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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
701,654 | KOU-19.106 | Kourovka Notebook Problem 19.106 | Let $G$ be a uniformly locally finite group (which means that there is a function $f$ on the natural numbers such that any subgroup generated by $n$ elements has size at most $f(n)$). Suppose that for any two definable subgroups $H$ and $K$, the intersection $H\cap K$ has finite index either in $H$ or in $K$. Is $G$ ne... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.106.
Proposed by: F. Wagner | 2 | open | F. Wagner | 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 |
701,655 | KOU-19.107 | Kourovka Notebook Problem 19.107 | Let $\mathscr{AE}$ be the smallest class of locally compact groups such that
\begin{itemize}
\item[(1)] $\mathscr{AE}$ contains the compact groups and the discrete amenable groups;
\item[(2)] $\mathscr{AE}$ is closed under taking closed subgroups, Hausdorff quotients, directed unions of open subgroups, and group extens... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.107.
Proposed by: P. Wesolek | 2 | open | P. Wesolek | 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 |
701,656 | KOU-19.108 | Kourovka Notebook Problem 19.108 | Let $P$ be a finite $p$-group, where $p$ is an odd prime. Let $\chi$ be a
complex irreducible character of $P$. If $\chi(x)\ne 0$ for some
$x\in P$, is it true that the order of $x$ must divide $|P|/\chi(1)^2$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.108.
Proposed by: T. Wilde | 2 | open | T. Wilde | 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 |
701,657 | KOU-19.109 | Kourovka Notebook Problem 19.109 | A subgroup $H$ of a finite group $G$ is called pronormal if for any $g\in G$ the subgroups $H$ and $H^g$ are conjugate by an element of $\langle H,H^g\rangle$. A maximal subgroup of a maximal subgroup is called second maximal.
Is it true that in a non-abelian finite simple group $G$ all maximal subgroups are Hall subgr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.109.
Proposed by: V. I. Zenkov
Published resolution:
No, not always, for example, in $SL_2(2^{11})$ every second maximal subgroup is pronormal (V. N. Tyutyanov, *Letter of 23 N... | 2 | solved | V. I. Zenkov | 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 | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always, for example, in $SL_2(2^{11})$ every second maximal subgroup is pronormal (V. N. Tyutyanov, *Letter of 23 November 2018*)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,658 | KOU-19.110 | Kourovka Notebook Problem 19.110 | (G. M. Bergman and A. Magidin). Do there exist varieties of groups in which the relatively free
group of rank 2 is finite, and the relatively free group of rank 3 is
infinite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.110.
Proposed by: P. Zusmanovich | 2 | open | P. Zusmanovich | 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 |
701,659 | KOU-19.111 | Kourovka Notebook Problem 19.111 | Do there exist infinite groups
all of whose proper subgroups are cyclic of order $p$ that do not satisfy any
nontrivial group identity except $x^p = 1$ and its consequences?
Note
that there exist infinite
groups all of whose proper subgroups are infinite cyclic that do not satisfy
any nontrivial group identity (due to... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.111.
Proposed by: P. Zusmanovich | 2 | open | P. Zusmanovich | 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 |
701,660 | KOU-20.1 | Kourovka Notebook Problem 20.1 | For $k\geq1$, a group $G$ is said to be *totally $k$-closed* if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\mathrm{Sym}(\Omega)$ which leaves invariant each of the $G$-orbits in the induced action on the set of ordered $k$-tuples $\Omega^k$.
Are there an... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.1.
Proposed by: M. Arezoomand, M. A. Iranmanesh, C. E. Praeger, G. Tracey
Discussion and literature:
[Solved] Yes, there are (AI solution confirmed by G. Tra... | 2 | solved | M. Arezoomand, M. A. Iranmanesh, C. E. Praeger, G. Tracey | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (AI solution confirmed by G. Tracey: \\ M. Kinyon, J. Urban, *Preprint*, 2026,\nhttps://k... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,661 | KOU-20.2 | Kourovka Notebook Problem 20.2 | Are there any nonabelian simple groups of Lie type which are totally $3$-closed?
There are exactly six nonabelian simple totally $2$-closed groups --- all are sporadic groups, the largest being the Monster. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.2.
Proposed by: M. Arezoomand, M. A. Iranmanesh, C. E. Praeger, G. Tracey
Discussion and literature:
[Solved] Yes, there are (T. Gong, M. R. Zeng, Y. Yang, *... | 2 | solved | M. Arezoomand, M. A. Iranmanesh, C. E. Praeger, G. Tracey | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (T. Gong, M. R. Zeng, Y. Yang, *Preprint*, 2026, https://arxiv.org/abs/2608.17878)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,662 | KOU-20.4 | Kourovka Notebook Problem 20.4 | A finite group $G$ is said to be *cut* (or *inverse semi-rational*) if $\langle x \rangle = \langle y \rangle$ implies that $x$ is conjugate to $y$ or to $y^{-1}$ for all $x, y \in G$.
a) Let $\mathbb{Q}(G)$ denote the field extension of the rationals obtained by adjoining all entries of the ordinary character table o... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.4.
Proposed by: A. B{\"a
Discussion and literature:
chle} | 2 | open | A. B{\"a | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,663 | KOU-20.5 | Kourovka Notebook Problem 20.5 | (Well-known problem). Let $G$ be a finite group, and $\mathrm{V}(\mathbb{Z}G)$ the group of normalized units of the integral group ring of $G$. Do the spectra of $G$ and $\mathrm{V}(\mathbb{Z} G)$ coincide? That is, is it true that, for any integer $n$, there is an element of order $n$ in $\mathrm{V}(\mathbb{Z} G)$ if ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.5.
Proposed by: A. B{\"a
Discussion and literature:
chle, L. Margolis} | 2 | open | A. B{\"a | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,664 | KOU-20.6 | Kourovka Notebook Problem 20.6 | (W. Kimmerle).
The prime graph (or Gruenberg--Kegel graph) $\Gamma(X)$ of a group $X$ has vertices labeled by primes appearing as orders of elements in $X$; two distinct primes $p$ and $q$ are adjacent in $\Gamma(X)$ if and only if $X$ contains an element of order $pq$.
Denote by $\mathrm{V}(\mathbb{Z}G)$ the group of ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.6.
Proposed by: A. B{\"a
Discussion and literature:
chle, L. Margolis} | 2 | open | A. B{\"a | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,665 | KOU-20.7 | Kourovka Notebook Problem 20.7 | (W. Boone and G. Higman) Does every finitely generated group with solvable word problem embed into a finitely presented simple group? It is known that every such group embeds into a simple subgroup of a finitely presented group (W. Boone, G. Higman, *J. Austral. Math. Soc.*, **18**, no. 1 (1974), 41--53). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.7.
Proposed by: J. Belk | 2 | open | J. Belk | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,666 | KOU-20.8 | Kourovka Notebook Problem 20.8 | Suppose $K < H < F$ are free groups of finite rank such that
$\operatorname{rank}(H) < \operatorname{rank}(K)$, but all proper subgroups of $H$ which contain $K$ have ranks ${}\geq \operatorname{rank}(K)$. Then is the inclusion of $H$ in $F$ the only homomorphism $H\rightarrow F$ fixing all elements of $K$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.8.
Proposed by: G. M. Bergman | 2 | open | G. M. Bergman | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,667 | KOU-20.9 | Kourovka Notebook Problem 20.9 | In the group algebra of a free group over a field, does every element whose
support in the group has cardinality more than 1 generate a proper 2-sided ideal?
This question is one of several related
questions in (G. M. Bergman, *Commun. Algebra*, **49**, no. 9 (2021), 3760--3776). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.9.
Proposed by: G. M. Bergman | 2 | open | G. M. Bergman | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,668 | KOU-20.10 | Kourovka Notebook Problem 20.10 | (a) If ${\mathscr U}$ and ${\mathscr U'}$ are nonprincipal ultrafilters on $\mathbb{N}$, can every group
which can be written as a homomorphic image of an
ultraproduct of groups with respect to ${\mathscr U}$ also be written as a homomorphic image of an
ultraproduct of groups with respect to ${\mathscr U'}$?
(b)
If th... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.10.
Proposed by: G. M. Bergman
Discussion and literature:
[Solved] (b) The affirmative answer is consistent with the ZFC axioms of set theory (S. M. Corson, ... | 2 | partially_solved | G. M. Bergman | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] (b) The affirmative answer is consistent with the ZFC axioms of set theory (S. M. Corson, *Pre... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,669 | KOU-20.11 | Kourovka Notebook Problem 20.11 | Let $F \leq H$ be free groups such that there exists a free group
$G$ of finite rank with $F\leq G\leq H$, and let $r$
be the least of the ranks of such groups $G$. Which,
if any, of the following statements must hold?
(i) There is a largest $G$ of rank $r$ between $F$ and $H$.
(i$'$) For any two $G_1$, $G_2$ of rank... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.11.
Proposed by: G. M. Bergman | 2 | open | G. M. Bergman | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,670 | KOU-20.12 | Kourovka Notebook Problem 20.12 | Let us say that a group $G$ has the *unique $n$-fold product* property (u.-$n$-p.)
if for every $n$-tuple of finite nonempty subsets $A_1, \dots , A_n\subseteq G$ there
exists $g\in G$ which can be written in one and only one way as $g = a_1\cdots a_n$ with $a_i\in A_i$. It is easy to see that u.-$n$-p. implies u.-$m$-... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.12.
Proposed by: G. M. Bergman | 2 | open | G. M. Bergman | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,671 | KOU-20.13 | Kourovka Notebook Problem 20.13 | {(a)} If an abelian group can be written as a homomorphic
image of a nonprincipal countable ultraproduct of *not necessarily* abelian groups $G_i$, must
it be a homomorphic image of a nonprincipal countable ultraproduct of abelian groups? See (G. M. Bergman, *Pacific J. Math.*, **274** (2015), 451--495).
(b) If an abe... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.13.
Proposed by: G. M. Bergman
Discussion and literature:
[Solved] (a) Yes, it must (S. M. Corson, *Preprint*, 2025, https://arxiv.org/pdf/2503.09228).
[Sol... | 2 | solved | G. M. Bergman | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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) Yes, it must (S. M. Corson, *Preprint*, 2025, https://arxiv.org/pdf/2503.09228).\n\n[Solved] (b) Yes... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,672 | KOU-20.14 | Kourovka Notebook Problem 20.14 | (a) Do there exist a variety $\mathfrak{V}$ of groups and a
group $G\in \mathfrak{V}$ such that the coproduct in $\mathfrak{V}$ of two copies of $G$ is embeddable in $G$, but the coproduct of three such copies is not? See (G. M. Bergman, *Indag. Math.*, **18** (2007), 349--403).
Given an embedding $G *_\mathfrak{V} G ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.14.
Proposed by: G. M. Bergman | 2 | open | G. M. Bergman | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,673 | KOU-20.15 | Kourovka Notebook Problem 20.15 | Let $\kappa$ be an infinite cardinal. If a residually finite
group $G$ is embeddable in the full permutation group of a set of cardinality $\kappa$, must it be
embeddable in the direct product of $\kappa$ finite groups? See (G. M. Bergman, *Indag. Math.*, **18** (2007), 349--403).
The converse is true: any such direct... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.15.
Proposed by: G. M. Bergman | 2 | open | G. M. Bergman | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,674 | KOU-20.16 | Kourovka Notebook Problem 20.16 | By Hartley's theorem (based on CFSG), if a simple locally finite group does not
contain a particular finite group as a section, then it is isomorphic to a group of Lie type over a locally finite field.
a) Is the same true for an infinite
definably simple locally finite group? A group is said to be *definably simple* i... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.16.
Proposed by: A. V. Borovik | 2 | open | A. V. Borovik | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,675 | KOU-20.17 | Kourovka Notebook Problem 20.17 | The *normal covering number* $\gamma(G)$ of a finite non-cyclic group $G$ is the minimum number of proper subgroups of $G$ such that $G$ is the union of their conjugates.
(ae) What is the exact value of \ $\displaystyle\liminf_{n\ \mathrm{even}}\gamma(S_n)/n$?
(ao) What is the exact value of \ $\displaystyle\liminf_{... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.17.
Proposed by: D. Bubboloni, C. E. Praeger, P. Spiga
Discussion and literature:
[Solved] (ae) It is $1/6$ (S. Eberhard, C. Mellon, *Bull. London Math. Soc.... | 2 | partially_solved | D. Bubboloni, C. E. Praeger, P. Spiga | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] (ae) It is $1/6$ (S. Eberhard, C. Mellon, *Bull. London Math. Soc.* (2025), https://doi.org/10... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,676 | KOU-20.18 | Kourovka Notebook Problem 20.18 | Let $R_{p^k}$ be the variety of class $2$ nilpotent groups of exponent
$p^k$, where $p$ is a prime number and $k\geq 2$. It is true that for every $p$ and $k$ there are infinitely many subquasivarieties of $R_{p^k}$ each of which is generated by a finite group with derived subgroup of exponent $p^k$ and does not have a... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.18.
Proposed by: A. I. Budkin | 2 | open | A. I. Budkin | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,677 | KOU-20.19 | Kourovka Notebook Problem 20.19 | A subgroup $H$ of a group $G$ is called *commensurated* if for all $g\in G$, the index $|H : H \cap gHg^{-1}|$ is finite. Can a non-abelian free group (or a non-elementary hyperbolic group) contain two infinite commensurated subgroups $A$, $B$ with a trivial intersection? The answer is negative if $A$ or $B$ is normal. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.19.
Proposed by: P.-E. Caprace | 2 | open | P.-E. Caprace | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,678 | KOU-20.20 | Kourovka Notebook Problem 20.20 | Let $E(l,m,n)$ be a group on generators $a,b$ with the presentation
$\langle a,b\mid a^l= 1,\;(ab)^m=b^n\rangle$, where $l,m,n$ satisfy $1/l + 1/m + 1/n < 1$.
Are there any values for $l,m,n$ such that $E(l,m,n)$ is shortlex automatic?
It is known that $E(6,2,6)$ is not shortlex automatic. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.20.
Proposed by: C. Chalk | 2 | open | C. Chalk | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,679 | KOU-20.21 | Kourovka Notebook Problem 20.21 | (G. Verret).
Does there exist a finite group $G$ with two normal subgroups $K$ and $L$, each with index 12 in $G$,
such that $K$ is isomorphic to $L$, but $G/K$ is isomorphic to $C_{12}$, while $G/L$ is isomorphic to $A_4$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.21.
Proposed by: M. Conder | 2 | open | M. Conder | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,680 | KOU-20.22 | Kourovka Notebook Problem 20.22 | Let $G$ be a non-free and non-cyclic one-relator group in which every subgroup of infinite index is free. Is $G$ a surface group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.22.
Proposed by: B. Fine, G. Rosenberger, L. Wienke
Discussion and literature:
[Solved] Yes, it is (H. Wilton, *Preprint*, 2024, https://arxiv.org/abs/2406.0... | 2 | solved | B. Fine, G. Rosenberger, L. Wienke | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (H. Wilton, *Preprint*, 2024, https://arxiv.org/abs/2406.02121)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,681 | KOU-20.23 | Kourovka Notebook Problem 20.23 | Is there a constant $\delta$ with $0 < \delta < 1$ and an integer $N$
such that whenever $A$, $B$, and $C$ are conjugacy classes in the
alternating group $Alt(n)$ each of size at least $|Alt(n)|^{\delta}$
with $n \geq N$, then $ABC = Alt(n)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.23.
Proposed by: M. Garonzi, A. Mar\'oti
Discussion and literature:
[Solved] Yes, there are (D. Dona, *Preprint*, 2025, https://arxiv.org/abs/2505.06012). | 2 | solved | M. Garonzi, A. Mar\'oti | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (D. Dona, *Preprint*, 2025, https://arxiv.org/abs/2505.06012)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,682 | KOU-20.24 | Kourovka Notebook Problem 20.24 | Does the group $SL_2(\mathbb{Z} [\sqrt 2])$ admit a faithful transitive
amenable action?
This is the same as asking if it admits any co-amenable subgroup except
the finite index subgroups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.24.
Proposed by: Y. Glasner, N. Monod | 2 | open | Y. Glasner, N. Monod | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,683 | KOU-20.25 | Kourovka Notebook Problem 20.25 | Let $n=k^2$. Let $\sigma\in S_n$ be the product of $k$ disjoint cycles, of lengths
$1,3,5,\dots , 2k-1$. Find the limiting proportion of odd entries in the
$\sigma$-column of the character table of $S_n$.
For some background, see pages 1005--1006 in (D. Gluck, *Proc. Amer. Math. Soc.*, **147** (2019), 1005--1011). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.25.
Proposed by: D. Gluck | 2 | open | D. Gluck | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,684 | KOU-20.26 | Kourovka Notebook Problem 20.26 | Let $\phi$ be the Euler totient function. Does there exist a constant $a > 0$ such that $| Aut(G)| \geq \phi (|G|)^a$
a) for every finite group $G$?
b) for every finite nilpotent group $G$?
If such a constant $a$ exists, then $a\le \frac{40}{41}$ (J. Gonz\'alez-S\'anchez, A. Jaikin-Zapirain, *Forum Math. Sigma*, **3... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.26.
Proposed by: J. Gonz\'alez-S\'anchez, A. Jaikin-Zapirain | 2 | open | J. Gonz\'alez-S\'anchez, A. Jaikin-Zapirain | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,685 | KOU-20.27 | Kourovka Notebook Problem 20.27 | Let $G$ be a finite group, $p$ a prime number, and let $|x^G|_p$ denote the maximum power of $p$ that divides the class size of an element $x\in G$. Suppose that there exists a $p$-element $g\in G$ such that $|g^G|_p=\max_{x\in G}|x^G|_p$. Is it true that $G$ has a normal $p$-complement?
A partial answer is in (https:... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.27.
Proposed by: I. B. Gorshkov
Discussion and literature:
[Solved] No, not necessarily, a counterexample is given by \verb#SmallGroup(192,945)#
(B. Sambale,... | 2 | solved | I. B. Gorshkov | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, a counterexample is given by \\verb#SmallGroup(192,945)#\n(B. Sambale, *Letter of 1... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,686 | KOU-20.28 | Kourovka Notebook Problem 20.28 | Let $L$ be a non-abelian finite simple group, and let $H(L)=M(L).L$ be the universal perfect central extension, where $M(L)$ is the Schur multiplier of $L$. Suppose that $G$ is a finite group such that the set of class sizes of $G$ is the same as the set of class sizes of $H(L)$. Is it true that $G\simeq H(L)\times A$,... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.28.
Proposed by: I. B. Gorshkov | 2 | open | I. B. Gorshkov | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,687 | KOU-20.29 | Kourovka Notebook Problem 20.29 | Let $S$ be a non-abelian finite simple group. Is it true that for any $n\in \mathbb{N}$, if the set of class sizes of a centreless finite group $G$ is the same as the set of class sizes of the direct power $S^n$, then $G\simeq S^n$?
This is proved for $S=Alt_5^2$ in (*J. Algebra Appl.*, **21**, no. 11 (2022), Article ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.29.
Proposed by: I. B. Gorshkov | 2 | open | I. B. Gorshkov | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,688 | KOU-20.30 | Kourovka Notebook Problem 20.30 | (P. M. Neumann and M. R. Vaughan-Lee).
Let $G$ be a perfect and centreless finite group, and let $n$ be the maximum size of a conjugate class in $G$. Is it true that $|G|\leq n^2$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.30.
Proposed by: I. B. Gorshkov
Discussion and literature:
[Solved] No, not always (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.co... | 2 | solved | I. B. Gorshkov | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/wp-content/up... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,689 | KOU-20.31 | Kourovka Notebook Problem 20.31 | Let $\omega(G)$ denote the set of element orders of a finite group $G$,
and $h(G)$ the number of pairwise nonisomorphic finite groups $H$ with $\omega(H)=\omega(G)$. Find $h(L)$, where
a)\; $L$ is the symmetric group of degree $10$;
b)\; $L$ is the automorphism group of the simple sporadic
Janko group $J_2$;
c)\; $P... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.31.
Proposed by: M. A. Grechkoseeva, A. V. Vasil'ev | 2 | open | M. A. Grechkoseeva, A. V. Vasil'ev | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,690 | KOU-20.32 | Kourovka Notebook Problem 20.32 | (E. Rapaport Strasser). The Lov\'asz conjecture states that a vertex-transitive connected graph is Hamiltonian. In the special case of Cayley graphs, one can ask the following. Consider a set $A$ which generates a group $G$ and is symmetric ($x\in A$ implies $x^{-1} \in A$). Is there a list $a_1,a_2,\dots, a_n$ of elem... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.32.
Proposed by: B. Green | 2 | open | B. Green | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,691 | KOU-20.33 | Kourovka Notebook Problem 20.33 | (A. Bauer).
Does Higman's Embedding Theorem relativize in the following way? Is it the case that for every subset $X \subseteq \mathbb{N}$, there is a finitely generated group $G_X$ that has an $X$-computable presentation (that is, there is a finite generating set relative to which the set of relations is computably en... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.33.
Proposed by: J. Grochow | 2 | open | J. Grochow | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,692 | KOU-20.34 | Kourovka Notebook Problem 20.34 | Let $f$ be an inner screen of a saturated formation $\mathfrak{F}$ and suppose that a finite group $A$ acts faithfully and $f$-hypercentrally on a finite group $G$. Is it true that $G\rtimes A\in \mathfrak{F}$ for all $G\in \mathfrak{F}$ if and only if $\mathfrak{F}$ is a Fitting class? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.34.
Proposed by: W. Guo | 2 | open | W. Guo | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,693 | KOU-20.35 | Kourovka Notebook Problem 20.35 | (Well-known problem).
Let $S$ be a connected orientable hyperbolic surface of finite type and complexity at least 2. Does its mapping class group ${MCG}(S)$ have a non-elementary hyperbolic quotient?
An affirmative answer is known when $S$ is a closed genus-2 surface (https://arxiv.org/pdf/2005.00567.pdf), and in the ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.35.
Proposed by: M. Hagen | 2 | open | M. Hagen | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,694 | KOU-20.36 | Kourovka Notebook Problem 20.36 | The free Burnside group of exponent four on three generators,
$B(3,4)$, has order $2^{69}$ as shown by Bayes, Kautsky, and Wamsley (1974).
Their proof is based on a theorem of Sanov (1940) which shows
that $B(n,4)$ is finite. Sanov's proof for $B(3,4)$ uses more than $2^{32}$ fourth powers,
because the subgroup of $B(3... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.36.
Proposed by: G. Havas, M. F. Newman | 2 | open | G. Havas, M. F. Newman | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,695 | KOU-20.37 | Kourovka Notebook Problem 20.37 | Is it true that for every finite group $G$ and every factorization $|G|=ab$ there exist subsets $A,B\subseteq G$ with $|A|=a$ and $|B|=b$ such that $G=AB$? Cf. 19.35 in Archive.
*Comment of 2025*:
any minimal counterexample $G$ must be a simple group without prime-index subgroups (R. R. Bildanov, V. A. Goryachenko, A.... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.37.
Proposed by: M. H. Hooshmand | 2 | open | M. H. Hooshmand | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,696 | KOU-20.38 | Kourovka Notebook Problem 20.38 | Suppose that $H$ is an almost simple group of Lie type and $G$ is a finite group such that
$G$ and $H$ have the same sets of degrees of irreducible complex characters. Must there exist an abelian normal subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$?
*Comment of 2025*: The conjecture was confirmed for projec... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.38.
Proposed by: A. Iranmanesh | 2 | open | A. Iranmanesh | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,697 | KOU-20.39 | Kourovka Notebook Problem 20.39 | Let $F$ be a non-abelian finitely generated free group, $1\ne w\in F$, and $n\ge 1$. Is the group $\langle F,t \mid t^n=w \rangle $ linear of degree 2 over a field of characteristic 0 if $w$ is not a proper power in $F$?
This question is motivated by the following well-known question: is it true that the free $\mathbb... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.39.
Proposed by: A. Jaikin-Zapirain | 2 | open | A. Jaikin-Zapirain | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,698 | KOU-20.40 | Kourovka Notebook Problem 20.40 | Let $G$ be a $\kappa$-existentially closed group of cardinality $\lambda > \kappa$, where $\kappa$ is a regular cardinal. Is it true that $|Aut(G)|=2^{\lambda}$?
The answer is known to be affirmative if $\lambda=\kappa$ (B. Kaya, M. Kuzucuo\u{g}lu, P. Longobardi, M. Maj,
*J. Algebra*, **666** (2025), 840--849). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.40.
Proposed by: B. Kaya, M. Kuzucuo\u{g
Discussion and literature:
lu} | 2 | open | B. Kaya, M. Kuzucuo\u{g | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,699 | KOU-20.41 | Kourovka Notebook Problem 20.41 | (L. Ciobanu, B. Fine, G. Rosenberger). Suppose that $G$ is a non-cyclic residually finite group in which every subgroup of finite index (including the group itself) is defined by a single defining relation, while all infinite index subgroups are free. Is it true that $G$ is either free or isomorphic to the fundamental ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.41.
Proposed by: D. Kielak
Discussion and literature:
[Solved] Yes, it is true (H. Wilton, *Preprint*, 2024, https://arxiv.org/abs/2406.02121). | 2 | solved | D. Kielak | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (H. Wilton, *Preprint*, 2024, https://arxiv.org/abs/2406.02121)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,700 | KOU-20.42 | Kourovka Notebook Problem 20.42 | (Y. Cornulier, A. Mann, A. Thom).
Does there exist a finitely generated residually finite group that is not (elementary) amenable and satisfies a nontrivial group law? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.42.
Proposed by: S. Kionke | 2 | open | S. Kionke | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,701 | KOU-20.43 | Kourovka Notebook Problem 20.43 | Is every family of finite groups satisfying a common group law uniformly amenable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.43.
Proposed by: S. Kionke, E. Schesler | 2 | open | S. Kionke, E. Schesler | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,702 | KOU-20.44 | Kourovka Notebook Problem 20.44 | The definition of \(\mathrm{CT}(\mathbb{Z})\) is given in 17.57. Is it true that a finitely generated subgroup of \(\mathrm{CT}(\mathbb{Z})\) either has
only finitely many orbits on \(\mathbb{Z}\) or there is a set of representatives for its orbits
on \(\mathbb{Z}\) which has positive density? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.44.
Proposed by: S. Kohl | 2 | open | S. Kohl | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,703 | KOU-20.45 | Kourovka Notebook Problem 20.45 | Let \(n\) be a positive integer, and let \(G \leq \mathrm{GL}(n,\mathbb{Z})\)
be finitely generated. Given a bound \(b \in \mathbb{N}\), let \(e_b\) be
the number of elements of \(G\) all of whose matrix entries have absolute value
\(\leq b\). Does the limit \ \(\displaystyle\lim_{b \rightarrow \infty} \ln{e_b}/\ln{b}\... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.45.
Proposed by: S. Kohl | 2 | open | S. Kohl | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,704 | KOU-20.46 | Kourovka Notebook Problem 20.46 | A group action on a compact space is said to be *topologically free* if the set of points with trivial stabilizer is dense. Let $G$ be a locally compact group, and $\partial_{sp} G$ its Furstenberg boundary (the largest minimal and strongly proximal compact $G$-space). Let $\Gamma_1$ and $\Gamma_2$ be two lattices in $... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.46.
Proposed by: A. Le Boudec | 2 | open | A. Le Boudec | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,705 | KOU-20.47 | Kourovka Notebook Problem 20.47 | Let $G = F_n$ be a finitely generated free group. Let $X$ be a compact $G$-space on which the $G$-action is faithful, minimal, and strongly proximal. Does it follow that the action is topologically free? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.47.
Proposed by: A. Le Boudec, N. Matte Bon | 2 | open | A. Le Boudec, N. Matte Bon | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,706 | KOU-20.48 | Kourovka Notebook Problem 20.48 | Suppose that $P$ is a finite $p$-group with a non-trivial partition (which is equivalent to having proper Hughes subgroup $H_p(P):=\langle g\in P\mid g^p\ne 1\rangle\ne P$). If $P$ admits a fixed-point-free automorphism of prime order, must $P$ be of exponent $p$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.48.
Proposed by: M. Lewis | 2 | open | M. Lewis | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,707 | KOU-20.49 | Kourovka Notebook Problem 20.49 | Is it true that any finite group contains a 2-generated subgroup with the
same exponent?
An affirmative answer is known for soluble groups. It is also known that any finite group contains a 3-generated subgroup with the same exponent (E. Detomi, A. Lucchini, *J. London Math. Soc. (2)*, **87**, no. 3 (2013), 689--706). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.49.
Proposed by: A. Lucchini | 2 | open | A. Lucchini | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,708 | KOU-20.50 | Kourovka Notebook Problem 20.50 | For a positive integer $m$, let $G_m$ be the largest group generated by $m$ involutions such that $(xy)^4=1$ for any two involutions $x,y\in G_m$. What is the order of $G_4$?
It is known that the order of $G_3$ is equal to $2^{11}$. Also cf. 18.58. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.50.
Proposed by: D. V. Lytkina | 2 | open | D. V. Lytkina | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,709 | KOU-20.51 | Kourovka Notebook Problem 20.51 | Is it true that for every odd integer $t > 3$ there exists a finite non-abelian group $G$ of odd order with exactly $t$ conjugacy classes? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.51.
Proposed by: D. MacHale | 2 | open | D. MacHale | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,710 | KOU-20.52 | Kourovka Notebook Problem 20.52 | A famous result of Burnside states that if $k(G)$ is the number of conjugacy classes of a finite group $G$ of odd order, then $|G|- k(G)$ is divisible by 16. Is it true that for every integer $m > 0$ there exists a finite non-abelian group $G(m)$ of odd order such that $|G(m)| - k(G(m)) = 16m$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.52.
Proposed by: D. MacHale | 2 | open | D. MacHale | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,711 | KOU-20.53 | Kourovka Notebook Problem 20.53 | A nonisotropic unitary graph $\Gamma$ is distance-regular with intersection array $\{q(q-1), (q+1)(q-2), q+1; 1, 1, q(q-2)\}$ for some prime power $q$. The group $G=Aut (\Gamma)$ acts transitively on the vertex set and on the edge set of $\Gamma$.
It is known that $\Gamma$ is distance-transitive if $q=3$. Does there ex... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.53.
Proposed by: A. A. Makhnev | 2 | open | A. A. Makhnev | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,712 | KOU-20.54 | Kourovka Notebook Problem 20.54 | A distance-regular graph of diameter 3 with the second eigenvalue $\theta_1=a_3$ is called a *Shilla graph*.
For a Shilla graph $\Gamma$ the number $a=a_3$ divides $k$ and we set $b=b(\Gamma)=k/a$. Koolen and Park proved that there are 12 feasible intersection arrays of Shilla graphs with $b=3$. At present it is proved... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.54.
Proposed by: A. A. Makhnev, D. O. Revin | 2 | open | A. A. Makhnev, D. O. Revin | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,713 | KOU-20.55 | Kourovka Notebook Problem 20.55 | Are there soluble finite groups $G$ and $H$, of derived lengths 2 and 4 and having identical character tables?
Pairs of nonisomorphic soluble finite groups with identical character tables and with derived
lengths $n$ and $n+1$ for any $n\ge 2$ were constructed in (S. Mattarei, *J. Algebra*, **175** (1995) 157--178). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.55.
Proposed by: S. Mattarei | 2 | open | S. Mattarei | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,714 | KOU-20.56 | Kourovka Notebook Problem 20.56 | Is a periodic group $G$ locally finite if it is generated by involutions and the centralizer of every involution in $G$ is locally finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.56.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,715 | KOU-20.57 | Kourovka Notebook Problem 20.57 | Is a periodic group $G$ locally finite if every finite subgroup of $G$ is contained in a subgroup of $G$ isomorphic to a finite alternating group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.57.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,716 | KOU-20.58 | Kourovka Notebook Problem 20.58 | Let $\omega(G)$ denote the set of element orders of a finite group $G$.
A finite group $G$ is said to be *recognizable* (*by spectrum*)
if every finite group $H$ with $\omega(H)=\omega(G)$ is isomorphic to $G$.
a)
Is it true that for every $n$ there is a recognizable group that is the $n$-th direct power of a nonabeli... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.58.
Proposed by: V. D. Mazurov, A. V. Vasil'ev
Discussion and literature:
[Solved] a) Yes, it is true (N. Yang, I. Gorshkov, A. Staroletov, A. V. Vasil'ev, *... | 2 | partially_solved | V. D. Mazurov, A. V. Vasil'ev | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] a) Yes, it is true (N. Yang, I. Gorshkov, A. Staroletov, A. V. Vasil'ev, *Annali Matem. Pura A... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,717 | KOU-20.59 | Kourovka Notebook Problem 20.59 | A subgroup $H$ is called a *virtual retract* of a group $G$ if $H$ is a retract of a finite-index subgroup of $G$. Is it true that every finitely generated subgroup of a finitely generated virtually free group is a virtual retract?
For motivation and partial results see (A. Minasyan, *Int. Math. Res. Notes*, **2021**,... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.59.
Proposed by: A. Minasyan | 2 | open | A. Minasyan | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,718 | KOU-20.60 | Kourovka Notebook Problem 20.60 | We say that $G$ is a *virtually compact special group* if $G$ has a finite-index subgroup which is isomorphic to the fundamental group of a compact special complex (in the sense of F. Haglund, D. T. Wise, *Geom. Funct. Anal.*, **17**, no. 5 (2008), 1551--1620).
Let $G$ be a virtual retract of a finitely generated right... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.60.
Proposed by: A. Minasyan | 2 | open | A. Minasyan | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,719 | KOU-20.61 | Kourovka Notebook Problem 20.61 | Suppose that $G$ is a virtually compact special group. Is it true that the centralizer of any element in $G$ is itself virtually compact special? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.61.
Proposed by: A. Minasyan | 2 | open | A. Minasyan | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,720 | KOU-20.62 | Kourovka Notebook Problem 20.62 | (J. Dixmier)
A group is said to be *unitarisable* if its every uniformly bounded representation on a Hilbert space is
unitarisable. Is every unitarisable
group amenable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.62.
Proposed by: N. Monod | 2 | open | N. Monod | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,721 | KOU-20.63 | Kourovka Notebook Problem 20.63 | a) Prove that every unitarisable group has trivial cost.
This is known to be true for residually finite groups (I. Epstein, N. Monod, *Int. Math. Res. Notes*, **2009**, no. 22 (2009), 4336--4353).
b) At least prove that every unitarisable group has vanishing
first $L^2$ Betti number. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.63.
Proposed by: N. Monod | 2 | open | N. Monod | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,722 | KOU-20.64 | Kourovka Notebook Problem 20.64 | Let $G$ be a non-amenable group.
a) Prove that $G^n$ is non-unitarisable for some $n$.
b) At least prove that $G^{\infty}$, the
direct sum (restricted product), is non-unitarisable. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.64.
Proposed by: N. Monod | 2 | open | N. Monod | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,723 | KOU-20.65 | Kourovka Notebook Problem 20.65 | Given a complex irreducible character $\chi\in \rm{Irr}(G)$ of a finite group $G$, let $\mathbb{Q}(\chi)$ denote the field extension of $\mathbb{Q}$ obtained by adjoining to $\mathbb{Q}$ all the values of $\chi$. We say that a finite group $G$ is $k$-*rational* if $|\mathbb{Q}(\chi):\mathbb{Q}|$ divides $k$ for every $... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.65.
Proposed by: A. Moret\'o | 2 | open | A. Moret\'o | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,724 | KOU-20.66 | Kourovka Notebook Problem 20.66 | A Schmidt $(p, q)$-group is a finite non-nilpotent group all of whose proper subgroups are nilpotent and whose Sylow $p$-subgroup is normal.
The $N$-critical graph $\Gamma_{Nc}(G)$ of a finite group $G$ is a directed graph on the vertex set
of all prime divisors of $|G|$ in which $(p, q)$ is an edge of $\Gamma_{Nc}(G)$... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.66.
Proposed by: V. I. Murashka, A. F. Vasil'ev | 2 | open | V. I. Murashka, A. F. Vasil'ev | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,725 | KOU-20.67 | Kourovka Notebook Problem 20.67 | Suppose that $G$ is a finite group, $p$ is
a prime, and $B$ is a Brauer $p$-block of $G$
with defect group $D$. Let $\mathrm{cd}(B)$ be the
set of degrees of the irreducible complex characters in $B$.
a) Is it true that the derived length of $D$ is bounded by $|\mathrm{cd}(B)|$?
b) (A. Jaikin-Zapirain). Is is even tr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.67.
Proposed by: G. Navarro | 2 | open | G. Navarro | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,726 | KOU-20.68 | Kourovka Notebook Problem 20.68 | The *submonoid membership problem* for a group $G$ generated by an alphabet $A$ asks, for given words $x_1, x_2, \dots , x_n$ and a word $w$ over $A$, whether $w$ belongs to the submonoid of $G$ generated by the $x_i$. Does there exist a hyperbolic one-relator group with undecidable submonoid membership problem? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.68.
Proposed by: C.-F. Nyberg-Brodda | 2 | open | C.-F. Nyberg-Brodda | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,727 | KOU-20.69 | Kourovka Notebook Problem 20.69 | Is the submonoid membership problem decidable for every one-relator group with torsion? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.69.
Proposed by: C.-F. Nyberg-Brodda | 2 | open | C.-F. Nyberg-Brodda | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,728 | KOU-20.70 | Kourovka Notebook Problem 20.70 | As a strengthening of the Burnside restriction, for every pair $(k,n)$ of positive integers, let a group $G$ satisfy condition $C_{k,n}$ if every $k$-generated subgroup of $G$ is finite of order at most $n$.
a) Does there exist $k\ge 2$ such that for any $n$ all groups with condition $C_{k,n}$ are locally finite?
b) ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.70.
Proposed by: A. Yu. Olshanskii | 2 | open | A. Yu. Olshanskii | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,729 | KOU-20.71 | Kourovka Notebook Problem 20.71 | (J. Lauri).
A *card* of a finite simple undirected graph $G$ of order $n=|V(G)|$ is an induced subgraph of order $n-1$. For a connected graph $G$ with $k$ isomorphism types of cards, can $Aut(G)$ have more than $k$ orbits on the vertex set $V(G)$
a) for $k=2$?
b) for $k=3$?
c)
for $k=4$?
If a graph $G$ has $k$ isom... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.71.
Proposed by: V. Pannone
Discussion and literature:
[Solved] c) Yes, it can for $k=4$ (J. Prajapati, *Preprint*, 2026,
https://kourovkanotebookorg.wordpr... | 2 | partially_solved | V. Pannone | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] c) Yes, it can for $k=4$ (J. Prajapati, *Preprint*, 2026,\n\nhttps://kourovkanotebookorg.wordp... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,730 | KOU-20.72 | Kourovka Notebook Problem 20.72 | Is there a variety of groups $\Theta$ such that the group of automorphisms of the category of free finitely generated groups $\Theta^0$ contains outer automorphisms? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.72.
Proposed by: E. Plotkin
Discussion and literature:
[Solved] Yes, there is (R. Barbosa-Fernandes, A. Tsurkov, *Preprint*, 2019, https://arxiv.org/abs/1909... | 2 | solved | E. Plotkin | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, there is (R. Barbosa-Fernandes, A. Tsurkov, *Preprint*, 2019, https://arxiv.org/abs/1909.05955)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,731 | KOU-20.73 | Kourovka Notebook Problem 20.73 | Can every countable group $G$ be factorized $G=AB$ into infinite subsets $A, B$ such that every
element $g\in G$ has a unique representation $g = ab$ for $a\in A$, $b \in B$?
This is true if $G$ is topologizable. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.73.
Proposed by: I. V. Protasov | 2 | open | I. V. Protasov | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,732 | KOU-20.74 | Kourovka Notebook Problem 20.74 | a) *Conjecture*: there are only finitely many nonabelian finite simple groups $G$ that have a normal subset $S$ closed under inversion such that $|S | > |G|/\log_2|G|$ and $S^2\ne G$. *Comment of 2024*: This is true for the class of alternating groups (M. Larsen, P. H. Tiep, *Preprint*, 2023, arXiv:2305.04806) and for ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.74.
Proposed by: L. Pyber
Discussion and literature:
[Solved] a) No, there are infinitely many such groups (O. Sheinfeld, *Preprint*, 2026,
https://arxiv.org... | 2 | solved | L. Pyber | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] a) No, there are infinitely many such groups (O. Sheinfeld, *Preprint*, 2026,\nhttps://arxiv.org/abs/260... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,733 | KOU-20.75 | Kourovka Notebook Problem 20.75 | Let $G$ be a finite $p$-group and assume that all abelian normal
subgroups of $G$ can be generated by $k$ elements.
Is it true that every abelian subgroup of $G$ can be generated by $2k$ elements?
The $k$-th direct power of $D_{16}$ shows that this bound would be best
possible. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.75.
Proposed by: L. Pyber
Discussion and literature:
[Solved] No, it is not (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/wp-co... | 2 | solved | L. Pyber | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it is not (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/wp-content/upl... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,734 | KOU-20.76 | Kourovka Notebook Problem 20.76 | Let $G$ be a finite $p$-group and assume that all abelian normal
subgroups of $G$ have order at most $p^k$.
Is it true that every abelian subgroup of $G$ has order at most $p^{2k}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.76.
Proposed by: L. Pyber
Discussion and literature:
[Solved] No, it is not (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/wp-co... | 2 | solved | L. Pyber | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it is not (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/wp-content/upl... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,735 | KOU-20.77 | Kourovka Notebook Problem 20.77 | (O. I. Tavgen').
A group is said to be *boundedly generated* if it is a
product of finitely many cyclic groups.
Is it true that every boundedly generated residually finite group is linear over a field
of characteristic zero?
This is true for residually finite-soluble groups (L. Pyber, D. Segal, *J. Reine Angew. Math.*... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.77.
Proposed by: L. Pyber | 2 | open | L. Pyber | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,736 | KOU-20.78 | Kourovka Notebook Problem 20.78 | For an irreducible complex character $\chi$ of a finite group
$G$, the *codegree* of $\chi$ is defined by $\mathrm{cod}(\chi)=|G:\ker\chi|/\chi(1)$. Let $\mathrm{Cod}(G)$ be the set of
irreducible character codegrees of $G$.
*Conjecture*: If $G$ has an
element of order $m$, then $m$ divides some member of $\mathrm{Cod}... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.78.
Proposed by: G. Qian | 2 | open | G. Qian | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,737 | KOU-20.79 | Kourovka Notebook Problem 20.79 | *Conjecture*: Suppose that $G$ is a non-abelian simple group and $H$ is a
finite group such that $\mathrm{Cod}(G)=\mathrm{Cod}(H)$. Then $G\cong
H$.
*Comment of 2025*: The conjecture is proved for
$G$ of type $\mathrm{PSL}(2,q)$ (A. Bahri, Z. Akhlaghi, B. Khosravi, *Bull. Austral. Math. Soc.*, **104**, no. 2 (2021), 2... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.79.
Proposed by: G. Qian | 2 | open | G. Qian | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,738 | KOU-20.80 | Kourovka Notebook Problem 20.80 | Let $G$ be a finite group, and $\chi$ an irreducible complex
character of $G$. We call $\chi$ a $\mathcal{P}$-character if
$\chi$ is a constituent of $(1_H)^G$ for some maximal subgroup $H$
of $G$; and $\chi$ is said to be monomial if $\chi$ is induced by a
linear character of a subgroup of $G$.
*Conjecture*: $G$ is s... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.80.
Proposed by: G. Qian | 2 | open | G. Qian | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,739 | KOU-20.81 | Kourovka Notebook Problem 20.81 | Does there exist a class $\mathfrak{X}$ of finite groups satisfying the following conditions:
(1) $\mathfrak{X}$ is closed with respect to taking subgroups and homomorphic images;
(2) the product of two normal $\mathfrak{X}$-subgroups of an arbitrary group is always an $\mathfrak{X}$-group;
(3) for every positive in... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.81.
Proposed by: D. O. Revin | 2 | open | D. O. Revin | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,740 | KOU-20.82 | Kourovka Notebook Problem 20.82 | Two groups are said to be *isospectral* if they have the same set of element
orders. Suppose that $G$ is a finite group such that every finite group isospectral to $G$ is isomorphic to $G$. Is it true that the quotient of $G$ by its socle is solvable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.82.
Proposed by: D. O. Revin | 2 | open | D. O. Revin | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,741 | KOU-20.83 | Kourovka Notebook Problem 20.83 | Let $\mathfrak{X}$ be a class of finite groups that is closed with respect to taking subgroups, homomorphic images, and extensions. A subgroup $H$ of a finite group $G$ is said to be \emph{$\mathfrak{X}$-submaximal} if there exists an embedding of $G$ into a group $G^*$ such that $G$ is subnormal in $G^*$ and $H$ coinc... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.83.
Proposed by: D. O. Revin, A. V. Zavarnitsine | 2 | open | D. O. Revin, A. V. Zavarnitsine | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,742 | KOU-20.84 | Kourovka Notebook Problem 20.84 | Does there exist a nilpotent group of class 3 with a non-trivial 5-th dimension subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.84.
Proposed by: E. Rips | 2 | open | E. Rips | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,743 | KOU-20.85 | Kourovka Notebook Problem 20.85 | Let $F$ be a free group. An element $\omega\in F$ is said to be *primitive* if there is a minimal generating system of $F$ that contains $\omega$, *almost primitive* if it is primitive in each finitely generated proper subgroup of $F$ containing $\omega$, *tame almost primitive* if, whenever $\omega^\alpha$ is containe... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.85.
Proposed by: G. Rosenberger, L. Wienke | 2 | open | G. Rosenberger, L. Wienke | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,744 | KOU-20.86 | Kourovka Notebook Problem 20.86 | Suppose $G$ is a finite group and $p$ a prime such that the number $s_p(G)=1+kp$ of Sylow $p$-subgroups $P$ of $G$ is greater than 1. By a theorem of Frobenius the set $G_p$ of all $p$-elements in $G$ has cardinality $|G_p|=|P|\cdot f_p(G)$ for some positive integer $f_p(G)$, and one easily gets that
$f_p(G)=1+\ell(p-1... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.86.
Proposed by: P. Schmid | 2 | open | P. Schmid | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,745 | KOU-20.87 | Kourovka Notebook Problem 20.87 | What are the non-abelian composition factors of finite groups in which the order of every element is divisible by at most two primes?
For the case of one prime, see (https://arxiv.org/pdf/2003.09445.pdf). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.87.
Proposed by: W. J. Shi | 2 | open | W. J. Shi | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,746 | KOU-20.88 | Kourovka Notebook Problem 20.88 | Let $u$ be an element of a free group $F_r$. Is it true that there is $v \in F_r$ (that depends on $u$) that cannot be a subword of any cyclically reduced word $\varphi(u)$, where $\varphi$ is an automorphism of $F_r$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.88.
Proposed by: V. Shpilrain
Discussion and literature:
[Solved] Yes, it is true (L. Koch-Hyde, S. O'Connor, E. Olive, V. Shpilrain, *Preprint*, 2025, https... | 2 | solved | V. Shpilrain | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (L. Koch-Hyde, S. O'Connor, E. Olive, V. Shpilrain, *Preprint*, 2025, https://arxiv.org/... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,747 | KOU-20.89 | Kourovka Notebook Problem 20.89 | An element $g$ of a group $G$ is said to be *almost Engel* if there is a finite subset $E(g)$ of $G$ such that for every $x\in G$ all sufficiently long commutators $[...[[x, g], g], \dots , g]$ belong to $E(g)$, that is,
there is a positive integer $n(x, g)$ such that $[...[[x, g], g], \dots ,g]\in E(g)$ if $g$ is repe... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.89.
Proposed by: P. Shumyatsky | 2 | open | P. Shumyatsky | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,748 | KOU-20.90 | Kourovka Notebook Problem 20.90 | A profinite group in which all centralizers of non-trivial elements are pronilpotent is called a CN-group. Find an example of a finitely generated infinite profinite
CN-group which is not prosoluble.
The structure of profinite CN-groups is described in (P. Shumyatsky,
*Israel J. Math.*, **235**, no. 1 (2020), 325--347... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.90.
Proposed by: P. Shumyatsky
Discussion and literature:
[Solved] An example is found (AI solution confirmed by P. Shumyatsky: M. Kinyon, J. Urban, *Preprin... | 2 | solved | P. Shumyatsky | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] An example is found (AI solution confirmed by P. Shumyatsky: M. Kinyon, J. Urban, *Preprint*, 2026,\nhtt... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,749 | KOU-20.91 | Kourovka Notebook Problem 20.91 | A group $G$ is said to be *stable* if for any first-order formula $\phi(\bar{x}, \bar{y})$ (in the first-order language of groups $\{ \cdot , {}^{-1}, 1\}$) there exists a natural number $n$ such that whenever there exist sequences of tuples of $G$, $(a_i)_{i<m}$, $(b_i)_{i<m}$ with $G\models \phi(a_i, b_j)$ if and onl... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.91.
Proposed by: R. Sklinos | 2 | open | R. Sklinos | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,750 | KOU-20.92 | Kourovka Notebook Problem 20.92 | This question is about existence of an analogue of the Lazard correspondence for pre-Lie algebras and braces. A *pre-Lie algebra* $A$ is a vector space with a bilinear operation $(x, y) \rightarrow xy$ satisfying $(xy)z -x(yz) = (yx)z - y(xz)$ for every $x,y,z\in A$.
A pre-Lie algebra $A$ is said to be *left nilpotent*... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.92.
Proposed by: A. Smoktunowicz
Discussion and literature:
[Unverified AI claim] a) Yes, it is (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 202... | 2 | partially_solved | A. Smoktunowicz | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] a) Yes, it is (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,\n\n... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,751 | KOU-20.93 | Kourovka Notebook Problem 20.93 | A group $G$ is called a *Shunkov group* if for any finite subgroup $H\leq G$ any two conjugate elements of prime order in $N_G(H)/H$ generate a finite subgroup. Are the following well-known results of the theory of finite groups true in the class of (periodic) Shunkov groups?
a) The Baer--Suzuki theorem (see 11.11 in ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.93.
Proposed by: A. I. Sozutov | 2 | open | A. I. Sozutov | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,752 | KOU-20.94 | Kourovka Notebook Problem 20.94 | Does there exist an infinite periodic simple group saturated (see the definition in 14.101) with finite Frobenius groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.94.
Proposed by: A. I. Sozutov | 2 | open | A. I. Sozutov | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,753 | KOU-20.95 | Kourovka Notebook Problem 20.95 | Is a periodic group a Frobenius group (see 6.53) if it is saturated with finite Frobenius groups, contains an involution, and does not contain non-cyclic subgroups of order 4? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 20.95.
Proposed by: A. I. Sozutov | 2 | open | A. I. Sozutov | 2,022 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | 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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