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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
701,454 | KOU-18.25 | Kourovka Notebook Problem 18.25 | Let $\mathrm{form}(G)$ be the formation generated by a finite group $G$. Suppose that $G$ has a unique composition series
$ 1\vartriangleleft G_1\vartriangleleft G_2\vartriangleleft G$ and the consecutive factors of this series are $Z_p$, $X$, $Z_q$,
where $p$ and $q$ are primes and $X$ is a non-abelian simple group. I... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.25.
Proposed by: V. P. Burichenko | 2 | open | V. P. Burichenko | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,455 | KOU-18.26 | Kourovka Notebook Problem 18.26 | Suppose that a finite group $G$ has a normal series $ 1< G_1<G_2<G $
such that the groups $G_1$ and $G_2/G_1$
are elementary abelian $p$-groups, $G/G_2\cong A_5\times A_5$
(where $A_5$ is the alternating group of degree 5), $G_1$ and
$G_2/G_1$ are minimal normal subgroups of $G$ and $G/G_1$, respectively.
Is it true th... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.26.
Proposed by: V. P. Burichenko | 2 | open | V. P. Burichenko | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,456 | KOU-18.27 | Kourovka Notebook Problem 18.27 | Does there exist an algorithm that determines whether there are finitely many subformations in $\mathrm{form}(G)$ for a given finite group $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.27.
Proposed by: V. P. Burichenko | 2 | open | V. P. Burichenko | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,457 | KOU-18.28 | Kourovka Notebook Problem 18.28 | Is it true that any subformation of every one-generator formation $\mathrm{form}(G)$ is also one-generator? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.28.
Proposed by: V. P. Burichenko | 2 | open | V. P. Burichenko | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,458 | KOU-18.29 | Kourovka Notebook Problem 18.29 | Let $\mathfrak {F}$ be a Fitting class of finite soluble groups which contains every soluble group $G=AB$, where $A$ and $B$ are abnormal $\mathfrak {F}$-subgroups of $G$. Is $\mathfrak {F}$ a formation? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.29.
Proposed by: A. F. Vasil'ev | 2 | open | A. F. Vasil'ev | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,459 | KOU-18.30 | Kourovka Notebook Problem 18.30 | A subgroup $H$ of a group $G$ is called \emph{$\mathbb{P}$-subnormal} in $G$ if either $H=G$, or there is a chain of subgroups $H_0\subset H_1\subset \cdots\subset H_{n-1}\subset H_{n} =G$ such that $|H_i : H_{i-1}|$ is a prime for all $i=1,\ldots, n$. Must a finite group be soluble if every Shmidt subgroup of it is $\... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.30.
Proposed by: A. F. Vasil'ev, T. I. Vasil'eva, V. N. Tyutyanov
Published resolution:
Yes, it must (V. N. Tyutyanov, *Probl. Fiz. Mat. Tekhn.*, **2015**, no. 1 (22), 88--91 (... | 2 | solved | A. F. Vasil'ev, T. I. Vasil'eva, V. N. Tyutyanov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it must (V. N. Tyutyanov, *Probl. Fiz. Mat. Tekhn.*, **2015**, no. 1 (22), 88--91 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,460 | KOU-18.31 | Kourovka Notebook Problem 18.31 | Let $\pi$ be a set of primes. Is it true that in any
$D_\pi$-group $G$ (see Archive, 3.62) there are three Hall $\pi$-subgroups whose intersection coincides with $O_\pi(G)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.31.
Proposed by: E. P. Vdovin, D. O. Revin
Published resolution:
No, it is not true, for example, for $G$ being an extension of $L_2(27)$ by a field automorphism of order $3$, ... | 2 | solved | E. P. Vdovin, D. O. Revin | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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, for example, for $G$ being an extension of $L_2(27)$ by a field automorphism of order $3$, in which $H$ is an extension of a Borel subgroup o... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,461 | KOU-18.32 | Kourovka Notebook Problem 18.32 | Is every Hall subgroup of a finite group pronormal in its normal closure? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.32.
Proposed by: E. P. Vdovin, D. O. Revin
Published resolution:
No, not always (M. N. Nesterov, *Siberian Math. J.*, **58**,\ no. 1 (2017),
128--133). | 2 | solved | E. P. Vdovin, D. O. Revin | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 (M. N. Nesterov, *Siberian Math. J.*, **58**,\\ no. 1 (2017),\n128--133)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,462 | KOU-18.33 | Kourovka Notebook Problem 18.33 | A group in which the derived subgroup of every 2-generated subgroup is cyclic is called an *Alperin group*. Is there a bound for the derived length of finite Alperin groups?
G. Higman proved that finite Alperin groups are soluble, and finite Alperin $p$-groups have bounded derived length, see Archive 17.46. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.33.
Proposed by: B. M. Veretennikov
Published resolution:
Yes, there is: it is at most 6. Indeed, in (P. Longobardi, M. Maj, H. Smith, *Rend. Semin. Mat. Univ. Padova*, **115**... | 2 | solved | B. M. Veretennikov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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: it is at most 6. Indeed, in (P. Longobardi, M. Maj, H. Smith, *Rend. Semin. Mat. Univ. Padova*, **115** (2006), 29--40) it was proved that finite ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,463 | KOU-18.34 | Kourovka Notebook Problem 18.34 | Let $G$ be a topological group and $S={G_0\leq \dots \leq G_n=G}$ a subnormal series of closed subgroups. The *infinite-length* of $S$ is the cardinality of the set of infinite factors $G_i/G_{i-1}$. The *virtual length* of $G$ is the supremum of the infinite lengths taken over all such series of $G$. It is known that ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.34.
Proposed by: N. Gavioli, V. Monti, C. M. Scoppola | 2 | open | N. Gavioli, V. Monti, C. M. Scoppola | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,464 | KOU-18.35 | Kourovka Notebook Problem 18.35 | A family ${\mathscr F}$ of group homomorphisms $A\to B$ is *separating* if for every nontrivial $a\in A$ there is $f\in {\mathscr F}$ such that $f(a)\ne 1$, and *discriminating* if for any finitely many nontrivial elements $a_1,\dots ,a_n\in A$ there is $f\in {\mathscr F}$ such that $f(a_i)\ne 1$ for all $i=1,\dots , n... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.35.
Proposed by: A. Gaglione, D. Spellman | 2 | open | A. Gaglione, D. Spellman | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,465 | KOU-18.36 | Kourovka Notebook Problem 18.36 | There are groups of cardinality at most $2^{\aleph_{0}}$, even nilpotent of
class $2$, that cannot
be embedded in $S:=Sym(\mathbb{N})$ (V. A. Churkin,
*Algebra and model theory* (Novosibirsk State Tech. Univ.), **5** (2005), 39--43 (Russian)).
One condition which might characterize subgroups of $S$ is the
following. Co... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.36.
Proposed by: J. D. Dixon
Discussion and literature:
[Solved] No, there exists an uncountable connected Polish group whose every abstract homomorphism int... | 2 | solved | J. D. Dixon | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 exists an uncountable connected Polish group whose every abstract homomorphism into $S$ is tri... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,466 | KOU-18.37 | Kourovka Notebook Problem 18.37 | (Well-known problem). Is every locally graded group of finite rank almost locally soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.37.
Proposed by: M. Dixon | 2 | open | M. Dixon | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,467 | KOU-18.38 | Kourovka Notebook Problem 18.38 | Let $E_\pi$ denote the class of finite groups that contain a Hall $\pi$-subgroup. Does the inclusion $E_{\pi_1} \cap E_{\pi_2} \subseteq E_{\pi_1 \cap \pi_2}
$ hold for arbitrary sets of primes $\pi_1$ and $\pi_2$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.38.
Proposed by: A. V. Zavarnitsine
Discussion and literature:
[Solved] Yes, it does (A. Buturlakin, N. Yang, *Preprint*, 2025,
https://arxiv.org/abs/2501.05... | 2 | solved | A. V. Zavarnitsine | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it does (A. Buturlakin, N. Yang, *Preprint*, 2025,\nhttps://arxiv.org/abs/2501.05865)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,468 | KOU-18.39 | Kourovka Notebook Problem 18.39 | *Conjecture*: Let $G$ be a hyperbolic group. Then every 2-dimensional rational homology class is virtually represented by a sum of closed surface subgroups, that is, for any $\alpha \in H_2(G;\mathbb{Q})$ there are finitely many closed oriented surfaces $S_i$ and injective homomorphisms $\rho_i:\pi_1(S_i) \to G$ such t... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.39.
Proposed by: D. Calegari | 2 | open | D. Calegari | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,469 | KOU-18.40 | Kourovka Notebook Problem 18.40 | The commutator length $cl(g)$ of $g \in [G,G]$ is the least number of commutators in $G$ whose product is $g$, and the stable commutator length is $scl(g):=
\lim_{n \to \infty} cl(g^n)/n$.
*Conjecture*: Let $G$ be a hyperbolic group. Then the stable commutator length takes on rational values on $[G,G]$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.40.
Proposed by: D. Calegari | 2 | open | D. Calegari | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,470 | KOU-18.41 | Kourovka Notebook Problem 18.41 | Let $F$ be a free group of rank 2.
a) It is known that $scl(g)=0$ only for $g=1$, and $scl(g)\ge 1/2$ for all $1\ne g\in [F,F]$. Is $1/2$ an isolated value?
b) Are there any intervals $J$ in ${\mathbb{R}$ such that the set of values of $scl$ on $[F,F]$ is dense in $J$?
c) Is there some $T\in {\mathbb{R}$ such that e... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.41.
Proposed by: D. Calegari | 2 | open | D. Calegari | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,471 | KOU-18.42 | Kourovka Notebook Problem 18.42 | For an orientation-preserving homeomorphism $f:S^1\to S^1$ of a unit circle $S^1$, let $\tilde f$ be its lifting to a homeomorphism of ${\mathbb{R}$; then the *rotation number* of $f$ is defined to be the limit $\lim _{n\to \infty} (\tilde f^n(x)-x)/n$ (which is independent of the point $x\in S^1$). Let $F$ be a free g... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.42.
Proposed by: D. Calegari, A. Walker | 2 | open | D. Calegari, A. Walker | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,472 | KOU-18.43 | Kourovka Notebook Problem 18.43 | a) Do there exist elements $u,v$ of a 2-generator free group $F(a,b)$ such that $u,v$ are not conjugate in $F(a,b)$ but for any matrices $A,B\in GL(3, \mathbb{C})$ we have $trace(u(A,B))= trace(v(A,B))$?
b) The same question if we replace $GL(3,\mathbb{C})$ by $SL(3,\mathbb{C})$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.43.
Proposed by: I. Kapovich | 2 | open | I. Kapovich | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,473 | KOU-18.44 | Kourovka Notebook Problem 18.44 | Let $G$ be a finite non-abelian group and $V$ a finite faithful irreducible $G$-module. Suppose that $M=|G/G'|$ is the largest orbit size of $G$ on $V$, and among orbits of $G$ on $V$ there are exactly two orbits of size $M$.
Does this imply that $G$ is dihedral of order 8, and $|V|=9$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.44.
Proposed by: T. M. Keller
Discussion and literature:
[Solved] Yes, it does (N. A. Jones, T. M. Keller, *Ann. Mat. Pura Appl.*, **199** (2020), 2323--2340... | 2 | solved | T. M. Keller | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it does (N. A. Jones, T. M. Keller, *Ann. Mat. Pura Appl.*, **199** (2020), 2323--2340)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,474 | KOU-18.45 | Kourovka Notebook Problem 18.45 | Let $G$ be a finite $p$-group of maximal class such that all its irreducible characters are induced from linear characters of normal subgroups. Let $S$ be the set of the derived
subgroups of the members of the central series of $G$. Is there a logarithmic bound for the derived length of $G$
in terms of $|S|$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.45.
Proposed by: T. M. Keller | 2 | open | T. M. Keller | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,475 | KOU-18.46 | Kourovka Notebook Problem 18.46 | Every finite group $G$ can be embedded in a group $H$ in such a way
that every element of $G$ is a square of an element of $H$.
The overgroup $H$ can be chosen such that
$|H| \le 2|G|^2$.
Is this estimate sharp?
It is known that the best possible estimate cannot be better than $|H| \le |G|^2$ (D. V. Baranov, Ant. A. K... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.46.
Proposed by: Ant. A. Klyachko | 2 | open | Ant. A. Klyachko | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,476 | KOU-18.47 | Kourovka Notebook Problem 18.47 | Is it algorithmically decidable whether a group generated by three given
class transpositions (for the definition, see 17.57)
a) has only finite orbits on \(\mathbb{Z}\)?
b) acts transitively on the set of nonnegative integers in its support?
A difficult case is the group
\(\langle \tau_{1(2),4(6)}, \tau_{1(3),2(6)}... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.47.
Proposed by: S. Kohl | 2 | open | S. Kohl | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,477 | KOU-18.48 | Kourovka Notebook Problem 18.48 | Is it true that there are only finitely many integers which occur as orders of products
of two class transpositions? (For the definition, see 17.57.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.48.
Proposed by: S. Kohl
Discussion and literature:
[Solved] Yes, it is true (A. Iskra, *Preprint*, 2026, https://arxiv.org/abs/2610.01793). | 2 | solved | S. Kohl | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (A. Iskra, *Preprint*, 2026, https://arxiv.org/abs/2610.01793)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,478 | KOU-18.49 | Kourovka Notebook Problem 18.49 | Let \(n \in \mathbb{N}\). Is it true that for any \(a, b, c \in \mathbb{N}\) satisfying
\(1 < a, b, c \leq n-2\) the symmetric group \(\mathrm{S}_n\) has elements of order \(a\)
and \(b\) whose product has order \(c\)? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.49.
Proposed by: S. Kohl
Published resolution:
Yes, it is
(G. A. Miller, *Amer. J. Math.*, **22**, no. 2 (1900), 185--190).
Another solution is in (J. K\"onig, *Eur. J. Comb.*,... | 2 | solved | S. Kohl | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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(G. A. Miller, *Amer. J. Math.*, **22**, no. 2 (1900), 185--190).\nAnother solution is in (J. K\\\"onig, *Eur. J. Comb.*, **57** (2016), 50--56)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,479 | KOU-18.50 | Kourovka Notebook Problem 18.50 | Let \(n \in \mathbb{N}\). Is it true that for every \(k \in \{1, \dots, n!\}\) there is
some group \(G\) and pairwise distinct elements \(g_1, \dots, g_n \in G\) such that the set
\(\{g_{\sigma(1)} \cdots g_{\sigma(n)} \ | \ \sigma \in \mathrm{S}_n\}\)
of all products of the \(g_i\) obtained by permuting the factors ha... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.50.
Proposed by: S. Kohl
Discussion and literature:
[Solved] Yes, it is true (AI solution confirmed by S. Kohl: \ P. Monticone, *Preprint*, 2026, https://kou... | 2 | solved | S. Kohl | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (AI solution confirmed by S. Kohl: \\ P. Monticone, *Preprint*, 2026, https://kourovkano... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,480 | KOU-18.51 | Kourovka Notebook Problem 18.51 | Given a prime \(p\) and \(n \in \mathbb{N}\), let \(f_p(n)\) be the smallest
number such that there is a group of order \(p^{f_p(n)}\) into which every group of
order \(p^n\) embeds. Is it true that \(f_p(n)\) grows faster than polynomially
but slower than exponentially when \(n\) tends to infinity? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.51.
Proposed by: S. Kohl | 2 | open | S. Kohl | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,481 | KOU-18.52 | Kourovka Notebook Problem 18.52 | Is every finite simple group generated by two elements of prime-power orders $m$, $n$? (Here numbers $m,n$ may depend on the group.) The work of many authors shows that it remains to verify this property for a small number of finite simple groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.52.
Proposed by: J. Krempa
Published resolution:
Yes, it is (mod CFSG); moreover, every finite simple group is generated by an involution and
an element of prime order (C. S. H... | 2 | solved | J. Krempa | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (mod CFSG); moreover, every finite simple group is generated by an involution and\nan element of prime order (C. S. H. King, *J. Algebra*, **478** (20... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,482 | KOU-18.53 | Kourovka Notebook Problem 18.53 | Is there a non-linear simple locally finite group in which every
centralizer of a non-trivial element is almost soluble, that is, has a
soluble subgroup of finite index? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.53.
Proposed by: M. Kuzucuo\u{g
Discussion and literature:
lu} | 2 | open | M. Kuzucuo\u{g | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,483 | KOU-18.54 | Kourovka Notebook Problem 18.54 | Can a group be equal to the union of conjugates of a proper finite nonabelian simple subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.54.
Proposed by: G. Cutolo | 2 | open | G. Cutolo | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,484 | KOU-18.55 | Kourovka Notebook Problem 18.55 | a)
Can a locally finite $p$-group $G$ of finite exponent be the union of conjugates of an abelian proper subgroup?
b) Can this happen when $G$ is of exponent $p$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.55.
Proposed by: G. Cutolo | 2 | open | G. Cutolo | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,485 | KOU-18.56 | Kourovka Notebook Problem 18.56 | Let $G$ be a finite 2-group, of order greater than 2, such that $|H/H_G| \le 2$ for all $H\le G$, where $H_G$ denotes the largest normal subgroup of $G$ contained in $H$. Must $G$ have an abelian subgroup of index 4? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.56.
Proposed by: G. Cutolo | 2 | open | G. Cutolo | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,486 | KOU-18.57 | Kourovka Notebook Problem 18.57 | Let $G$ be a finite 2-group generated by involutions in which
$[x,u,u]=1$ for every $x\in G$ and every involution $u\in G$.
Is the derived length of $G$ bounded? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.57.
Proposed by: D. V. Lytkina
Published resolution:
No, it is not (A. Abdollahi,
*J. Group Theory*, **18**, no. 1 (2015), 111--114). | 2 | solved | D. V. Lytkina | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 (A. Abdollahi,\n*J. Group Theory*, **18**, no. 1 (2015), 111--114)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,487 | KOU-18.58 | Kourovka Notebook Problem 18.58 | Let $G$ be a group generated by finite number $n$ of involutions
in which $(uv)^4=1$ for all involutions $u,v\in G$. Is it true
that $G$ is finite? is a 2-group?
This is true for $n\leq 3$.
*Editors' comment of 2021*: the previously published claim
of an affirmative solution of this problem proved to be erroneous. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.58.
Proposed by: D. V. Lytkina | 2 | open | D. V. Lytkina | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,488 | KOU-18.59 | Kourovka Notebook Problem 18.59 | Does there exist a periodic group $G$ such that $G$ contains an
involution, all involutions in $G$ are conjugate, and the
centralizer of every involution $i$ is isomorphic to $\langle i\rangle \times L_2(P)$,
where $P$ is some infinite locally finite field of characteristic $2$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.59.
Proposed by: D. V. Lytkina
Discussion and literature:
[Solved] No, there are no such groups (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg... | 2 | solved | D. V. Lytkina | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, there are no such groups (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,489 | KOU-18.60 | Kourovka Notebook Problem 18.60 | Let $V$ be an infinite countable elementary abelian additive 2-group.
Does $\mathrm{Aut} V$ contain a subgroup $G$ such that
(a) $G$ is transitive on the set of non-zero elements of $V$, and
(b) if $H$ is the stabilizer in $G$ of a non-zero element $v\in V$,
then $V=\langle v\rangle \oplus V_v$, where $V_v$ is
$H$-in... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.60.
Proposed by: D. V. Lytkina
Discussion and literature:
[Solved] Yes, there are no such groups (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookor... | 2 | solved | D. V. Lytkina | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 are no such groups (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.co... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,490 | KOU-18.61 | Kourovka Notebook Problem 18.61 | Is a 2-group nilpotent if all its finite subgroups are nilpotent
of class at most 3? This is true if 3 is replaced by 2. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.61.
Proposed by: D. V. Lytkina. | 2 | open | D. V. Lytkina. | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,491 | KOU-18.62 | Kourovka Notebook Problem 18.62 | The *spectrum* of a finite group is the
set of orders of its elements. Let $\omega$ be a finite set of positive integers. A group $G$ is
said to be
*$\omega$-critical*
if the spectrum of $G$ coincides with $\omega$, but the spectrum
of every proper
section of $G$
is not equal to $\omega$.
a) Does there exist a number ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.62.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,492 | KOU-18.63 | Kourovka Notebook Problem 18.63 | Let $G$ be a periodic group generated by two fixed-point-free
automorphisms of order 5 of an abelian group. Is $G$ finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.63.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,493 | KOU-18.64 | Kourovka Notebook Problem 18.64 | (K. Harada). *Conjecture*: Let $G$ be a finite group, $p$ a prime, and
$B$ a $p$-block of $G$. If $J$ is a non-empty subset of $Irr(B)$
such that $\sum\limits_{\chi \in J}\chi (1)\chi (g)=0$
for
every $p$-singular element $g\in G$, then $J=Irr(B)$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.64.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,494 | KOU-18.65 | Kourovka Notebook Problem 18.65 | (R. Guralnick, G. Malle). *Conjecture*: Let $p$ be a prime different from 5, and $C$ a class of conjugate $p$-elements in a finite group $G$. If $[c,d]$ is a $p$-element for any $c,d\in C$, then $C\subseteq O_p(G)$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.65.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,495 | KOU-18.66 | Kourovka Notebook Problem 18.66 | Suppose that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that $GF$ is also a Frobenius group with kernel $G$ and complement $F$.
Is the derived length of $G$ bounded in terms of $|H|$ and the derived length of $C_G(H)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.66.
Proposed by: N. Yu. Makarenko, E. I. Khukhro, P. Shumyatsky | 2 | open | N. Yu. Makarenko, E. I. Khukhro, P. Shumyatsky | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,496 | KOU-18.67 | Kourovka Notebook Problem 18.67 | Suppose that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that $C_G(F)=1$. Is the exponent of $G$ bounded in terms of $|F|$ and the exponent of $C_G(H)$?
This was proved when $F$ is cyclic (E. I. Khukhro, N. Y. Makarenko, P. Shumyatsky, *Forum Math.*,
**26**... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.67.
Proposed by: N. Yu. Makarenko, E. I. Khukhro, P. Shumyatsky | 2 | open | N. Yu. Makarenko, E. I. Khukhro, P. Shumyatsky | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,497 | KOU-18.68 | Kourovka Notebook Problem 18.68 | What are the nonabelian composition factors of a finite nonsoluble group all of whose maximal subgroups have complements?
Note that finite simple groups with all maximal subgroups having complements are, up to isomorphism, $L_2(7)$, $L_2(11)$, and $L_5(2)$ (V. M. Levchuk, A. G. Likharev, *Siberian Math. J.*, **47**, n... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.68.
Proposed by: N. V. Maslova, D. O. Revin
Discussion and literature:
[Unverified AI claim] These factors are described (unverified AI solution: \ R. Sater,... | 2 | open | N. V. Maslova, D. O. Revin | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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] These factors are described (unverified AI solution: \\ R. Sater, *Preprint*, 202... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,498 | KOU-18.69 | Kourovka Notebook Problem 18.69 | Does there exist a relatively free group $G $
containing a free subsemigroup and having $[G,G]$ finitely generated?
Note that $G$ cannot be locally graded (*Publ. Math. Debrecen*, **81**, no. 3-4 (2012), 415--420.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.69.
Proposed by: O. Macedo\'nska | 2 | open | O. Macedo\'nska | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,499 | KOU-18.70 | Kourovka Notebook Problem 18.70 | Is every finitely generated Coxeter group
conjugacy separable?
Some special cases were considered in
(P.-E. Caprace, A. Minasyan, *Illinois J. Math.*, **57**, no. 2 (2013), 499--523). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.70.
Proposed by: A. Minasyan | 2 | open | A. Minasyan | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,500 | KOU-18.71 | Kourovka Notebook Problem 18.71 | (N. Aronszajn). Suppose that $W(x, y) = 1$ in an open subset of $G\times G$, where $G$ is a connected topological group. Must $W(x, y) = 1$ for all $(x, y)\in G\times G$?
For locally compact groups the answer is yes. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.71.
Proposed by: J. Mycielski
Discussion and literature:
[Solved] No, not necessarily: for any odd integer $n>10^{10}$ there is a connected topological group... | 2 | solved | J. Mycielski | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not necessarily: for any odd integer $n>10^{10}$ there is a connected topological group such that th... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,501 | KOU-18.72 | Kourovka Notebook Problem 18.72 | Is it true that an existentially closed subgroup of a nonabelian free group of finite rank is a nonabelian free factor of this group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.72.
Proposed by: A. G. Myasnikov, V. A. Roman'kov | 2 | open | A. G. Myasnikov, V. A. Roman'kov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,502 | KOU-18.73 | Kourovka Notebook Problem 18.73 | a) Does every finitely generated solvable group of derived length $l\ge 2$ embed
into a $2$-generated solvable group of length $l+1$?
*Comment of 2021*: It is proved that any countable solvable group of derived length $l$ with torsion-free abelianization embeds in a 2-generated solvable group of derived length $l+1$ (... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 18.73.
Proposed by: A. Yu. Olshanskii
Archive of Solved Problems entry:
b) It is proved that any finitely generated solvable group of derived length $l$ can ... | 2 | partially_solved | A. Yu. Olshanskii | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (b) It is proved that any finitely generated solvable group of derived length $l$ can be embedded in a $4$-generated solvable group of derived length $l + 1$ (V. ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,503 | KOU-18.74 | Kourovka Notebook Problem 18.74 | Let $G$ be a finitely generated elementary amenable group which
is not virtually nilpotent. Is there a finitely generated metabelian non-virtually-nilpotent section
in $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.74.
Proposed by: A. Yu. Olshanskii | 2 | open | A. Yu. Olshanskii | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,504 | KOU-18.75 | Kourovka Notebook Problem 18.75 | Does every finite solvable group $G$ have the following property: there is a number
$d=d(G)$ such that $G$ is a homomorphic image of every group with $d$ generators and one relation?
This property holds for finite nilpotent groups and does not hold for every
non-solvable finite group; see (S. A. Zaĭtsev,
*Moscow Univ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.75.
Proposed by: A. Yu. Olshanskii
Published resolution:
Yes, it does (N. Nikolov, D. Segal, *Bull. London Math. Soc.*, **39**, no. 2 (2007), 209--213). | 2 | solved | A. Yu. Olshanskii | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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. Nikolov, D. Segal, *Bull. London Math. Soc.*, **39**, no. 2 (2007), 209--213)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,505 | KOU-18.76 | Kourovka Notebook Problem 18.76 | Let $A$ be a division ring, $G$ a subgroup of the
multiplicative group of $A$, and $E$ an extension of the additive
group of $A$ by $G$ such that $G$ acts by multiplication in $A$.
Is it true that $E$ splits? This is true if $G$ is finite. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.76.
Proposed by: E. A. Palyutin.
Discussion and literature:
[Solved] No, not always (AI solutions confirmed by B.A.F.Wehrfritz: \ Achyuth Jayadevan, *Preprin... | 2 | solved | E. A. Palyutin. | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (AI solutions confirmed by B.A.F.Wehrfritz: \\ Achyuth Jayadevan, *Preprint of 14 Septemb... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,506 | KOU-18.77 | Kourovka Notebook Problem 18.77 | Let $G$ be a finite $p$-group and let $p^e$ be the largest
degree of an irreducible complex representation of $G$. If $p>e$, is it
necessarily true that $\bigcap \ker \Theta= 1$,
where the intersection runs over all irreducible complex representations $\Theta$ of
$G$ of degree $p^e$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.77.
Proposed by: D. S. Passman | 2 | open | D. S. Passman | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,507 | KOU-18.78 | Kourovka Notebook Problem 18.78 | Let $K^tG$ be a twisted group algebra of the finite group $G$ over the field $K$.
If $K^tG$ is a simple $K$-algebra, is $G$ necessarily solvable? This is known to be
true if $K^tG$ is central simple. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.78.
Proposed by: D. S. Passman | 2 | open | D. S. Passman | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,508 | KOU-18.79 | Kourovka Notebook Problem 18.79 | Let $K[G]$ be the group algebra of the finitely generated group $G$
over the field $K$. Is the Jacobson radical
${\mathcal{J}}K[G]$ equal to the join of all nilpotent ideals of the ring? This is
known to be true if $G$ is solvable or linear. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.79.
Proposed by: D. S. Passman | 2 | open | D. S. Passman | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,509 | KOU-18.80 | Kourovka Notebook Problem 18.80 | (I. Kaplansky). For $G\neq 1$, show that
the augmentation ideal of the group algebra $K[G]$ is equal to the Jacobson radical of the ring
if and only if $\mathrm{char} K=p>0$ and
$G$ is a locally finite $p$-group. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.80.
Proposed by: D. S. Passman | 2 | open | D. S. Passman | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,510 | KOU-18.81 | Kourovka Notebook Problem 18.81 | Let $G$ be a finitely generated $p$-group that is residually finite.
Are all maximal subgroups of $G$ necessarily normal? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.81.
Proposed by: D. S. Passman | 2 | open | D. S. Passman | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,511 | KOU-18.82 | Kourovka Notebook Problem 18.82 | Is there a function $f:{\mathbb{N} \to {\mathbb{N}$ such that for any prime $p$, if $p^{f(n)}$ divides the order of a finite group $G$, then $p^n$ divides the order of $\mathrm{Aut} G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.82.
Proposed by: R. M. Patne
Published resolution:
Yes, there is. This was established in (J. A. Green, *Proc. Roy. Soc. London Ser. A.*, **237** (1956), 574--581), which impro... | 2 | solved | R. M. Patne | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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. This was established in (J. A. Green, *Proc. Roy. Soc. London Ser. A.*, **237** (1956), 574--581), which improved a previous result with a functio... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,512 | KOU-18.83 | Kourovka Notebook Problem 18.83 | A generating system $X$ of a group $G$ is *fast* if there is an integer $n$
such that every element of $G$ can be expressed as a product of at most
$n$ elements of $X$ or their inverses. If not, we say that it is *slow*.
For instance, in $(\mathbb{Z} ,+)$, the squares are fast, but the powers of 2 are slow.
Do there e... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.83.
Proposed by: B. Poizat | 2 | open | B. Poizat | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,513 | KOU-18.84 | Kourovka Notebook Problem 18.84 | Let $\pi$ be a set of primes. We say that a finite group is a $BS_\pi$-group if every conjugacy class in this group any two elements of which generate a $\pi$-subgroup itself generates a
$\pi$-subgroup. Is every normal subgroup of a $BS_\pi$-group a $BS_\pi$-group?
In the case
$2\not\in\pi$, an affirmative answer foll... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.84.
Proposed by: D. O. Revin | 2 | open | D. O. Revin | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,514 | KOU-18.85 | Kourovka Notebook Problem 18.85 | A subset of a group is said to be *rational* if it can be obtained from finite subsets by finitely many rational operations, that is, taking union, product, and the submonoid generated by a set.
*Conjecture*: every finitely generated solvable group in which all rational subsets form a Boolean algebra is virtually abel... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.85.
Proposed by: V. A. Roman'kov | 2 | open | V. A. Roman'kov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,515 | KOU-18.86 | Kourovka Notebook Problem 18.86 | Is the group $G = \langle a, b \mid [[a,b],b]=1\rangle$, which is isomorphic to the group of all unitriangular automorphisms of the free group of rank 3, linear? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.86.
Proposed by: V. A. Roman'kov
Published resolution:
Yes, it is linear, since it embeds in the holomorph $\mathrm{Hol} F_2$ of the free group $F_2$, which can be seen by addi... | 2 | solved | V. A. Roman'kov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 linear, since it embeds in the holomorph $\\mathrm{Hol} F_2$ of the free group $F_2$, which can be seen by adding a new generator $c=[a,b]$, so that $... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,516 | KOU-18.87 | Kourovka Notebook Problem 18.87 | A system of equations with coefficients in a group $G$ is said to be *independent* if the matrix composed of the sums of exponents of the unknowns has rank equal to the number of equations.
a) The *Kervaire--Laudenbach Conjecture (KLC)*: every independent system of equations with coefficients in an arbitrary group $G$... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.87.
Proposed by: V. A. Roman'kov | 2 | open | V. A. Roman'kov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,517 | KOU-18.88 | Kourovka Notebook Problem 18.88 | Can a finitely generated infinite group of finite exponent be the quotient of a residually finite group by a locally finite normal subgroup?
If not, then there exists a hyperbolic group that is not residually finite. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.88.
Proposed by: M. Sapir | 2 | open | M. Sapir | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,518 | KOU-18.89 | Kourovka Notebook Problem 18.89 | Consider the set of balanced presentations $\langle x_1,\dots ,x_n\mid r_1,\dots, r_n\rangle $ with fixed $n$ generators $x_1,\dots, x_n$. By definition, the group ${AC}_n$ of Andrews--Curtis moves on this set of balanced presentations is generated by the Nielsen transformations together with conjugations of relators. ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.89.
Proposed by: J. Swan, A. Lisitsa
Discussion and literature:
[Solved] The group ${AC}_2$ is not finitely presented (S. Krsti\'c, J. McCool, *J. London Mat... | 2 | solved | J. Swan, A. Lisitsa | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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] The group ${AC}_2$ is not finitely presented (S. Krsti\\'c, J. McCool, *J. London Math. Soc. (2)*, **56*... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,519 | KOU-18.90 | Kourovka Notebook Problem 18.90 | Prove that the group
$Y(m,n) =\langle a_{1},a_{2},\dots,a_{m} \mid
a_{k}^{n} =1 \text{ }(1\leq k\leq m)\text{, } ( a_{k}^{i}a_{l}^{i}) ^{2} =e\text{ }(1\leq k<l\leq m\text{, }
1\leq i\leq \frac{n}{2})\rangle $
is finite for every pair $( m,n) $.
Some known cases are (where $C_n$ denotes a cyclic group of order $n$): $... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.90.
Proposed by: S. Sidki | 2 | open | S. Sidki | 2,014 | 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 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,520 | KOU-18.91 | Kourovka Notebook Problem 18.91 | A subgroup $H$ of a group $G$ is said to be *propermutable* in $G$
if there is a subgroup $B\leq G$ such that $G=N_{G}(H)B$ and $H$
permutes with every subgroup of $B$.
a) Is there a finite group $G$ with subgroups $A\leq B\leq G$ such that $A$ is propermutable in $G$ but $A$ is not propermutable in $B$?
b) Let $P$ b... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.91.
Proposed by: A. N. Skiba
Published resolution:
a) Yes, there is
(A. A. Pypka, D. Yu. Storozhenko,
*Dopov. Nac. Akad. Nauk Ukrain.*, **2017**, no. 7, 18--20 (Ukrainian)).
... | 2 | solved | A. N. Skiba | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) Yes, there is\n\n(A. A. Pypka, D. Yu. Storozhenko,\n*Dopov. Nac. Akad. Nauk Ukrain.*, **2017**, no. 7, 18--20 (Ukrainian)).\n\nb) Yes, it is (Kh. A. Al-Sharo,... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,521 | KOU-18.92 | Kourovka Notebook Problem 18.92 | A non-empty set $\theta$ of formations is called a *complete lattice* of
formations if the intersection of any set of formations in $\theta$
belongs to $\theta$ and $\theta$ has the largest element (with respect to inclusion). If $L$ is a complete lattice, then an element $a\in L$ is
said to be compact if $ a\leq \bigv... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.92.
Proposed by: A. N. Skiba
Discussion and literature:
[Solved] a), b) Yes, there are, in both cases (AI solution confirmed by A. N. Skiba: \ M. Kinyon, J. ... | 2 | solved | A. N. Skiba | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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) Yes, there are, in both cases (AI solution confirmed by A. N. Skiba: \\ M. Kinyon, J. Urban, *Pre... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,522 | KOU-18.93 | Kourovka Notebook Problem 18.93 | Let $\mathcal{M}$ be a one-generated saturated formation, that is, the
intersection of all saturated formations containing some fixed finite
group. Let $\mathcal{F}$ be a subformation of $\mathcal{M}$ such that $\mathcal{F}
\ne {\mathcal{F}}{\mathcal{F}}$.
a) Is it true that then $\mathcal{F}$ can be written in the fo... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.93.
Proposed by: A. N. Skiba | 2 | open | A. N. Skiba | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,523 | KOU-18.94 | Kourovka Notebook Problem 18.94 | Let $G$ be a group without involutions, $a$ an element of it that is not a square of any element of $G$, and $n$ an odd positive integer.
Is it true that the quotient $G/\langle (a^{n})^{G}\rangle $ does not contain involutions? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.94.
Proposed by: A. I. Sozutov
Published resolution:
No, not always, as shown by an example of V. I. Trofimov; another example: the quotient of $G=\langle a,b\mid a^2=b^2\rangl... | 2 | solved | A. I. Sozutov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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, as shown by an example of V. I. Trofimov; another example: the quotient of $G=\\langle a,b\\mid a^2=b^2\\rangle$ by $\\langle a^G\\rangle$ has or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,524 | KOU-18.95 | Kourovka Notebook Problem 18.95 | Suppose that a group $G=AB$ is a product of an abelian subgroup $A$ and a locally quaternion group $B$ (that is, $B$ is a union of an increasing chain of finite generalized quaternion groups). Is $G$ soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.95.
Proposed by: A. I. Sozutov
Published resolution:
Yes, it is (B. Amberg, Ya. Sysak, On products of groups with abelian subgroups of small index, *J. Group Theory*, **20**, n... | 2 | solved | A. I. Sozutov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (B. Amberg, Ya. Sysak, On products of groups with abelian subgroups of small index, *J. Group Theory*, **20**, no. 6 (2017), 1061--1072)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,525 | KOU-18.96 | Kourovka Notebook Problem 18.96 | Suppose that a periodic group $G$ contains involutions and the centralizer of each involution is locally finite. Is it true that $G$ has a nontrivial locally finite normal subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.96.
Proposed by: N. M. Suchkov | 2 | open | N. M. Suchkov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,526 | KOU-18.97 | Kourovka Notebook Problem 18.97 | Let $G$ be a periodic Zassenhaus group, that is, a two-transitive permutation group with trivial stabilizer of every three points. Suppose that the stabilizer of a point is a Frobenius group with locally finite kernel $U$ containing an involution. Is it true that $U$ is a $2$-group? This was proved for finite groups by... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.97.
Proposed by: N. M. Suchkov | 2 | open | N. M. Suchkov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,527 | KOU-18.98 | Kourovka Notebook Problem 18.98 | The work of many authors shows that most finite
simple groups are generated by two elements of orders 2 and 3; for example, see the survey http://math.nsc.ru/conference/groups2013/slides/MaximVsemirnov_slides.pdf.
Which finite simple groups cannot be generated by two elements of orders 2 and 3?
In particular, is it tr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.98.
Proposed by: M. C. Tamburini | 2 | open | M. C. Tamburini | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,528 | KOU-18.99 | Kourovka Notebook Problem 18.99 | Let $\mathrm{cd}(G)$ be the set of all complex irreducible character degrees of a finite group $G$. *Huppert's Conjecture:*
if $H$ is a nonabelian simple group and $G$ is any finite group such that $\mathrm{cd}(G)=\mathrm{cd(H)}$, then $G\cong H\times A$, where $A$ is an abelian group. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.99.
Proposed by: H. P. Tong-Viet | 2 | open | H. P. Tong-Viet | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,529 | KOU-18.100 | Kourovka Notebook Problem 18.100 | For a given class of groups $\mathfrak{X}$, let $(\mathfrak{X},\infty)^{\ast }$ denote the class of groups in which every infinite subset contains two distinct elements $x$ and $y$ that satisfy $\langle x,x^{y}\rangle\in\mathfrak{X}$.
Let $G$ be a finitely generated soluble-by-finite group in the class $(\mathfrak{X},\... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.100.
Proposed by: N. Trabelsi | 2 | open | N. Trabelsi | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,530 | KOU-18.102 | Kourovka Notebook Problem 18.102 | (E. C. Dade). Let $C$ be a Carter subgroup of a finite solvable group $G$, and let $\ell(C)$ be the number of primes dividing $|C|$ counting multiplicities. It was proved in (E. C. Dade, *Illinois J. Math.*, **13** (1969), 449--514) that there is an exponential function $f$ such that the nilpotent length of $G$ is at m... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.102.
Proposed by: A. Turull | 2 | open | A. Turull | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,531 | KOU-18.103 | Kourovka Notebook Problem 18.103 | A group is said to be *minimax* if it has a finite subnormal series each of whose factors satisfies either the minimum or the maximum condition on subgroups.
Is it true that in the class of nilpotent minimax groups only finitely generated groups may have faithful irreducible primitive representations over a finitely ge... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.103.
Proposed by: A. V. Tushev | 2 | open | A. V. Tushev | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,532 | KOU-18.104 | Kourovka Notebook Problem 18.104 | Construct an example of a pro-$p$ group $G$ and a proper abstract normal subgroup $K$ such that $G/K$ is perfect. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.104.
Proposed by: John S. Wilson | 2 | open | John S. Wilson | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,533 | KOU-18.105 | Kourovka Notebook Problem 18.105 | Let $R$ be the formal power series algebra in commuting indeterminates $x_1,\dots x_n, \dots$ over the field with $p$ elements, with $p$ a prime. (Thus its elements are (in general infinite) linear combinations of monomials $x_1^{a_1}\dots x_n^{a_n}$ with $n, a_1,\dots,a_n$ non-negative integers.)
Let $I$ be the maxima... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.105.
Proposed by: John S. Wilson | 2 | open | John S. Wilson | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,534 | KOU-18.106 | Kourovka Notebook Problem 18.106 | Let $R$ be a group that acts coprimely on the finite group $G$.
Let $p$ be a prime, let $P$ be the unique maximal $RC_{G}(R)$-invariant $p$-subgroup
of $G$ and assume that $C_{G}(O_{p}(G)) \leq O_{p}(G)$.
If $p>3$ then $P$ contains a nontrivial characteristic subgroup that is normal in $G$ (P. Flavell, *J. Algebra*, **... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.106.
Proposed by: P. Flavell | 2 | open | P. Flavell | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,535 | KOU-18.107 | Kourovka Notebook Problem 18.107 | Is there a finitely generated infinite
residually finite $p$-group such that every subgroup of infinite
index is cyclic?
The answer is known to be negative
for $p=2$. Note that in (M. Ershov, A. Jaikin-Zapirain,
*J. Reine Angew. Math.*, **677** (2013), 71--134) it is shown that for every prime $p$ there is a finitely ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.107.
Proposed by: A. Jaikin-Zapirain | 2 | open | A. Jaikin-Zapirain | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,536 | KOU-18.108 | Kourovka Notebook Problem 18.108 | A group $G$ is said to have *property $(\tau )$* if
its trivial representation is an isolated point in the subspace of irreducible representations with finite images (in the topological unitary dual space of $G$).
Is it true that there exists a finitely generated group satisfying property $(\tau )$ that homomorphically... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.108.
Proposed by: A. Jaikin-Zapirain | 2 | open | A. Jaikin-Zapirain | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,537 | KOU-18.109 | Kourovka Notebook Problem 18.109 | Is it true that a group satisfying Kazhdan's property (T) cannot homomorphically map onto infinitely many simple groups of Lie type of rank 1?
It is known that a group which maps onto $PSL_2(q)$ for infinitely many $q$ does not have Kazhdan's property (T). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.109.
Proposed by: A. Jaikin-Zapirain | 2 | open | A. Jaikin-Zapirain | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,538 | KOU-18.110 | Kourovka Notebook Problem 18.110 | The *non-$p$-soluble length* of a finite group $G$ is the number of non-$p$-soluble factors in a shortest normal series each of whose factors either is $p$-soluble or is a direct product of non-abelian simple groups of order divisible by $p$. For a given prime $p$ and a given proper group variety ${\mathfrak{V}}$, is t... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.110.
Proposed by: E. I. Khukhro, P. Shumyatsky | 2 | open | E. I. Khukhro, P. Shumyatsky | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,539 | KOU-18.111 | Kourovka Notebook Problem 18.111 | Let $G$ be a discrete countable group, given as a central extension $0\to {\mathbb{Z}}\to G\to Q\to 0$. Assume that either $G$ is quasi-isometric to $\mathbb{Z}\times Q$, or that $\mathbb{Z}\to G$ is a quasi-isometric embedding. Does that imply that $G$ comes from a bounded cocycle on $Q$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.111.
Proposed by: I. Chatterji, G. Mislin
Discussion and literature:
[Solved] No, it does not, in both cases (R. Frigerio, A. Sisto, *Ann. Henri Lebesgue*, *... | 2 | solved | I. Chatterji, G. Mislin | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it does not, in both cases (R. Frigerio, A. Sisto, *Ann. Henri Lebesgue*, **6** (2023), 225--258)."
... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,540 | KOU-18.112 | Kourovka Notebook Problem 18.112 | Is it true that the orders of all elements of a finite group $G$ are powers of primes if, for every divisor $d$ ($d>1$) of $|G|$ and for every subgroup $H$ of $G$
of order coprime to $d$, the order $|H|$
divides the number of elements of $G$ of order $d$?
The converse is true (W. J. Shi, *Math. Forum (Vladikavkaz)*, *... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.112.
Proposed by: W. J. Shi
Published resolution:
Yes, it is true (A. A. Buturlakin, R. Shen, W. Shi, *Siberian Math. J.*, **58**, no. 3 (2017), 405--407). | 2 | solved | W. J. Shi | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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. A. Buturlakin, R. Shen, W. Shi, *Siberian Math. J.*, **58**, no. 3 (2017), 405--407)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,541 | KOU-18.113 | Kourovka Notebook Problem 18.113 | Let $\mathfrak{M}$ be a finite set of finite simple nonabelian
groups. Is it true that a periodic group saturated with groups from $\mathfrak{M}$ (see 14.101) is isomorphic to one of groups in $\mathfrak{M}$? The case where $\mathfrak{M}$ is one-element is of special interest. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.113.
Proposed by: A. K. Shl\"epkin | 2 | open | A. K. Shl\"epkin | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,542 | KOU-18.114 | Kourovka Notebook Problem 18.114 | Does there exist an irreducible $5'$-subgroup $G$ of $GL(V)$ for some finite ${\mathbb{F} _5$-space $V$ such that the number of conjugacy classes of the semidirect product $VG$ is equal to $|V|$ but $G$ is not cyclic?
Such a subgroup exists if 5 is replaced by $p=2, 3$ but does not exist for primes $p>5$ (*J. Group Th... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.114.
Proposed by: P. Schmid | 2 | open | P. Schmid | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,543 | KOU-18.115 | Kourovka Notebook Problem 18.115 | Let $G$ be a finite simple group, and let $X, Y$ be isomorphic simple maximal subgroups of $G$. Are $X$ and $Y$ conjugate in $\mathrm{Aut} G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.115.
Proposed by: P. Schmid
Published resolution:
No, not always. For
example, there are two $\mathrm{Aut} G$-classes of $M_{12}$ in $E_6(5)$ (P. B. Kleidman, R. A. Wilson, *J.... | 2 | solved | P. Schmid | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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. For\nexample, there are two $\\mathrm{Aut} G$-classes of $M_{12}$ in $E_6(5)$ (P. B. Kleidman, R. A. Wilson, *J. London Math.\nSoc.*, **42** (199... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,544 | KOU-18.116 | Kourovka Notebook Problem 18.116 | Given a finite $2$-group $G$ of order $2^n$, does there exist a finite set $S$ of rational primes such that $|S|\le n$ and $G$ is a quotient group of the absolute Galois group
of the maximal $2$-extension of $\mathbb{Q}$ unramified outside $S\cup \{\infty \}$?
This is true for $p$-groups, $p$ odd, as noted by Serre. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.116.
Proposed by: P. Schmid | 2 | open | P. Schmid | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,545 | KOU-18.117 | Kourovka Notebook Problem 18.117 | Is every element of a nonabelian finite simple group a commutator of two elements of coprime orders?
The answer is known to be ``yes'' for the alternating groups (P. Shumyatsky, *Forum Math.*, **27**, no. 1 (2015), 575--583) and for $PSL(2,q)$ (M. A. Pellegrini, P. Shumyatsky, *Arch. Math.*, **99** (2012), 501--507). ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.117.
Proposed by: P. Shumyatsky | 2 | open | P. Shumyatsky | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,546 | KOU-18.118 | Kourovka Notebook Problem 18.118 | Let $G$ be a profinite group such that $G/Z(G)$ is periodic. Is $[G,G]$ necessarily periodic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.118.
Proposed by: P. Shumyatsky | 2 | open | P. Shumyatsky | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,547 | KOU-18.119 | Kourovka Notebook Problem 18.119 | A multilinear
commutator word is any commutator of weight $n$ in $n$ distinct variables. Let $w$ be a multilinear commutator word and let $G$ be a finite group. Is it true that every Sylow $p$-subgroup of the verbal subgroup $w(G)$ is generated by $w$-values? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.119.
Proposed by: P. Shumyatsky | 2 | open | P. Shumyatsky | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,548 | KOU-18.120 | Kourovka Notebook Problem 18.120 | Let $P=AB$ be a finite $p$-group
factorized by an abelian subgroup $A$ and a class-two subgroup $B$. Suppose, if
necessary, $A \cap B=1$. Then is it true that $ \langle A, [B,B]
\rangle =AB_{0}$, where $B_{0}$ is an abelian subgroup of $ B$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.120.
Proposed by: E. Jabara | 2 | open | E. Jabara | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,549 | KOU-19.1 | Kourovka Notebook Problem 19.1 | An element $g$ of a group $G$ is a
*non-near generator* of $G$ if for every subset $S\subseteq G$ such that
$|G:\langle g,S\rangle|<\infty$ it follows that $|G:\langle S\rangle |<\infty$. The set of all non-near generators of $G$ forms a characteristic subgroup of $G$ called the
*lower near Frattini subgroup* of $G$, d... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.1.
Proposed by: M. K. Azarian | 2 | open | M. K. Azarian | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,550 | KOU-19.2 | Kourovka Notebook Problem 19.2 | For a subgroup $L$ of a group $G$, let $L_G$ denote the largest normal subgroup of $G$ contained in $L$. If $M\vartriangleleft G$, then we say that $G$ *nearly splits*
over $M$ if there is a subgroup $N\leq G$ such that $|G:N|=\infty$,
$|G:MN|<\infty$, and $(M\cap N)_G=1$.
Let $G$ be any group, and $H$ a normal subgro... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.2.
Proposed by: M. K. Azarian | 2 | open | M. K. Azarian | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,551 | KOU-19.3 | Kourovka Notebook Problem 19.3 | Let $G=A\ast_{H}B$ be the generalized free product of groups $A$ and $B$ with amalgamated subgroup $H$. Is $\psi(G)=1$ in the following cases?
\begin{itemize}
\item[{a)}] $H$ is finite cyclic and $H_G=1$.
\item[{b)}] $H$ is finite cyclic and either $\lambda(A)\cap H_G=1$ or $\lambda(B)\cap H_G=1$.
\item[{c)}] $H_G=1$... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.3.
Proposed by: M. K. Azarian | 2 | open | M. K. Azarian | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,552 | KOU-19.4 | Kourovka Notebook Problem 19.4 | a) If $G$ is the free product of infinitely many finitely
generated free groups with cyclic amalgamation, then
is $\psi(G)=1$?
b) If $G$ is the free product of infinitely many finitely
generated free abelian groups with cyclic amalgamation, then
is $\psi(G)=1$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.4.
Proposed by: M. K. Azarian | 2 | open | M. K. Azarian | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,553 | KOU-19.5 | Kourovka Notebook Problem 19.5 | If $G$ is the free product of infinitely many finitely generated
abelian groups with amalgamated subgroup $H$, then is $\psi(G)$
equal to the torsion subgroup of $H$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.5.
Proposed by: M. K. Azarian | 2 | open | M. K. Azarian | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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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