F_n$ and $\varphi\in \mathrm{Aut }F_n$ are such
that $\langle w,\varphi (w),\dots,\varphi^{n}(w)\rangle= F_n$, then $\langle w,\varphi (w),\dots,\varphi^{n-1}(w)\rangle=F_n$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.32.
Proposed by: O. V. Bogopolski
Discussion and literature:
[Solved] No, it is not true (V. Ionin, A. Semidetnov, *Preprint*, 2026, https://arxiv.org/pdf/26... | 2 | solved | O. V. Bogopolski | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it is not true (V. Ionin, A. Semidetnov, *Preprint*, 2026, https://arxiv.org/pdf/2608.29219; \\ O. K... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,335 | KOU-17.33 | Kourovka Notebook Problem 17.33 | Can the quasivariety generated by the group $\langle a,b\mid
a^{-1}ba=b^{-1}\rangle$ be defined by a set of quasi-identities in
a finite set of variables? The answer is known for all other
groups with one defining relation. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.33.
Proposed by: A. I. Budkin
Discussion and literature:
[Unverified AI claim] No, it cannot (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,
... | 2 | open | A. I. Budkin | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, it cannot (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,\n\n... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,336 | KOU-17.34 | Kourovka Notebook Problem 17.34 | Let $N_c$ be the quasivariety of nilpotent torsion-free groups of
class at most $c$. Is it true that the dominion in $N_c$ (see the
definition in 16.20) of a divisible subgroup in every group
in $N_c$ is equal to this subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.34.
Proposed by: A. I. Budkin
Discussion and literature:
[Unverified AI claim] Yes, it is true (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026,
... | 2 | open | A. I. Budkin | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: Achyuth Jayadevan, *Preprint*, 2026,\n\n... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,337 | KOU-17.35 | Kourovka Notebook Problem 17.35 | Suppose we have a finite two-colourable
triangulation of the sphere, with triangles each coloured either
black or white so that no pair of triangles with the same colour
share an edge. On each vertex we write a generator, and we assume
the generators commute. We use these generators to generate an
abelian group $G_W$ w... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.35.
Proposed by: I. M. Wanless, N. J. Cavenagh
Published resolution:
Conjecture is proved
(S. R. Blackburn, T. A. McCourt,
*Combinatorica*, **34**, no. 5 (2014), 527--546). | 2 | solved | I. M. Wanless, N. J. Cavenagh | 2,010 | 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": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Conjecture is proved\n(S. R. Blackburn, T. A. McCourt,\n*Combinatorica*, **34**, no. 5 (2014), 527--546)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,338 | KOU-17.36 | Kourovka Notebook Problem 17.36 | Two groups are called *isospectral* if
they have the same set of element orders. Find all finite
non-abelian simple groups $G$ for which there is a finite group
$H$ isospectral to $G$ and containing a proper normal subgroup
isomorphic to $G$. For every simple group $G$ determine all groups
$H$ satisfying this condition... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.36.
Proposed by: A. V. Vasil'ev
Published resolution:
All such groups are determined: for exceptional groups of Lie type in (M. A. Zvezdina, *Algebra Logic*,
**55**, no. 5 (201... | 2 | solved | A. V. Vasil'ev | 2,010 | 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: All such groups are determined: for exceptional groups of Lie type in (M. A. Zvezdina, *Algebra Logic*,\n**55**, no. 5 (2016), 354--366);\nfor the other simple g... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,339 | KOU-17.37 | Kourovka Notebook Problem 17.37 | Is there an integer $n$ such that for all $m
> n$ the alternating group $A_{m}$ has no
non-trivial $A_{m}$-permutable subgroups?
(See the definition in 17.112.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.37.
Proposed by: A. F. Vasil'ev, A. N. Skiba
Discussion and literature:
[Solved] Yes, there is (N. Yang, A. Galt, *Bull. Malays. Math. Sci. Soc. (2)*, **46**... | 2 | solved | A. F. Vasil'ev, A. N. Skiba | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
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"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": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, there is (N. Yang, A. Galt, *Bull. Malays. Math. Sci. Soc. (2)*, **46**, no. 5 (2023), Paper no. 17... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,340 | KOU-17.38 | Kourovka Notebook Problem 17.38 | A formation $\mathfrak{F}$ is called {\it radical in a class
$\mathfrak{H}$} if $\mathfrak{F} \subseteq \mathfrak{H}$ and in
every $\mathfrak{H}$-group the product of any two normal $\mathfrak{F}$-subgroups belongs to $\mathfrak{F}$. Let $\mathfrak{M}$ be the
class of all saturated hereditary formations of finite group... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.38.
Proposed by: A. F. Vasil'ev, L. A. Shemetkov | 2 | open | A. F. Vasil'ev, L. A. Shemetkov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,341 | KOU-17.39 | Kourovka Notebook Problem 17.39 | Is there a positive integer $n$ such that the hypercenter of any
finite soluble group coincides with the intersection of $n$ system
normalizers of that group? What is the least number with this
property?
*Comment of 2017*:
This is proved for groups with trivial Frattini subgroup (A. Ballester-Bolinches, J. Cossey, S. ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.39.
Proposed by: A. F. Vasil'ev, L. A. Shemetkov
Discussion and literature:
[Unverified AI claim] No, there is no such number (unverified AI solution: M. Kin... | 2 | open | A. F. Vasil'ev, L. A. Shemetkov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, there is no such number (unverified AI solution: M. Kinyon, J. Urban, *Prepri... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,342 | KOU-17.40 | Kourovka Notebook Problem 17.40 | Let $N$ be a nilpotent subgroup of a finite group $G$. Do there
always exist elements $x,y\in G$ such that $N\cap N^x\cap N^y\leq
F(G)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.40.
Proposed by: E. P. Vdovin
Published resolution:
Yes, such elements always exist, mod CFSG (V. I. Zenkov, *Siberian Math. J.*, **62**, no. 4 (2021), 621--637). | 2 | solved | E. P. Vdovin | 2,010 | 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, such elements always exist, mod CFSG (V. I. Zenkov, *Siberian Math. J.*, **62**, no. 4 (2021), 621--637)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,343 | KOU-17.41 | Kourovka Notebook Problem 17.41 | Let $S$ be a solvable subgroup of a finite group $G$ that has no
nontrivial solvable normal subgroups.
(a) (L. Babai, A. J. Goodman, L. Pyber). Do
there always exist seven conjugates of $S$ whose intersection is
trivial?
(b)
Do there always exist five conjugates of $S$
whose intersection is trivial?
*Editors' commen... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.41.
Proposed by: E. P. Vdovin | 2 | open | E. P. Vdovin | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,344 | KOU-17.42 | Kourovka Notebook Problem 17.42 | Let $\overline{G}$ be a simple algebraic group of adjoint type
over the algebraic closure $\overline{F}_p$ of a finite field
$F_p$ of prime order $p$, and $\sigma$ a Frobenius map (that is, a
surjective homomorphism such that
$\overline{G}_\sigma=C_{\overline{G}}(\sigma)$ is finite). Then
$G=O^{p'}(\overline{G}_\sigma)... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.42.
Proposed by: E. P. Vdovin
Discussion and literature:
[Solved] A definitive answer for a stronger question on the size of a base has been obtained in (T. ... | 2 | solved | E. P. Vdovin | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 definitive answer for a stronger question on the size of a base has been obtained in (T. C. Burness, A... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,345 | KOU-17.43 | Kourovka Notebook Problem 17.43 | Let $\pi$ be a set of primes. Find all finite simple
$D_\pi$-groups (see Archive, 3.62) in which
(a) every subgroup is a $D_\pi$-group (H. Wielandt);
(b) every subgroup possessing a Hall $\pi$-subgroup is a $D_\pi$-group.
All finite simple $D_\pi$-groups are known (D. O. Revin, *Algebra Logic*, **47**, no. 3 (2008),... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.43.
Proposed by: E. P. Vdovin, D. O. Revin | 2 | open | E. P. Vdovin, D. O. Revin | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,346 | KOU-17.44 | Kourovka Notebook Problem 17.44 | Let $\pi$ be a set of primes. A finite group is called a *$C_\pi$-group* if it possesses exactly one class of conjugate Hall
$\pi$-subgroups. A finite group is called a {\it $D_{\pi} $-group} if any two of its maximal $\pi
$-subgroups are conjugate.
a) In a $C_\pi$-group, is an overgroup of a Hall $\pi$-subgroup alway... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.44.
Proposed by: E. P. Vdovin, D. O. Revin
Published resolution:
a) Yes, it is
(E. P. Vdovin, D. O. Revin, *Siberian Math. J.*, **54**, no. 1 (2013), 22--28).
b) Yes, it is (N... | 2 | solved | E. P. Vdovin, D. O. Revin | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) Yes, it is\n(E. P. Vdovin, D. O. Revin, *Siberian Math. J.*, **54**, no. 1 (2013), 22--28).\n\nb) Yes, it is (N. Ch. Manzaeva, https://arxiv.org/pdf/1504.0313... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,347 | KOU-17.45 | Kourovka Notebook Problem 17.45 | c) A subgroup $H$ of a group $G$ is called *pronormal* if $H$
and $H^g$ are conjugate in $\langle H,H^g\rangle$ for every $g\in
G$. We say that $H$ is *strongly pronormal* if $L^g$ is
conjugate to a subgroup of $H$ in $\langle H,L^g\rangle$ for every
$L\leq H$ and $g\in G$.
In a finite group, is a Hall subgroup with a... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 17.45.
Proposed by: E. P. Vdovin, D. O. Revin
Archive of Solved Problems entry:
a) Yes, they are (E. P. Vdovin, D. O. Revin, *Siberian Math. J.*,
**53**, no.... | 2 | partially_solved | E. P. Vdovin, D. O. Revin | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, they are (E. P. Vdovin, D. O. Revin, *Siberian Math. J.*,\n**53**, no. 3 (2012), 419--430).\n\nb) No, not always (M. N. Nesterov, *Siberian Math. J.*, **... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,348 | KOU-17.46 | Kourovka Notebook Problem 17.46 | Let $G$ be a finite $p$-group in which every
2-generator subgroup has cyclic derived subgroup. Is the derived
length of $G$ bounded?
If $p\ne 2$, then $G^{(2)}=1$ (J. Alperin),
but for $p=2$ there are examples with $G^{(2)}$ cyclic and
elementary abelian of arbitrary order. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.46.
Proposed by: B. M. Veretennikov
Published resolution:
Yes, it is at most 4 (B. Wilkens, *J. Group Theory*, **17**, no. 1 (2014), 151--174). | 2 | solved | B. M. Veretennikov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 at most 4 (B. Wilkens, *J. Group Theory*, **17**, no. 1 (2014), 151--174)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,349 | KOU-17.47 | Kourovka Notebook Problem 17.47 | Let $G$ be a nilpotent group in which every
coset $x[G,G]$ for $x\not\in [G,G]$ coincides with the conjugacy
class $x^G$. Is there a bound for the nilpotency class of $G$?
If
$G$ is finite, then its class is at most 3 (R. Dark,
C. Scoppola, *J. Algebra*, **181**, no. 3 (1996), 787--802). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.47.
Proposed by: S. H. Ghate, A. S. Muktibodh
Discussion and literature:
[Solved] No, there is no such bound (V. Ionin, A. Semidetnov, *Preprint*, 2026, http... | 2 | solved | S. H. Ghate, A. S. Muktibodh | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 is no such bound (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 |
701,350 | KOU-17.48 | Kourovka Notebook Problem 17.48 | Does the free product of two groups with stable
first-order theory also have stable first-order theory? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.48.
Proposed by: E. Jaligot | 2 | open | E. Jaligot | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,351 | KOU-17.49 | Kourovka Notebook Problem 17.49 | Is it consistent with ZFC that every
non-discrete topological group contains a nonempty nowhere dense
subset without isolated points? Under Martin's Axiom, there are
countable non-discrete topological groups in which each nonempty
nowhere dense subset has an isolated point. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.49.
Proposed by: E. G. Zelenyuk | 2 | open | E. G. Zelenyuk | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,352 | KOU-17.50 | Kourovka Notebook Problem 17.50 | Is it true that for every finite group $G$,
there is a finite group $F$ and a surjective homomorphism $f: F
\to G$ such that for each nontrivial subgroup $H$ of $F$, the
restriction $f|_H$ is not injective?
It is known that for every finite group $G$
there is a finite group $F$ and a surjective homomorphism $f: F
\to ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.50.
Proposed by: E. G. Zelenyuk
Published resolution:
Yes, it is
(V. P. Burichenko, *Math. Notes*, **92**, no. 3 (2012), 327--332). | 2 | solved | E. G. Zelenyuk | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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(V. P. Burichenko, *Math. Notes*, **92**, no. 3 (2012), 327--332)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,353 | KOU-17.51 | Kourovka Notebook Problem 17.51 | Is it true that every non-discrete topological
group containing no countable open subgroup of exponent 2 can be
partitioned into two (infinitely many) dense subsets? If the group
is Abelian or countable, the answer is yes. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.51.
Proposed by: E. G. Zelenyuk, I. V. Protasov | 2 | open | E. G. Zelenyuk, I. V. Protasov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
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