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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
701,354 | KOU-17.52 | Kourovka Notebook Problem 17.52 | (N. Eggert). Let $R$ be a commutative
associative finite-dimensional nilpotent algebra over a finite
field $F$ of characteristic $p$. Let $R^{(p)}$ be the subalgebra
of all elements of the form $r^p$ ($r\in R$). Is it true that
$\dim R\geq p \dim R^{(p)}$?
An affirmative answer gives a
solution to 11.44 for the case w... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.52.
Proposed by: L. S. Kazarin | 2 | open | L. S. Kazarin | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,355 | KOU-17.53 | Kourovka Notebook Problem 17.53 | A finite group $G$ is called *simply
reducible* (SR-group) if every element of $G$ is conjugate to
its inverse, and the tensor product of any two irreducible
representations of $G$
decomposes
into a sum of irreducible representations of $G$ with coefficients
$0$ or $1$.
a) Is it true that the nilpotent length of a
sol... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.53.
Proposed by: L. S. Kazarin | 2 | open | L. S. Kazarin | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,356 | KOU-17.54 | Kourovka Notebook Problem 17.54 | Does there exist a non-hereditary local formation ${\mathfrak{F}}$ of
finite groups such that in every finite group the set of all
${\mathfrak{F}}$-subnormal subgroups is a sublattice of the subgroup
lattice?
A negative answer would solve 9.75, since all the hereditary local
formations with the same property are alrea... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.54.
Proposed by: S. F. Kamornikov | 2 | open | S. F. Kamornikov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,357 | KOU-17.55 | Kourovka Notebook Problem 17.55 | Does there exist an absolute constant $k$ such that for any
prefrattini subgroup $H$ in any finite soluble group $G$ there
exist $k$ conjugates of $H$ whose intersection is $\Phi (G)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.55.
Proposed by: S. F. Kamornikov
Published resolution:
Yes, such a constant exists and is equal to 3 (S. F. Kamornikov, *Int. J. Group Theory*, **6**, no. 2 (2017), 1--5). | 2 | solved | S. F. Kamornikov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, such a constant exists and is equal to 3 (S. F. Kamornikov, *Int. J. Group Theory*, **6**, no. 2 (2017), 1--5)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,358 | KOU-17.56 | Kourovka Notebook Problem 17.56 | Suppose that a subgroup $H$ of a finite group
$G$ is such that $HM=MH$ for every minimal non-nilpotent subgroup
$M$ of $G$. Must $H/H_G$ be nilpotent, where $H_G$ is the largest
normal subgroup of $G$ contained in $H$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.56.
Proposed by: V. N. Knyagina, V. S. Monakhov
Discussion and literature:
[Solved] No, not always (A. Ballester-Bolinches, R. Esteban-Romero, S. F. Kamornik... | 2 | solved | V. N. Knyagina, V. S. Monakhov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (A. Ballester-Bolinches, R. Esteban-Romero, S. F. Kamornikov, V. P\\'erez-Calabuig, *J. A... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,359 | KOU-17.57 | Kourovka Notebook Problem 17.57 | Let $r(m)=\{r+km\mid k\in \mathbb{Z}\}$ for integers
$0\leq r<m$. For \(r_1(m_1)\cap r_2(m_2) = \varnothing\) define the
*class transposition* \(\tau_{r_1(m_1),r_2(m_2)}\) as the
involution which interchanges \(r_1+km_1\) and \(r_2+km_2\) for
each integer \(k\) and fixes everything else. The group \(\mathrm{CT}(\mathbb... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.57.
Proposed by: S. Kohl | 2 | open | S. Kohl | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,360 | KOU-17.58 | Kourovka Notebook Problem 17.58 | Does \(\mathrm{CT}(\mathbb{Z})\) have subgroups of
intermediate (word-) growth? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.58.
Proposed by: S. Kohl | 2 | open | S. Kohl | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,361 | KOU-17.59 | Kourovka Notebook Problem 17.59 | \ A permutation of \(\mathbb{Z}\) is called
*residue-class-wise affine* if there is a positive integer
\(m\) such that its restrictions to the residue classes (mod
\(m\)) are all affine. Is \(\mathrm{CT}(\mathbb{Z})\) the group of
all residue-class-wise affine permutations of \(\mathbb{Z}\) which
fix the nonnegative in... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.59.
Proposed by: S. Kohl | 2 | open | S. Kohl | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,362 | KOU-17.60 | Kourovka Notebook Problem 17.60 | Given a set \(\mathcal{P}\) of odd primes, let
\(\mathrm{CT}_{\mathcal{P}}(\mathbb{Z})\) denote the subgroup of
\(\mathrm{CT}(\mathbb{Z})\) which is generated by all class
transpositions which interchange residue classes whose moduli have
only prime factors in \(\mathcal{P} \cup \{2\}\). The groups
\(\mathrm{CT}_{\math... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.60.
Proposed by: S. Kohl | 2 | open | S. Kohl | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,363 | KOU-17.61 | Kourovka Notebook Problem 17.61 | The group \(\mathrm{CT}_{\mathcal{P}}(\mathbb{Z})\)
is finitely generated if and only if \(\mathcal{P}\) is finite. If
\(\mathcal{P} = \varnothing\), then it is isomorphic to the
finitely presented (first) Higman--Thompson group
(J. P. McDermott, see Remark 1.4 in S. Kohl, *J. Group Theory*, **20**, no. 5 (2017), 1025-... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.61.
Proposed by: S. Kohl | 2 | open | S. Kohl | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,364 | KOU-17.62 | Kourovka Notebook Problem 17.62 | Given a free group $F_n$ and a proper
characteristic subgroup $C$, is it ever possible to generate the
quotient $F_n/C$ by fewer than $n$ elements? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.62.
Proposed by: J. Conrad
Discussion and literature:
[Solved] Yes, it is possible; moreover, for all $n\geq 2$, the free group $F_n$ admits continuum many p... | 2 | solved | J. Conrad | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is possible; moreover, for all $n\\geq 2$, the free group $F_n$ admits continuum many pairwise n... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,365 | KOU-17.63 | Kourovka Notebook Problem 17.63 | Prove that if $p$ is an odd prime and $s$ is a
positive integer, then there are only finitely many $p$-adic space
groups of finite coclass with point group of coclass $s$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.63.
Proposed by: C. R. Leedham-Green | 2 | open | C. R. Leedham-Green | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,366 | KOU-17.64 | Kourovka Notebook Problem 17.64 | Say that a group $G$ is an *$n$-approximation*
to the Nottingham group $J=N(p)$ (as defined in Archive, 12.24) if
$G$ is an infinite pro-$p$ group, and $G/\gamma_n(G)$ is
isomorphic to $J/\gamma_n(J)$. Does there exist a function $f(p)$
such that, if $G$ is an $f(p)$-approximation to the Nottingham
group, then $\gamma_... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.64.
Proposed by: C. R. Leedham-Green | 2 | open | C. R. Leedham-Green | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,367 | KOU-17.65 | Kourovka Notebook Problem 17.65 | If Problem 17.64 has an affirmative answer,
does it follow that, for some function $g(p)$, the number of
isomorphism classes of $g(p)$-approximations to $J$ is
a) countable?
b) one?
Note that if, for some $g(p)$, there are only finitely many
isomorphism classes, then for some $h(p)$ there is only one. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.65.
Proposed by: C. R. Leedham-Green | 2 | open | C. R. Leedham-Green | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,368 | KOU-17.66 | Kourovka Notebook Problem 17.66 | (R. Guralnick). Does there exist a
positive integer $d$ such that $\dim H^1(G,V)\leq d$ for any
faithful absolutely irreducible module $V$ for any finite group
$G$? Cf. 16.55 in Archive. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.66.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,369 | KOU-17.67 | Kourovka Notebook Problem 17.67 | (H. Zassenhaus). *Conjecture*: Every
invertible element of finite order
of the integral group ring ${\mathbb{Z}}G$ of a
finite group $G$ is conjugate in the rational group ring ${\mathbb{Q}}G$ to an element of $\pm G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.67.
Proposed by: V. D. Mazurov
Published resolution:
A counterexample was constructed (F. Eisele, L. Margolis,
*Adv. Math.* **339** (2018), 599--641). | 2 | solved | V. D. Mazurov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: A counterexample was constructed (F. Eisele, L. Margolis,\n*Adv. Math.* **339** (2018), 599--641)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,370 | KOU-17.68 | Kourovka Notebook Problem 17.68 | Let $C$ be a cyclic subgroup of a group $G$ and
$G=CFC$ where $F$ is a finite cyclic subgroup. Is it true that
$|G:C|$ is finite? Cf. 12.62. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.68.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,371 | KOU-17.69 | Kourovka Notebook Problem 17.69 | Let $G$ be a group of prime exponent acting
freely on a non-trivial abelian group. Is $G$ cyclic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.69.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,372 | KOU-17.70 | Kourovka Notebook Problem 17.70 | Let $\alpha$ be an automorphism of prime order
$q$ of an infinite free Burnside group $G=B(q,p)$ of prime
exponent $p$ such that $\alpha$ cyclically permutes the free
generators of $G$. Is it true that $\alpha$ fixes some non-trivial
element of $G$?
*Editors' comment*:
this is proved for $q=2$ in (V. S. Atabekyan, H. ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.70.
Proposed by: V. D. Mazurov
Discussion and literature:
[Solved] No, it does not for all $q\geq 3$ and all sufficiently large $p$ (V.S. Atabekyan, *Preprin... | 2 | solved | V. D. Mazurov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it does not for all $q\\geq 3$ and all sufficiently large $p$ (V.S. Atabekyan, *Preprint*, 2026, htt... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,373 | KOU-17.71 | Kourovka Notebook Problem 17.71 | b) Let $\alpha$ be a fixed-point-free automorphism
of prime order $p$ of a periodic group $G$. Suppose that $G$ does not contain a
non-trivial $p$-element. Is $\alpha$ a splitting automorphism?
[Archived solved part(s) of 17.71:]
a) Let $\alpha$ be a fixed-point-free automorphism
of prime order $p$ of a periodic group... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 17.71.
Proposed by: V. D. Mazurov
Archive of Solved Problems entry:
a) No, it may contain a non-trivial $p$-element (E. B. Durakov, A. I. Sozutov, *Algebra L... | 2 | partially_solved | V. D. Mazurov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (a) No, it may contain a non-trivial $p$-element (E. B. Durakov, A. I. Sozutov, *Algebra Logic*, **52**, no. 5 (2013), 422--425).); remaining part(s) in main sect... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,374 | KOU-17.72 | Kourovka Notebook Problem 17.72 | b) Let $AB$ be a Frobenius group with kernel $A$
and complement $B$. Suppose that $AB$ acts on a finite group $G$
so that $GA$ is also a Frobenius group with kernel $G$ and
complement $A$.
Is the exponent of $G$ bounded in terms of
$|B|$ and the exponent of $C_G(B)$?
[Archived solved part(s) of 17.72:]
a)
Let $AB$ be ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 17.72.
Proposed by: V. D. Mazurov
Archive of Solved Problems entry:
a) Yes, it is (N. Yu. Makarenko, P. Shumyatsky,
*Proc. Amer. Math. Soc.*, **138** (2010),... | 2 | partially_solved | V. D. Mazurov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (a) Yes, it is (N. Yu. Makarenko, P. Shumyatsky,\n*Proc. Amer. Math. Soc.*, **138** (2010), 3425--3436).); remaining part(s) in main section."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,375 | KOU-17.73 | Kourovka Notebook Problem 17.73 | Let $G$ be a finite simple group of Lie type
defined over a field of characteristic $p$, and $V$ an absolutely
irreducible $G$-module over a field of the same characteristic.
Is it true that in the following cases the split extension of $V$ by $G$ must contain an element whose order is distinct from the order of any el... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.73.
Proposed by: V. D. Mazurov
Published resolution:
Yes, it is true: a) (M. A. Grechkoseeva, S. V. Skresanov, *Siberian Electron. Math. Rep.*, **17** (2020), 585--589); \ b)--... | 2 | solved | V. D. Mazurov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is true: a) (M. A. Grechkoseeva, S. V. Skresanov, *Siberian Electron. Math. Rep.*, **17** (2020), 585--589); \\ b)--i) (M. A. Grechkoseeva, *J. Algebra A... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,376 | KOU-17.74 | Kourovka Notebook Problem 17.74 | Let $G$ be a finite simple group of Lie type
defined over a field of characteristic $p$ whose Lie rank is at
least three, and $V$ an absolutely irreducible $G$-module over a
field of characteristic that does not divide $p$. It is true that
the split extension of $V$ by $G$ must contain an element whose
order is distinc... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.74.
Proposed by: V. D. Mazurov
Published resolution:
Yes, it is true: for linear groups (A. V. Zavarnitsine, *Siberian Math. J.*, **49** (2008), 246--256); for other classical ... | 2 | solved | V. D. Mazurov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is true: for linear groups (A. V. Zavarnitsine, *Siberian Math. J.*, **49** (2008), 246--256); for other classical groups (M. A. Grechkoseeva, *J. Algebr... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,377 | KOU-17.75 | Kourovka Notebook Problem 17.75 | Can the Monster $M$ act on a nontrivial finite $3$-group $V$ so
that all elements of $M$ of orders $41$, $47$, $59$, $71$ have no
nontrivial fixed points in $V$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.75.
Proposed by: V. D. Mazurov
Discussion and literature:
[Solved] No, it cannot; moreover, $M$ cannot act on a $3$-group in such a way that an element of or... | 2 | solved | V. D. Mazurov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it cannot; moreover, $M$ cannot act on a $3$-group in such a way that an element of order 41 has no ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,378 | KOU-17.76 | Kourovka Notebook Problem 17.76 | Does there exist a finite group $G$, with $|G|
> 2$, such that there is exactly one element in $G$ which is not
a commutator? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.76.
Proposed by: D. MacHale
Discussion and literature:
[Solved] Yes, such group do exist, and there are infinitely many examples (S. V. Skresanov, *Siberian ... | 2 | solved | D. MacHale | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, such group do exist, and there are infinitely many examples (S. V. Skresanov, *Siberian Electron. M... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,379 | KOU-17.77 | Kourovka Notebook Problem 17.77 | Let $k$ be a positive integer such that there is an insoluble
finite group with exactly $k$ conjugacy classes. Is it true that a
finite group of maximal order with exactly $k$ conjugacy classes
is insoluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.77.
Proposed by: R. Heffernan, D. MacHale
Published resolution:
No, it is not; see Table 1 in (A. Vera-Lopez, J. Sangroniz, *Math. Nachr.*, **280**, no. 5--6 (2007), 676--694). | 2 | solved | R. Heffernan, D. MacHale | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it is not; see Table 1 in (A. Vera-Lopez, J. Sangroniz, *Math. Nachr.*, **280**, no. 5--6 (2007), 676--694)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,380 | KOU-17.78 | Kourovka Notebook Problem 17.78 | Does there exist a finitely generated group
without free subsemigroups generating a proper variety {containing
${\mathfrak{A}}_p{\mathfrak{A}}$?} | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.78.
Proposed by: O. Macedo\'nska | 2 | open | O. Macedo\'nska | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,381 | KOU-17.79 | Kourovka Notebook Problem 17.79 | Does there exist a finitely generated torsion
group of unbounded exponent generating a proper variety? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.79.
Proposed by: O. Macedo\'nska
Published resolution:
Yes, moreover, there is a continuum of such groups
(V. S. Atabekyan, *Infinite simple groups satisfying an identity*, Dep... | 2 | solved | O. Macedo\'nska | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, moreover, there is a continuum of such groups\n(V. S. Atabekyan, *Infinite simple groups satisfying an identity*, Dep. VINITI no. 5381-V86, Moscow, 1986 (Ru... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,382 | KOU-17.80 | Kourovka Notebook Problem 17.80 | Let
$[u, _nv]:=[u,\overbrace{v,\dots ,v}^{n} ]$. Is the group
$M_n=\langle x, y\mid x=[x, _ny],\; y=[y, _nx] \rangle$
infinite for every $n>2$? (See also 11.18.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.80.
Proposed by: O. Macedo\'nska | 2 | open | O. Macedo\'nska | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,383 | KOU-17.81 | Kourovka Notebook Problem 17.81 | Given normal subgroups $R_1,\dots, R_n$ of a
group $G$, let $\displaystyle [[R_1,\dots, R_n]]:=\prod
\Big[\bigcap_{i\in I}R_i,\bigcap_{j\in J}R_j\Big] $, where the
product is over all $I\cup J=\{1,\dots,n\}$, $ I\cap J=\varnothing$.
Let $G$ be a free group, and let $R_i=\langle r_i\rangle^G$ be the
normal closures of e... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.81.
Proposed by: R. Mikhailov | 2 | open | R. Mikhailov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,384 | KOU-17.82 | Kourovka Notebook Problem 17.82 | Is it true that in every finitely presented
group the intersection of the derived series has trivial
abelianization? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.82.
Proposed by: R. Mikhailov
Published resolution:
No, not always (V. A. Roman'kov, *Siberian Math. J.*, **52**, no. 2 (2011), 348--351). | 2 | solved | R. Mikhailov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always (V. A. Roman'kov, *Siberian Math. J.*, **52**, no. 2 (2011), 348--351)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,385 | KOU-17.83 | Kourovka Notebook Problem 17.83 | Does there exist a group such that every
central extension of it is residually nilpotent, but there exists
a central extension of a central extension of it which is not
residually nilpotent? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.83.
Proposed by: R. Mikhailov | 2 | open | R. Mikhailov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,386 | KOU-17.84 | Kourovka Notebook Problem 17.84 | An associative algebra $A$ is said to be *Calabi--Yau of
dimension $d$* (for short, CY$d$) if there is a natural
isomorphism of $A$-bimodules $\mathrm{Ext}_{A\text{-bimod}}^d(A, A\otimes A)\cong A$ and $\mathrm{Ext}_{A\text{-bimod}}^n(A, A\otimes A)=0$ for $n\ne d$. By
Kontsevich's theorem, the complex group algebra ${... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.84.
Proposed by: R. Mikhailov | 2 | open | R. Mikhailov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,387 | KOU-17.85 | Kourovka Notebook Problem 17.85 | Let $\mathcal{V}$ be a variety of groups such
that any free group in $\mathcal{V}$ has torsion-free integral
homology groups in all dimensions. Is it true that $\mathcal{V}$ is
abelian?
It is known that free metabelian groups may
have 2-torsion in the second homology groups, and free nilpotent
groups of class 2 may ha... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.85.
Proposed by: R. Mikhailov | 2 | open | R. Mikhailov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,388 | KOU-17.86 | Kourovka Notebook Problem 17.86 | (Simplest questions related to the
Whitehead asphericity conjecture). Let $\langle x_1,x_2,x_3\mid
r_1,r_2,r_3\rangle$ be a presentation of the trivial group.
a) Prove that the group $\langle
x_1,x_2,x_3\mid r_1,r_2\rangle$ is torsion-free.
[Archived solved part(s) of 17.86:]
(Simplest questions related to the
Whiteh... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 17.86.
Proposed by: R. Mikhailov
Archive of Solved Problems entry:
b) No, not always (V. A. Roman'kov, *Siberian Math. J.*, **52**, no. 2 (2011), 348--351). | 2 | partially_solved | R. Mikhailov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (b) No, not always (V. A. Roman'kov, *Siberian Math. J.*, **52**, no. 2 (2011), 348--351).); remaining part(s) in main section."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,389 | KOU-17.87 | Kourovka Notebook Problem 17.87 | Construct a group of intermediate growth with
finitely generated Schur multiplier. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.87.
Proposed by: R. Mikhailov | 2 | open | R. Mikhailov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,390 | KOU-17.88 | Kourovka Notebook Problem 17.88 | Compute $K_0({\mathbb{F}_2G)$ and $K_1({\mathbb{F}_2G)$, where $G$ is the first Grigorchuk group, ${\mathbb{F}_2G$
its group algebra over the field of two elements, and $K_0, K_1$
the zeroth and first $K$-functors. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.88.
Proposed by: R. Mikhailov | 2 | open | R. Mikhailov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,391 | KOU-17.89 | Kourovka Notebook Problem 17.89 | By Bousfield's theorem, the free pronilpotent
completion of a non-cyclic free group has uncountable Schur
multiplier. Is it true that the free prosolvable completion (that
is, the inverse limit of quotients by the derived subgroups) of a
non-cyclic free group has uncountable Schur multiplier? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.89.
Proposed by: R. Mikhailov | 2 | open | R. Mikhailov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,392 | KOU-17.90 | Kourovka Notebook Problem 17.90 | (G. Baumslag). A group is *parafree* if it is residually nilpotent and has the same lower
central quotients as a free group. Is it true that $H_2(G)=0$ for
any finitely generated parafree group $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.90.
Proposed by: R. Mikhailov | 2 | open | R. Mikhailov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,393 | KOU-17.91 | Kourovka Notebook Problem 17.91 | Let $d(X)$ denote the derived length of a group
$X$.
a) Does there exist an absolute constant $k$
such that $d(G)-d(M)\leq k$ for every finite soluble group $G$
and any maximal subgroup $M$?
b) Find the minimum $k$ with this property. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.91.
Proposed by: V. S. Monakhov | 2 | open | V. S. Monakhov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,394 | KOU-17.92 | Kourovka Notebook Problem 17.92 | What are the non-abelian composition factors of
a finite non-soluble group all of whose maximal subgroups are Hall
subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.92.
Proposed by: V. S. Monakhov
Published resolution:
Answer: $L_2(7)$, $L_2(11)$, $L_5(2)$
(N. V. Maslova, *Siberian Math. J.*, **53**, no. 5 (2012), 853--861). | 2 | solved | V. S. Monakhov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Answer: $L_2(7)$, $L_2(11)$, $L_5(2)$\n(N. V. Maslova, *Siberian Math. J.*, **53**, no. 5 (2012), 853--861)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,395 | KOU-17.93 | Kourovka Notebook Problem 17.93 | (Well-known problem). Let $G$ be a compact
topological group which has elements of arbitrarily high orders.
Must $G$ contain an element of infinite order? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.93.
Proposed by: J. Mycielski | 2 | open | J. Mycielski | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,396 | KOU-17.94 | Kourovka Notebook Problem 17.94 | (Well-known problem). Can the free product $Z*G$
of the infinite cyclic group $Z$ and a nontrivial group $G$ be the
normal closure of a single element?
*Editors' comments*:
the negation is also known as Kervaire's conjecture. It was proved for torsion-free groups (A. Klyachko, *Commun. Algebra*, **21**, no. 7 (1993), ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.94.
Proposed by: J. Mycielski | 2 | open | J. Mycielski | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,397 | KOU-17.95 | Kourovka Notebook Problem 17.95 | Let $G$ be a permutation group on the finite
set $\Omega$. A partition $\rho$ of $\Omega$
is said to be
$G$-*regular* if there exists
a subset $S$ of $\Omega$ such that $S^g$ is a
transversal
of $\rho$ for all $g \in G$. The group $G$ is said to be *synchronizing* if $|\Omega| > 2$
and there are no non-trivial proper $... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.95.
Proposed by: P. M. Neumann | 2 | open | P. M. Neumann | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,398 | KOU-17.96 | Kourovka Notebook Problem 17.96 | Does there exist a variety of groups
which contains only countably many subvarieties
but in which there is an infinite properly
descending chain of subvarieties? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.96.
Proposed by: P. M. Neumann | 2 | open | P. M. Neumann | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,399 | KOU-17.97 | Kourovka Notebook Problem 17.97 | Is every variety of groups of exponent $4$
finitely based? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.97.
Proposed by: P. M. Neumann | 2 | open | P. M. Neumann | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,400 | KOU-17.98 | Kourovka Notebook Problem 17.98 | A variety of groups is said to be *small*
if it contains only countably many non-isomorphic finitely
generated groups.
a) Is it true that if $G$ is a finitely
generated group and the variety
$\mathrm{Var}(G)$ it generates is
small then $G$ satisfies the
maximal condition on normal subgroups?
b) Is it true that a vari... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.98.
Proposed by: P. M. Neumann | 2 | open | P. M. Neumann | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,401 | KOU-17.99 | Kourovka Notebook Problem 17.99 | Consider the group $B = \langle a,b \mid
(bab^{-1}) a (bab^{-1})^{-1} = a^2\rangle $ introduced by Baumslag
in 1969. The same relation is satisfied by the functions $ f(x) =
2x$ and $g(x) = 2^x$ under the operation of composition in the
group of germs of monotonically increasing to $\infty $ continuous
functions on $(0... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.99.
Proposed by: A. Yu. Olshanskii
Discussion and literature:
[Solved] No, it is not (C.-F. Nyberg-Brodda, *Preprint*, 2026, https://arxiv.org/abs/2606.27408... | 2 | solved | A. Yu. Olshanskii | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it is not (C.-F. Nyberg-Brodda, *Preprint*, 2026, https://arxiv.org/abs/2606.27408)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,402 | KOU-17.100 | Kourovka Notebook Problem 17.100 | *Conjecture*: A finite group is not
simple if it has an irreducible complex character of odd degree
vanishing on a class of odd length.
If true, this implies the solvability of groups
of odd order, so a proof independent of CFSG is of special
interest. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.100.
Proposed by: V. Pannone | 2 | open | V. Pannone | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,403 | KOU-17.101 | Kourovka Notebook Problem 17.101 | According to P. Hall, a group $G$ is said to be
*homogeneous* if every isomorphism of its finitely generated
subgroups is induced by an automorphism of $G$. It is known
(Higman--Neumann--Neumann) that every group is embeddable into a
homogeneous one. Is the same true for the category of
representations of groups? A rep... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.101.
Proposed by: B. I. Plotkin
Discussion and literature:
[Unverified AI claim] Yes, it is true (unverified AI solutions: Achyuth Jayadevan, *Preprint of 11... | 2 | open | B. I. Plotkin | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] Yes, it is true (unverified AI solutions: Achyuth Jayadevan, *Preprint of 11 Sept... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,404 | KOU-17.102 | Kourovka Notebook Problem 17.102 | We say that two subsets $A, B$ of an infinite
group $G$ are *separated* if there exists an infinite subset
$X$ of $G$ such that $1\in X$, $X=X^{-1}$, and $XAX\cap
B=\varnothing$. Is it true that any two disjoint subsets $A, B$ of
an infinite group $G$ satisfying $|A|<|G|$, $|B|<|G|$ are
separated? This is so if $A,B$ a... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.102.
Proposed by: I. V. Protasov
Discussion and literature:
[Solved] No, not always (S. Wan, *Preprint*, 2026, https://arxiv.org/abs/2608.00504). | 2 | solved | I. V. Protasov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"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 (S. Wan, *Preprint*, 2026, https://arxiv.org/abs/2608.00504)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,405 | KOU-17.103 | Kourovka Notebook Problem 17.103 | Does there exist a continuum of sets $\pi$ of primes for which
every finite group possessing a Hall $\pi$-subgroup is a
$D_\pi$-group?
*Comment of 2025*:
It has been proved that, for any integer $x$, if $\pi$ is the set of all prime numbers $p>x$, then any finite group containing a Hall $\pi$-subgroup is a $D_\pi$-gro... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.103.
Proposed by: D. O. Revin | 2 | open | D. O. Revin | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,406 | KOU-17.104 | Kourovka Notebook Problem 17.104 | Let $\Gamma$ be a finite non-oriented graph on
the set of vertices $\{x_1,\dots,x_n\}$ and let
$S_\Gamma=\langle x_1,\dots,x_n\mid x_ix_j=x_jx_i \Leftrightarrow
(x_i,x_j) \in \Gamma;\; {\mathfrak{A}}^2\rangle$ be a presentation of
a partially commutative metabelian group $S_\Gamma$ in the variety
of all metabelian grou... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.104.
Proposed by: V. N. Remeslennikov, E. I. Timoshenko | 2 | open | V. N. Remeslennikov, E. I. Timoshenko | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,407 | KOU-17.105 | Kourovka Notebook Problem 17.105 | An equation over a pro-$p$-group $G$ is an
expression $v(x)=1$, where $v(x)$ is an element of the free
pro-$p$-product of $G$ and a free pro-$p$-group with basis $\{
x_1, \ldots , x_n \}$; solutions are sought in the affine space
$G^n$. Is it true that a free pro-$p$-group is equationally Noetherian,
that is, for any $... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.105.
Proposed by: N. S. Romanovski\u{\i | 2 | open | N. S. Romanovski\u{\i | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,408 | KOU-17.107 | Kourovka Notebook Problem 17.107 | Does $G=\mathrm{SL}_2(\mathbb{C})$ contain a
2-generated free subgroup that is conjugate in $G$ to a proper
subgroup of itself?
A 6-generated free subgroup with this property
has been found (D. Calegari, N. M. Dunfield,
*Proc. Amer. Math. Soc.*, **134**, no. 11 (2006), 3131--3136). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.107.
Proposed by: M. Sapir | 2 | open | M. Sapir | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,409 | KOU-17.108 | Kourovka Notebook Problem 17.108 | Is the group $\langle a,b,t \mid a^t=ab,\;
b^t=ba\rangle$ linear?
If not, this would be an easy example of a
non-linear hyperbolic group. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.108.
Proposed by: M. Sapir
Published resolution:
Yes, this group is linear. Indeed, as noticed by M. Sapir, this group is
the mapping torus of an irreducible atoroidal self-mon... | 2 | solved | M. Sapir | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, this group is linear. Indeed, as noticed by M. Sapir, this group is\nthe mapping torus of an irreducible atoroidal self-monomorphism of a free group; thus i... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,410 | KOU-17.109 | Kourovka Notebook Problem 17.109 | A non-trivial group word $w$ is *uniformly
elliptic* in a class $\mathcal{C}$ if there is a function
$f:\mathbb{N}\rightarrow\mathbb{N}$ such that the width of $w$ in
every $d$-generator $\mathcal{C}$-group $G$ is bounded by $f(d)$
(i. e. every element of the verbal subgroup $w(G)$ is equal to a
product of $f(d)$ value... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.109.
Proposed by: D. Segal | 2 | open | D. Segal | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,411 | KOU-17.110 | Kourovka Notebook Problem 17.110 | a) Is it true that for each word $w$, there is
a function $h:\mathbb{N}\times \mathbb{N}\rightarrow\mathbb{N}$
such that the width of $w$ in every finite $p$-group of Pr\"{u}fer
rank $r$ is bounded by $h(p,r)$?
b) If so, can $h(p,r)$ be made independent of
$p$?
An affirmative answer to a) would imply
A. Jaikin-Zapira... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.110.
Proposed by: D. Segal | 2 | open | D. Segal | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,412 | KOU-17.111 | Kourovka Notebook Problem 17.111 | Let $G$ be a finite group, and $p$ a prime
divisor of $|G|$. Suppose that every maximal subgroup of a Sylow
$p$-subgroup of $G$ has a $p$-soluble supplement in $G$. Must $G$
be $p$-soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.111.
Proposed by: A. N. Skiba
Published resolution:
Yes, it must (GuoHua Qian, *Science China Mathematics*, **56**, no. 5 (2013), 1015--1018). | 2 | solved | A. N. Skiba | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it must (GuoHua Qian, *Science China Mathematics*, **56**, no. 5 (2013), 1015--1018)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,413 | KOU-17.112 | Kourovka Notebook Problem 17.112 | A subgroup $A$ of a group $G$ is said to be
*$G$-permutable in $G$* if for every subgroup $B$ of $G$
there exists an element $x\in G $ such that $AB^x=B^xA$. A
subgroup $A$ is said to be *hereditarily $G$-permutable in
$G$* if $A$ is $E$-permutable in every subgroup $E$ of $G$
containing $A$. Which finite non-abelian s... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.112.
Proposed by: A. N. Skiba, V. N. Tyutyanov | 2 | open | A. N. Skiba, V. N. Tyutyanov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,414 | KOU-17.113 | Kourovka Notebook Problem 17.113 | Do there exist for $p>3$ \ 2-generator finite
$p$-groups
with deficiency zero (see 8.12) of arbitrarily high
nilpotency class? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.113.
Proposed by: J. Wiegold | 2 | open | J. Wiegold | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,415 | KOU-17.114 | Kourovka Notebook Problem 17.114 | Does every generalized free product (with amalgamation) of two
non-trivial groups have maximal subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.114.
Proposed by: J. Wiegold | 2 | open | J. Wiegold | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,416 | KOU-17.115 | Kourovka Notebook Problem 17.115 | Can a locally free non-abelian group have
non-trivial Frattini subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.115.
Proposed by: J. Wiegold | 2 | open | J. Wiegold | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,417 | KOU-17.116 | Kourovka Notebook Problem 17.116 | Let $n(G)$ be the maximum of positive integers
$n$ such that the $n$-th direct power of a finite simple group $G$
is 2-generated. Is it true that $n(G)\geq \sqrt{|G|}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 17th issue (2010). Original problem number: 17.116.
Proposed by: A. Erfanian, J. Wiegold
Published resolution:
Yes, it is, even with $n(G)>2\sqrt{|G|}$
(A. Mar\'oti, M. C. Tamburini, *Commun. Algebra*, **41**, no. 1 (2013),... | 2 | solved | A. Erfanian, J. Wiegold | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is, even with $n(G)>2\\sqrt{|G|}$\n(A. Mar\\'oti, M. C. Tamburini, *Commun. Algebra*, **41**, no. 1 (2013), 34--49)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,418 | KOU-17.117 | Kourovka Notebook Problem 17.117 | (Well-known problem). If groups $A$ and $B$
have decidable elementary theories $Th(A)$ and $Th(B)$, must
$Th(A*B)$ be decidable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.117.
Proposed by: O. Kharlampovich | 2 | open | O. Kharlampovich | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,419 | KOU-17.118 | Kourovka Notebook Problem 17.118 | Suppose that a finite $p$-group $G$ has a
subgroup of exponent $p$ and of index $p$. Must $G$ also have a
characteristic subgroup of exponent $p$ and of index bounded in
terms of $p$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.118.
Proposed by: E. I. Khukhro | 2 | open | E. I. Khukhro | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,420 | KOU-17.119 | Kourovka Notebook Problem 17.119 | Suppose that a finite soluble group $G$ admits a soluble group of automorphisms
$A$ of coprime order such that $C_G(A)$ has rank $r$. Let $|A|$ be the product of $l$ not necessarily distinct primes.
Is there a linear function $f$ such that $G/F_{f(l)}(G)$ has
$(|A|,r)$-bounded rank, where $F_{f(l)}(G)$ is the $f(l)$-th... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.119.
Proposed by: E. I. Khukhro | 2 | open | E. I. Khukhro | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,421 | KOU-17.120 | Kourovka Notebook Problem 17.120 | Is a residually finite group all of whose subgroups of infinite
index are finite necessarily cyclic-by-finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.120.
Proposed by: N. S. Chernikov | 2 | open | N. S. Chernikov | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,422 | KOU-17.121 | Kourovka Notebook Problem 17.121 | Let $ G $ be a group whose set of elements is
the real numbers and which is ``nicely definable'' (see below).
Does
$ G $ being $ { \aleph_\omega} $-free
imply it being free if ``nicely definable'' means
a) being $ F_ \sigma $?
b) being Borel?
c) being analytic?
d) being projective $ {\mathbb{L} [ {\mathbb{R} ]
$?
... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.121.
Proposed by: S. Shelah | 2 | open | S. Shelah | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,423 | KOU-17.122 | Kourovka Notebook Problem 17.122 | The same questions as 17.121 for an abelian
group $G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.122.
Proposed by: S. Shelah | 2 | open | S. Shelah | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,424 | KOU-17.123 | Kourovka Notebook Problem 17.123 | Do there exist finite groups $G_1, G_2$ such
that $\pi _e(G_1)=\pi _e(G_2)$, \ $h(\pi _e(G_1))<\infty$, and
each non-abelian composition factors of each of the groups
$G_1, G_2$ is not isomorphic to a section of the other?
For definitions see Archive, 13.63. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.123.
Proposed by: W. J. Shi | 2 | open | W. J. Shi | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,425 | KOU-17.124 | Kourovka Notebook Problem 17.124 | Is the set of finitely presented metabelian
groups recursively enumerable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.124.
Proposed by: V. Shpilrain
Discussion and literature:
[Unverified AI claim] Yes, it is (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026, http... | 2 | open | V. Shpilrain | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] Yes, it is (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026, https://... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,426 | KOU-17.125 | Kourovka Notebook Problem 17.125 | Does every finite group $G$ contain a pair of
conjugate elements $a,b$ such that $\pi(G)=\pi(\langle
a,b\rangle)$? This is true for soluble groups.
*Comment of 2013:*
it was proved in (A. Lucchini, M. Morigi, P. Shumyatsky, *Forum Math.*, **24** (2012), 875--887) that every finite group $G$ contains a 2-generator subg... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.125.
Proposed by: P. Shumyatsky | 2 | open | P. Shumyatsky | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,427 | KOU-17.126 | Kourovka Notebook Problem 17.126 | Suppose that $G$ is a residually finite group
satisfying the identity $[x,y]^n=1$. Must $[G,G]$ be locally
finite?
An equivalent question: let $G$ be a finite
soluble group satisfying the identity $[x,y]^n=1$; is the Fitting
height of $G$ bounded in terms of $n$? Cf. Archive, 13.34 | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.126.
Proposed by: P. Shumyatsky | 2 | open | P. Shumyatsky | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,428 | KOU-17.127 | Kourovka Notebook Problem 17.127 | Suppose that a finite soluble group $G$ of
derived length $d$ admits an elementary abelian $p$-group of
automorphisms $A$ of order $p^n$ such that $C_G(A)=1$. Must $G$
have a normal series of $n$-bounded length with nilpotent factors
of $(p,n,d)$-bounded nilpotency class?
This is true for $p=2$. An affirmative answer ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.127.
Proposed by: P. Shumyatsky | 2 | open | P. Shumyatsky | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,429 | KOU-17.128 | Kourovka Notebook Problem 17.128 | Let $T$ be a finite $p$-group admitting an
elementary abelian group of automorphisms $A$ of order $p^{2}$
such that in the semidirect product $P=TA$ every element of
$P\setminus T$ has order $p$. Does it follow that $T$ is of
exponent $p$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 17.128.
Proposed by: E. Jabara | 2 | open | E. Jabara | 2,010 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,430 | KOU-18.1 | Kourovka Notebook Problem 18.1 | Given a group $G$ of finite order $n$, does there necessarily
exist a bijection $f$ from $G$ onto a cyclic group of order $n$
such that for each element $x\in G$, the order of $x$ divides
the order of $f(x)$?
An affirmative answer in the case where $G$ is solvable was given by F. Ladisch. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.1.
Proposed by: I. M. Isaacs
Discussion and literature:
[Solved] Yes, it does (M. Amiri, *J. Pure Applied Algebra*, **228**, no. 7 (2024), 107632). | 2 | solved | I. M. Isaacs | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it does (M. Amiri, *J. Pure Applied Algebra*, **228**, no. 7 (2024), 107632)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,431 | KOU-18.2 | Kourovka Notebook Problem 18.2 | Let the group $G=AB$ be the product of two Chernikov subgroups $A$ and $B$ each of which
has an abelian subgroup of index at most 2. Is $G$ soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.2.
Proposed by: B. Amberg | 2 | open | B. Amberg | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,432 | KOU-18.3 | Kourovka Notebook Problem 18.3 | Let the group $G=AB$ be the product of two central-by-finite subgroups $A$ and $B$. By a theorem of N. Chernikov, $G$ is soluble-by-finite. Is $G$ metabelian-by- \text{finite?} \text{} | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.3.
Proposed by: B. Amberg | 2 | open | B. Amberg | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,433 | KOU-18.4 | Kourovka Notebook Problem 18.4 | (A. Rhemtulla, S. Sidki). a) Is a group of the form $G=ABA$ with cyclic subgroups $A$ and $B$ always soluble? (This is known to be true if $G$ is finite.)
b) The same question when in addition $A$ and $B$ are conjugate in $G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.4.
Proposed by: B. Amberg | 2 | open | B. Amberg | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,434 | KOU-18.5 | Kourovka Notebook Problem 18.5 | Let the (soluble) group $G=AB$ with finite
torsion-free rank $r_0(G)$ be the product of two subgroups $A$ and $B$.
Is the equation $r_0(G) = r_0(A) + r_0(B) - r_0(A \cap B)$ valid in this case?
It is known that the inequality $\leq$ always holds. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.5.
Proposed by: B. Amberg | 2 | open | B. Amberg | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,435 | KOU-18.6 | Kourovka Notebook Problem 18.6 | Is every finitely generated one-relator group residually amenable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.6.
Proposed by: G. N. Arzhantseva | 2 | open | G. N. Arzhantseva | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,436 | KOU-18.7 | Kourovka Notebook Problem 18.7 | Let $n\geq 665$ be an odd integer. Is it true that the group of outer automorphisms
$\mathrm{Out}(B(m,n))$ of the free Burnside group $B(m,n)$ for $m>2$ is complete, that is, has trivial center and all its automorphisms are inner? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.7.
Proposed by: V. S. Atabekyan | 2 | open | V. S. Atabekyan | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,437 | KOU-18.8 | Kourovka Notebook Problem 18.8 | Let $\mathcal{X}$ be a class of finite simple groups such that $\pi(\mathcal{X}) = char(\mathcal{X})$. A formation of finite groups $\mathfrak{F}$ is said to be $\mathcal{X}$-saturated if a finite group $G$ belongs to $\mathfrak{F}$ whenever the factor group $G/\Phi(O_{\mathcal{X}}(G))$ is in $\mathfrak{F}$, where $O_{... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.8.
Proposed by: A. Ballester-Bolinches | 2 | open | A. Ballester-Bolinches | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,438 | KOU-18.9 | Kourovka Notebook Problem 18.9 | Does there exist a subgroup-closed saturated formation $\mathfrak{F}$ of finite groups
properly contained in $\mathfrak{E}_{\pi}$, where $\pi = char(\mathcal{F})$, satisfying the following property: if $G \in \mathfrak{F}$, then there exists a prime $p$ (depending on the group $G$) such that the wreath product $C_p\wr ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.9.
Proposed by: A. Ballester-Bolinches
Published resolution:
Yes, there does
(S. F. Kamornikov, *Trudy. Inst. Mat. (Minsk)*, **24**, no. 1 (2016), 30--33). | 2 | solved | A. Ballester-Bolinches | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there does\n(S. F. Kamornikov, *Trudy. Inst. Mat. (Minsk)*, **24**, no. 1 (2016), 30--33)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,439 | KOU-18.10 | Kourovka Notebook Problem 18.10 | A formation $\mathfrak{F}$ of finite groups satisfies the *Wielandt property for residuals* if whenever $U$ and $V$ are subnormal subgroups of $\langle U, V\rangle $ in a finite group $G$, then the $\mathfrak{F}$-residual $\langle U, V\rangle ^{\mathfrak{F}}$ of $\langle U, V\rangle $ coincides with $\langle U^{\mathfr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.10.
Proposed by: A. Ballester-Bolinches | 2 | open | A. Ballester-Bolinches | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,440 | KOU-18.11 | Kourovka Notebook Problem 18.11 | Let $G$ and $H$ be subgroups of the automorphism group $Aut (F_n)$ of a free group $F_n$ of rank
$n \geq 2$. Is it true that the free product $G * H$ embeds in the automorphism group
$Aut (F_m)$ for some $m$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.11.
Proposed by: V. G. Bardakov
Discussion and literature:
[Solved] Yes, it is true (AI solution confirmed by V. G. Bardakov: \ Achyuth Jayadevan, *Preprint*... | 2 | solved | V. G. Bardakov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (AI solution confirmed by V. G. Bardakov: \\ Achyuth Jayadevan, *Preprint*, 2026,\nhttps... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,441 | KOU-18.12 | Kourovka Notebook Problem 18.12 | Let $G$ be a finitely generated group of intermediate growth. Is it true that there is a positive integer $m$ such that every element of the derived subgroup $G'$ is a product of at most $m$ commutators?
This assertion is valid for groups of polynomial growth, since they are almost nilpotent by Gromov's theorem. On th... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.12.
Proposed by: V. G. Bardakov | 2 | open | V. G. Bardakov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,442 | KOU-18.13 | Kourovka Notebook Problem 18.13 | (D. B. A. Epstein).
Is it true that the group $ H = (\mathbb{Z}_3 \times \mathbb{Z}) * (\mathbb{Z}_2 \times \mathbb{Z}) =
\langle x, y, z, t \mid x^3 = z^2 = [x, y] = [z, t] = 1 \rangle$ cannot be defined by three relators in the generators $x, y, z, t$?
It is known that the relation module of the group $H$ has rank 3... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.13.
Proposed by: V. G. Bardakov, M. V. Neshchadim | 2 | open | V. G. Bardakov, M. V. Neshchadim | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,443 | KOU-18.14 | Kourovka Notebook Problem 18.14 | For an automorphism $\varphi \in Aut(G)$ of a group $G$, let $[e]_{\varphi} =\{g^{-1}g^{\varphi}\mid g\in G\}$.
*Conjecture*: If $[e]_{\varphi}$ is a subgroup for every $\varphi \in Aut(G)$, then
the group $G$ is nilpotent. If in addition $G$ is finitely generated, then $G$ is abelian. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.14.
Proposed by: V. G. Bardakov, M. V. Neshchadim, T. R. Nasybullov
Discussion and literature:
[Solved] The conjecture is disproved (C. Nicotera, *Arch. Math... | 2 | solved | V. G. Bardakov, M. V. Neshchadim, T. R. Nasybullov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] The conjecture is disproved (C. Nicotera, *Arch. Math.*, **123**, no. 3 (2024), 225--232)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,444 | KOU-18.15 | Kourovka Notebook Problem 18.15 | For an automorphism $\varphi \in Aut(G)$ of a group $G$, let $[e]_{\varphi} =\{g^{-1}g^{\varphi}\mid g\in G\}$.
Is it true that if a group $G$ has trivial center, then there is an inner automorphism $\varphi$ such that
$[e]_{\varphi}$ is not a subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.15.
Proposed by: V. G. Bardakov, M. V. Neshchadim, T. R. Nasybullov
Published resolution:
No, not always; see V. V. Bludov's Example 2 in (D. Gonçalves, T. Nasybullov, *Commun.... | 2 | solved | V. G. Bardakov, M. V. Neshchadim, T. R. Nasybullov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always; see V. V. Bludov's Example 2 in (D. Gonçalves, T. Nasybullov, *Commun. Algebra*, **47**, no. 3 (2019), 930--944)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,445 | KOU-18.16 | Kourovka Notebook Problem 18.16 | Is it true that any definable endomorphism of any ordered abelian group
is of the form $x\mapsto rx$, for some rational number $r$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.16.
Proposed by: O. V. Belegradek
Discussion and literature:
[Solved] Yes, it is true (H. Ji, F. Xu, D. Qiu, *Preprint*, 2026, https://arxiv.org/abs/2608.178... | 2 | solved | O. V. Belegradek | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (H. Ji, F. Xu, D. Qiu, *Preprint*, 2026, https://arxiv.org/abs/2608.17861)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,446 | KOU-18.17 | Kourovka Notebook Problem 18.17 | Is there a torsion-free group which is finitely presented in the quasi-variety of torsion-free groups but not finitely presentable in the variety of
all groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.17.
Proposed by: O. V. Belegradek
Published resolution:
Yes, there is (A. I. Budkin, *Siberian Electron. Math. Rep.*, **14** (2017), 937--945, http://semr.math.nsc.ru). | 2 | solved | O. V. Belegradek | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there is (A. I. Budkin, *Siberian Electron. Math. Rep.*, **14** (2017), 937--945, http://semr.math.nsc.ru)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,447 | KOU-18.18 | Kourovka Notebook Problem 18.18 | Mal'cev proved that the set of sentences that hold in all finite groups is not
computably enumerable, although its complement is. Is it true that both
the set of sentences that hold in almost all finite groups and its complement
are not computably enumerable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.18.
Proposed by: O. V. Belegradek
Discussion and literature:
[Solved] Yes, it is true (AI solution confirmed by H. D. Macpherson and B. I. Zilber: \ M. Kinyo... | 2 | solved | O. V. Belegradek | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (AI solution confirmed by H. D. Macpherson and B. I. Zilber: \\ M. Kinyon, J. Urban, *Pr... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,448 | KOU-18.19 | Kourovka Notebook Problem 18.19 | Is any torsion-free, relatively free group of infinite rank not $\aleph_1$-homogeneous?
This is true for group varieties in which all free groups are residually finite
(O. Belegradek, *Arch. Math. Logic*, **51**
(2012), 781--787). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.19.
Proposed by: O. V. Belegradek | 2 | open | O. V. Belegradek | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,449 | KOU-18.20 | Kourovka Notebook Problem 18.20 | Characters $\varphi$ and $\psi$ of a finite group $G$
are said to be *semiproportional* if they are not proportional and there is a
normal subset $M$ of $G$ such that
$\varphi |_M$ is proportional to $\psi |_M$ and
$\varphi |_{G\setminus M}$ is proportional to $\psi |_{G\setminus M}$.
*Conjecture*: If $\varphi$ and $\... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.20.
Proposed by: V. A. Belonogov | 2 | open | V. A. Belonogov | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,450 | KOU-18.21 | Kourovka Notebook Problem 18.21 | (B. H. Neumann and H. Neumann).
Fix an integer $d\geq 2$. If $\mathfrak{V}$ is a variety of groups such that
all $d$-generated groups in $\mathfrak{V}$ are finite,
must $\mathfrak{V}$ be locally finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.21.
Proposed by: G. M. Bergman | 2 | open | G. M. Bergman | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,451 | KOU-18.22 | Kourovka Notebook Problem 18.22 | (H. Neumann).
Is the Kostrikin variety of all locally finite groups
of given prime exponent $p$
determined by finitely many identities? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.22.
Proposed by: G. M. Bergman | 2 | open | G. M. Bergman | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,452 | KOU-18.23 | Kourovka Notebook Problem 18.23 | The normal covering number of the symmetric group $S_n$ of degree $n$ is the minimum number $\gamma(S_n)$ of proper subgroups $H_1,\ldots,H_{\gamma(S_n)}$ of $S_n$
such that every element of $S_n$ is conjugate to an element of $H_i$, for some $i=1,\dots, \gamma(S_n)$.
Write $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ for p... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 18th issue (2014). Original problem number: 18.23.
Proposed by: D. Bubboloni, C. E. Praeger, P. Spiga
Published resolution:
A. Mar\'oti showed that the conjecture is incorrect for odd $n$; his counterexample is presented in... | 2 | solved | D. Bubboloni, C. E. Praeger, P. Spiga | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: A. Mar\\'oti showed that the conjecture is incorrect for odd $n$; his counterexample is presented in \\S 2 of (D. Bubboloni, C. E. Praeger, P. Spiga, *Monatsh. M... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,453 | KOU-18.24 | Kourovka Notebook Problem 18.24 | For a group $G$, a function $\phi : G\to {\mathbb{R}$ is a *quasimorphism* if there is a least non-negative number $D(\phi)$ (called the *defect*) such that $|\phi(gh) -\phi(g) - \phi(h)| \leq D(\phi)$ for all $g, h\in G$. A quasimorphism is *homogeneous* if in addition $\phi(g^n) = n\phi(g)$ for all $g\in G$.
Let $\ph... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 18.24.
Proposed by: M. Burger, D. Calegari | 2 | open | M. Burger, D. Calegari | 2,014 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
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