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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
701,554 | KOU-19.6 | Kourovka Notebook Problem 19.6 | Let $G=A\ast_{H}B$ be the generalized free product of groups $A$ and $B$ with amalgamated subgroup $H$.
a) *Conjecture :* If both $A$ and $B$ are
nilpotent, then $\mu(G)\leq H$.
b) *Conjecture :* If $G$ is residually finite, and $H$
satisfies a nontrivial identical relation, then $\lambda(G)\leq H$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.6.
Proposed by: M. K. Azarian | 2 | open | M. K. Azarian | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,555 | KOU-19.7 | Kourovka Notebook Problem 19.7 | The group of virtual pure braids $VP_n$, $n \geq 2$, is generated by elements $\lambda_{ij}$, $1 \leq i \not= j \leq n$, and is defined by the relations
$
\lambda_{ij}\lambda_{kl}=\lambda_{kl}\lambda_{ij}$, \ $\lambda_{ki}\lambda_{kj}\lambda_{ij}=\lambda_{ij}\lambda_{kj}\lambda_{ki}
$,
where different letters denote di... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.7.
Proposed by: V. G. Bardakov | 2 | open | V. G. Bardakov | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,556 | KOU-19.8 | Kourovka Notebook Problem 19.8 | A word in an alphabet $A = \{ a_1, a_2, \ldots, a_n \}$ is called a palindrome if it reads the same from left to right and from right to left. Let $k$ a non-negative integer. A word in the alphabet $A$ is called an *almost $k$-palindrome* if it can be transformed into a palindrome by changing $\leq k$ letters in it. (S... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.8.
Proposed by: V. G. Bardakov
Discussion and literature:
[Solved] No, there are no such integers (M. Staiger, *J. Algebra*, **659** (2024), 475--481). | 2 | solved | V. G. Bardakov | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, there are no such integers (M. Staiger, *J. Algebra*, **659** (2024), 475--481)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,557 | KOU-19.9 | Kourovka Notebook Problem 19.9 | Let $G$ be a finitely generated linear group.
a) Is it true that the non-abelian tensor square $G \otimes G$ is a linear group?
b) In particular, is this true for the braid group $B_n$ for $n > 3$?
It is known that $B_3 \otimes B_3$ is linear, and there is a countable group $G$ such that $G \otimes G$ is not linear.... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.9.
Proposed by: V. G. Bardakov, M. V. Neshchadim | 2 | open | V. G. Bardakov, M. V. Neshchadim | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,558 | KOU-19.10 | Kourovka Notebook Problem 19.10 | a)
Is it possible to embed a finitely generated non-abelian free pro-$p$ group as an open subgroup of a simple totally disconnected locally compact group?
b) If the answer is yes, can we require that the simple envelope is also compactly generated?
*Comment of 2025*: Partial progress on part b) is made in (Y. Barnea,... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.10.
Proposed by: Y. Barnea
Discussion and literature:
[Solved] a) Yes, it is possible (Y. Barnea, M. Ershov, A. Le Boudec, C. D. Reid,
M. Vannacci, T. Weigel... | 2 | partially_solved | Y. Barnea | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] a) Yes, it is possible (Y. Barnea, M. Ershov, A. Le Boudec, C. D. Reid,\nM. Vannacci, T. Weige... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,559 | KOU-19.11 | Kourovka Notebook Problem 19.11 | Does there exist a constant $c$ such that the
number of conjugacy classes in a finite group $G$ is always at least $c \log_2 |G|$?
*Editors' comment of 2021*: It is proved that every group $G$ contains at least $\varepsilon \log |G|/(\log \log |G|)^8$ conjugacy classes for some fixed $\varepsilon > 0$ (L. Pyber, *J. L... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.11.
Proposed by: E. Bertram | 2 | open | E. Bertram | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,560 | KOU-19.12 | Kourovka Notebook Problem 19.12 | A finite group $G$ is called conjugacy-expansive if for every normal
subset $N$ and conjugacy class $C$ of $G$ the normal set $NC$ contains at least
as many conjugacy classes of $G$ as $N$ does. Is it true that every finite
simple group is conjugacy-expansive? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.12.
Proposed by: M. Bezerra, Z. Halasi, A. Mar\'oti, S. Sidki | 2 | open | M. Bezerra, Z. Halasi, A. Mar\'oti, S. Sidki | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,561 | KOU-19.13 | Kourovka Notebook Problem 19.13 | The well-known Baer--Suzuki theorem
states that if every two conjugates of an element $a$ of
a finite group $G$ generate a finite $p
$-subgroup,
then $a$ is contained in a normal $p
$-subgroup. Does
such a theorem hold in the class of periodic groups for $p>2$?
A counterexample is known for $p=2$; see Archive 11.11a). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.13.
Proposed by: A. V. Borovik | 2 | open | A. V. Borovik | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,562 | KOU-19.14 | Kourovka Notebook Problem 19.14 | Let $G$ be a finite group such that the quasivariety generated by all its Sylow subgroups contains only finitely many subquasivarieties. Is it true that the quasivariety generated by $G$ also contains only finitely many subquasivarieties? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.14.
Proposed by: A. I. Budkin | 2 | open | A. I. Budkin | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,563 | KOU-19.15 | Kourovka Notebook Problem 19.15 | For a class of groups $M$, let $L(M)$ denote the class of all groups $G$ in which
the normal closure $\langle a^G\rangle $ of any element $a\in G$ is contained in $M$. Let $qA$ be the quasivariety generated by a finite nilpotent group $A$. Can the quasivariety $L(qA)$ contain a non-nilpotent group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.15.
Proposed by: A. I. Budkin | 2 | open | A. I. Budkin | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,564 | KOU-19.16 | Kourovka Notebook Problem 19.16 | (C. Drutu and M. Sapir). Is every free-by-cyclic group of the form $F_n \rtimes \mathbb{Z}$ linear? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.16.
Proposed by: J. O. Button | 2 | open | J. O. Button | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,565 | KOU-19.17 | Kourovka Notebook Problem 19.17 | (Well-known problem). Suppose that $G$ is a finitely presented group such that the set of first Betti numbers over all the
finite index subgroups of $G$ is unbounded above. (Here, the first Betti number is the maximum $n$ for which there is a surjective homomorphism onto $\mathbb{Z}^n$.)
a) Must $G$ have a finite inde... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.17.
Proposed by: J. O. Button | 2 | open | J. O. Button | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,566 | KOU-19.18 | Kourovka Notebook Problem 19.18 | Given an infinite set $\Omega$, define an
algebra $A$ (the *reduced incidence algebra of finite
subsets*) as follows. Let $V_n$ be the set of functions from the
set of $n $-element subsets of $\Omega$ to the rationals $\mathbb{Q}$.
Now let $A=\bigoplus V_n$, with multiplication as follows: for
$f\in V_n$, $g\in V_m$, a... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.18.
Proposed by: P. J. Cameron | 2 | open | P. J. Cameron | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,567 | KOU-19.19 | Kourovka Notebook Problem 19.19 | A finite transitive
permutation group $G$ is said to have the *road closure property* if,
given any orbit $O$ of $G$ on $2$-sets, and any proper block of imprimitivity
for $G$ acting on $O$, the graph with edge set $O\setminus B$ is connected.
Such a group must be primitive, and basic (not contained in a wreath product... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.19.
Proposed by: P. J. Cameron | 2 | open | P. J. Cameron | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,568 | KOU-19.20 | Kourovka Notebook Problem 19.20 | For a finite group $G$, let $\mathrm{End}(G)$ denote
the semigroup of endomorphisms of $G$, and $\mathrm{PIso}(G)$ the semigroup
of partial isomorphisms of $G$ (isomorphisms between subgroups of $G$). If
$G$ is abelian, then $|\mathrm{End}(G)|=|\mathrm{PIso}(G)|$. Is the converse
true? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.20.
Proposed by: P. J. Cameron
Discussion and literature:
[Solved] No, it is not (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/... | 2 | solved | P. J. Cameron | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it is not (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/wp-content/upl... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,569 | KOU-19.21 | Kourovka Notebook Problem 19.21 | Can a non-discrete, compactly generated, topologically simple, locally compact group be amenable and non-compact?
(A positive answer implies a negative answer to 19.107.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.21.
Proposed by: P.-E. Caprace | 2 | open | P.-E. Caprace | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,570 | KOU-19.22 | Kourovka Notebook Problem 19.22 | Let $\mathscr{S}$ denote the class of nondiscrete compactly generated, topologically simple totally disconnected locally compact groups.
Two topological groups are called *locally isomorphic* if they contain isomorphic open subgroups. Is the number of local isomorphism classes of groups in $\mathscr{S}$ uncountable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.22.
Proposed by: P.-E. Caprace
Discussion and literature:
[Solved] Yes, it is (I. Castellano, J. Fari\ {n}a-Asategui, M. E. Garciarena,
B. Marchionna, M. Qui... | 2 | solved | P.-E. Caprace | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "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 (I. Castellano, J. Fari\\ {n}a-Asategui, M. E. Garciarena,\nB. Marchionna, M. Quick, C. D. Re... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,571 | KOU-19.23 | Kourovka Notebook Problem 19.23 | For a group $G$, let $\operatorname{Tor}_{1}(G)$ be the normal closure of all torsion elements of $G$, and then by induction let $\operatorname{Tor}_{i+1}(G)$ be the inverse image of $\operatorname{Tor}_{1}(G/\operatorname{Tor}_{i}(G))$. The torsion length of $G$ is defined to be either the least positive integer $l$ s... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.23.
Proposed by: M. Chiodo, R. Vyas
Published resolution:
Yes, such groups exist (I. J. Leary, A. Minasyan, *J. Group Theory*, **26**, no. 4 (2023), 741--750). | 2 | solved | M. Chiodo, R. Vyas | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, such groups exist (I. J. Leary, A. Minasyan, *J. Group Theory*, **26**, no. 4 (2023), 741--750)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,572 | KOU-19.24 | Kourovka Notebook Problem 19.24 | For a group $G$, let $\operatorname{Tor}(G)$ be the normal closure of all torsion elements of $G$. Does there exist a finitely presented group $G$ such that $G/\operatorname{Tor}(G)$ is not finitely presented? Such a group must necessarily be non-hyperbolic. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.24.
Proposed by: M. Chiodo, R. Vyas
Published resolution:
Yes, such groups exist: one soluble example and another virtually torsion-free are constructed in (I. J. Leary, A. Min... | 2 | solved | M. Chiodo, R. Vyas | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, such groups exist: one soluble example and another virtually torsion-free are constructed in (I. J. Leary, A. Minasyan, *J. Group Theory*, **26**, no. 4 (20... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,573 | KOU-19.25 | Kourovka Notebook Problem 19.25 | Let $G$ and $H$ be finite groups of the same order with $\sum_{g\in G}\varphi (|g|) = \sum_{h\in H} \varphi (|h|)$, where $\varphi$ is the Euler totient function. Suppose that $G$ is simple. Is $H$ necessarily simple? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.25.
Proposed by: B. Curtin, G. R. Pourgholi
Discussion and literature:
[Solved] No, it need not be (AI solution confirmed by GAP: \ P. Monticone, *Preprint*,... | 2 | solved | B. Curtin, G. R. Pourgholi | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "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 need not be (AI solution confirmed by GAP: \\ P. Monticone, *Preprint*, 2026,\n\nhttps://kourovka... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,574 | KOU-19.26 | Kourovka Notebook Problem 19.26 | (Y. O. Hamidoune).
Suppose that $A$ and $B$ are finite subsets of a group $G$
such that $\vert A \vert \ge 2$ and $\vert B \vert \ge 2$, and let
$A\cdot_2B$ denote the set of elements of $G$ which can be expressed in the form
$ab$ for at least two different $(a,b) \in A \times B$. Is it true that
$\vert A \vert + \vert... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.26.
Proposed by: W. Dicks | 2 | open | W. Dicks | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,575 | KOU-19.27 | Kourovka Notebook Problem 19.27 | (Well-known question). A finitely generated group $G$ that acts on a tree in such a way that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag--Solitar group. The Bass--Serre theory gives finite presentations of such groups. Is the isomorphism problem soluble for
generalized Ba... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.27.
Proposed by: F. A. Dudkin | 2 | open | F. A. Dudkin | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,576 | KOU-19.28 | Kourovka Notebook Problem 19.28 | Let $G$ be a group, and $\varphi$ an automorphism of $G$. Elements $x,y\in G$ are said to be $\varphi$-conjugate if $x=z^{-1}y\varphi (z)$ for some $z\in G$. The $\varphi$-conjugacy is an equivalence relation; the number of $\varphi$-conjugacy classes is denoted by $R(\varphi )$.
*Conjecture*: If a finitely generated ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.28.
Proposed by: A. L. Fel'shtyn, E. V. Troitsky | 2 | open | A. L. Fel'shtyn, E. V. Troitsky | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,577 | KOU-19.29 | Kourovka Notebook Problem 19.29 | Let $\Phi \in \mathrm{Out } G = \mathrm{Aut} G/\mathrm{Inn} G$. Two automorphisms $\varphi,\psi\in\Phi$ are said to be isogradient if $\varphi=\hat g^{-1}\psi\hat g$ for some inner automorphism $\hat g$. The number of isogradiency classes in $\Phi$ is denoted by $S(\Phi )$.
*Conjecture*: If a finitely generated residu... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.29.
Proposed by: A. L. Fel'shtyn, E. V. Troitsky | 2 | open | A. L. Fel'shtyn, E. V. Troitsky | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,578 | KOU-19.30 | Kourovka Notebook Problem 19.30 | An element $g$ of a finite group $G$ is said to be vanishing if $\chi(g)=0$ for some irreducible complex character $\chi\in \mathrm{Irr}(G)$. Must a finite group and a finite simple group be isomorphic if they have equal orders and the same set of orders of vanishing elements? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.30.
Proposed by: M. Foroudi Ghasemabadi, A. Iranmanesh, | 2 | open | M. Foroudi Ghasemabadi, A. Iranmanesh, | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,579 | KOU-19.31 | Kourovka Notebook Problem 19.31 | Let $\omega(G,S)$ be the exponential growth rate of
a group $G$ with a finite generating set $S$ (see 14.7).
Let $MCG(\Sigma_g)$ be the mapping class group
of the orientable surface $\Sigma_g$ of genus $g$.
Is there a constant $C>1$ such that
$\omega(MCG(\Sigma_g), S) \ge C$
for every $g >0$ and every finite generating... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.31.
Proposed by: K. Fujiwara
Discussion and literature:
[Solved] No, there is no such constant that works for all genera; but if the genus $g$ is fixed, then... | 2 | solved | K. Fujiwara | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, there is no such constant that works for all genera; but if the genus $g$ is fixed, then such a cons... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,580 | KOU-19.32 | Kourovka Notebook Problem 19.32 | Let $p\geq 673$ be a prime and $r \geq 2$ be an integer. Let $B$ be the free group in the Burnside variety of exponent $p$ on $a_1, \dots , a_r$. Must every non-identical one-variable equation $w(a_1, \dots , a_r, x) = 1$ over $B$ have at most finitely many solutions in $B$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.32.
Proposed by: A. Gaglione | 2 | open | A. Gaglione | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,581 | KOU-19.33 | Kourovka Notebook Problem 19.33 | *Conjecture*: Let $G$ be a finite group, $p$ a prime number, and $P$ a Sylow $p$-subgroup of $G$. Suppose that an irreducible ordinary character $\chi$ of $G$ has degree divisible by $p$. If the restriction $\chi_P$ of $\chi$ to $P$ has a linear constituent, then $\chi_P$ has at least $p$ different linear constituents.... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.33.
Proposed by: E. Giannelli | 2 | open | E. Giannelli | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,582 | KOU-19.34 | Kourovka Notebook Problem 19.34 | Let $G$ be a finitely generated group such that for every element $g\in G$ the set of commutators $\{[x,g]\mid x\in G\}$ is a subgroup of $G$. Is it true that $G$ is residually nilpotent?
The answer is affirmative if $G$ is finite. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.34.
Proposed by: D. Gon\c{c
Discussion and literature:
alves, T. Nasybullov} | 2 | open | D. Gon\c{c | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,583 | KOU-19.35 | Kourovka Notebook Problem 19.35 | Let $G$ be a finite group of order $n$. Is it true that for every factorization $n=a_1 \cdots a_k$ there exist subsets $A_1,\dots ,A_k$ such that
$|A_1|=a_1,\dots ,|A_k|=a_k$ and $G=A_1 \cdots A_k$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.35.
Proposed by: M. H. Hooshmand
Published resolution:
No, it is not true. A counterexample with $k=3$ is given by the alternating group on 4 letters $G={A}_4$ and $(a_1, a_2, ... | 2 | solved | M. H. Hooshmand | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it is not true. A counterexample with $k=3$ is given by the alternating group on 4 letters $G={A}_4$ and $(a_1, a_2, a_3) =\n(2,3,2)$. (G. M. Bergman, *Lette... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,584 | KOU-19.36 | Kourovka Notebook Problem 19.36 | Let $G$ be a periodic group and let $\mathscr{I}=\{ x \in G \mid x^{2}=1 \not =x \}$ be the set of its involutions. Let $D$ be a non-empty set of odd integers greater than $1$; then $G$ is called a group with $D$-involutions if $G=\langle \mathscr{I} \rangle$ and for $x,y \in \mathscr{I}$ the order of $xy$ is in the se... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.36.
Proposed by: E. Jabara
Published resolution:
The conjecture is proved, even without using the hypothesis
that the group $G$ is periodic (E. Bettio, *J. Group Theory*, **24*... | 2 | solved | E. Jabara | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: The conjecture is proved, even without using the hypothesis\nthat the group $G$ is periodic (E. Bettio, *J. Group Theory*, **24** (2021), 1055--1067)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,585 | KOU-19.37 | Kourovka Notebook Problem 19.37 | {a)} Does there exist an absolute constant $k$ such that for any nilpotent injector $H$ of an arbitrary finite group $G$ there are $k$ conjugates of $H$ the intersection of which is equal to the Fitting subgroup $F(G)$ of $G$?
b) Can one choose $k=3$ as such a constant? This is true for finite soluble groups (D. S. Pa... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.37.
Proposed by: S. F. Kamornikov
Published resolution:
Yes, one can choose $k=3$ as such a constant, mod CFSG (V. I. Zenkov, *Siberian Math. J.*, **62**, no. 4 (2021), 621--63... | 2 | solved | S. F. Kamornikov | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, one can choose $k=3$ as such a constant, mod CFSG (V. I. Zenkov, *Siberian Math. J.*, **62**, no. 4 (2021), 621--637)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,586 | KOU-19.38 | Kourovka Notebook Problem 19.38 | Suppose that $H$ is a
subgroup of a finite soluble group $G$ that covers all Frattini chief factors of $G$ and avoids all complemented chief factors of $G$. Is it true that there are elements $x,y\in G$ such that $H\cap H^x\cap H^y=H_G$, where $H_G$ is the largest normal subgroup of $G$ contained in $H$?
This is true ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.38.
Proposed by: S. F. Kamornikov
Discussion and literature:
[Solved] Yes, it is true (S. F. Kamornikov, O. L. Shemetkova, *J. Algebra*, **641** (2024), 1--8... | 2 | solved | S. F. Kamornikov | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "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 (S. F. Kamornikov, O. L. Shemetkova, *J. Algebra*, **641** (2024), 1--8)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,587 | KOU-19.39 | Kourovka Notebook Problem 19.39 | A group is said to be $\mathfrak{X}$-critical if it does not belong to $\mathfrak{X}$ but all its proper subgroups belong to $\mathfrak{X}$.
Let $\mathfrak{F}$ be a soluble hereditary formation of finite groups for which all $\mathfrak{F}$-critical groups are either groups of prime order or $\mathfrak{U}$-critical grou... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.39.
Proposed by: S. F. Kamornikov | 2 | open | S. F. Kamornikov | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,588 | KOU-19.40 | Kourovka Notebook Problem 19.40 | Does Thompson's group $F$ (see 12.20) have the Howson property, that is, is the intersection of any two finitely generated subgroups of $F$ finitely generated? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.40.
Proposed by: I. Kapovich
Published resolution:
No, it does not. Otherwise any subgroup of $F$ would have this property. But the wreath product $\mathbb{Z}\wr \mathbb{Z}$, w... | 2 | solved | I. Kapovich | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it does not. Otherwise any subgroup of $F$ would have this property. But the wreath product $\\mathbb{Z}\\wr \\mathbb{Z}$, which does not have the Howson pro... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,589 | KOU-19.41 | Kourovka Notebook Problem 19.41 | Let $\phi$ be an automorphism of a free group $F_n$ of finite rank, and let $F_n\rtimes_\phi \mathbb{Z}$ be the split extension of $F_n$ by $\mathbb{Z}$ with $\mathbb{Z}$ acting as $\langle \phi\rangle$. Is the group $F_n\rtimes_\phi \mathbb{Z}$ conjugacy separable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.41.
Proposed by: I. Kapovich
Discussion and literature:
[Solved] Yes, it is (F. Dahmani, S. Hughes, M. Kudlinska, N. Touikan, *Preprint*, 2026, https://arxiv... | 2 | solved | I. Kapovich | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "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 (F. Dahmani, S. Hughes, M. Kudlinska, N. Touikan, *Preprint*, 2026, https://arxiv.org/abs/250... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,590 | KOU-19.42 | Kourovka Notebook Problem 19.42 | Suppose that $H$ is a word-hyperbolic subgroup of a word-hyperbolic group $G$ such that the inclusion of $H$ to $G$ extends to a continuous $H$-equivariant map $j:\partial H\to \partial G$ between their hyperbolic boundaries. If such an extension exists, it is unique and $j$ is called the *Cannon--Thurston map*.
a) Is... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.42.
Proposed by: I. Kapovich | 2 | open | I. Kapovich | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,591 | KOU-19.43 | Kourovka Notebook Problem 19.43 | Suppose that $\varphi $ is an automorphism of a finite soluble group $G$. Is the Fitting height of $G$ bounded in terms of $|\varphi |$ and $|C_G(\varphi )|$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.43.
Proposed by: E. I. Khukhro
Discussion and literature:
[Solved] Yes, it is (E. Khukhro,
*Bull. London Math. Soc.*, **58**, no. 1 (2026), Article ID e70206... | 2 | solved | E. I. Khukhro | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "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 (E. Khukhro,\n*Bull. London Math. Soc.*, **58**, no. 1 (2026), Article ID e70206)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,592 | KOU-19.44 | Kourovka Notebook Problem 19.44 | By definition a profinite group has finite rank at most $r$ if every subgroup of it can be (topologically) generated by $r$ elements.
Suppose that for every element $g$ of a profinite group $G$ there is a closed subgroup $E_g$ of finite rank such that for every $x\in G$ all sufficiently long Engel commutators $[x,g,\do... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.44.
Proposed by: E. I. Khukhro | 2 | open | E. I. Khukhro | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,593 | KOU-19.45 | Kourovka Notebook Problem 19.45 | Let \(G\) be a group generated by 3 class transpositions (see the definition in 17.57), and let \(m\) be the least common multiple of the moduli of the residue classes interchanged by the generators of \(G\). Assume that \(G\) does not setwisely stabilize any union of residue classes modulo \(m\) except for \(\varnothi... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.45.
Proposed by: S. Kohl | 2 | open | S. Kohl | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,594 | KOU-19.46 | Kourovka Notebook Problem 19.46 | Does the group \(\mathrm{CT}(\mathbb{Z})\) have finitely generated infinite periodic subgroups? (See the definition of \(\mathrm{CT}(\mathbb{Z})\) in 17.57). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.46.
Proposed by: S. Kohl | 2 | open | S. Kohl | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,595 | KOU-19.47 | Kourovka Notebook Problem 19.47 | Let $K$ be a finite extension of degree $n$ of a field $k$ of odd characteristic. The multiplicative group $K^{*}$ embeds into the group of all $k$-linear automorphisms $\mathrm{Aut}_{k}(K)$ by the rule $t(x) = tx$ for all $x\in K$. For a fixed basis of $K$ over $k$, the group $\mathrm{Aut}_k(K)$ is isomorphic to
$GL(n... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.47.
Proposed by: V. A. Ko\u{\i
Discussion and literature:
baev} | 2 | open | V. A. Ko\u{\i | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,596 | KOU-19.48 | Kourovka Notebook Problem 19.48 | A system of additive subgroups $\sigma_{ij}$, $1\leq i,j\leq
n$, of a field $K$ is called a *net* (or a *carpet*) of order
$n$ if $\sigma_{ir}\sigma_{rj}\subset\sigma_{ij}$ for all $i, r, j$.
A net that does not contain the diagonal is called an *elementary net*. A net $\sigma=(\sigma_{ij})$ is said to be *irreducible*... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.48.
Proposed by: V. A. Ko\u{\i
Discussion and literature:
baev} | 2 | open | V. A. Ko\u{\i | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,597 | KOU-19.49 | Kourovka Notebook Problem 19.49 | A *skew brace* is a set $B$ equipped with two operations $+$ and $\cdot$ such that $(B,+)$ is an additively written (but not necessarily abelian) group, $(B,\cdot )$ is a multiplicatively written group, and $a\cdot (b+c)=ab-a+ac$ for any $a,b,c\in B$.
Let $A$ be a skew brace with left-orderable multiplicative group. I... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.49.
Proposed by: V. Lebed, L. Vendramin
Published resolution:
No, not always (T. Nasybullov,
*J. Algebra*, **540** (2019), 156--167). | 2 | solved | V. Lebed, L. Vendramin | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always (T. Nasybullov,\n*J. Algebra*, **540** (2019), 156--167)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,598 | KOU-19.50 | Kourovka Notebook Problem 19.50 | A finite graph is said to be integral if all eigenvalues of its adjacency matrix are integers.
a) Let $G$ be a finite group generated by a normal subset $R$ consisting of involutions. Is it true that the Cayley graph $Cay(G,R)$ is integral?
b) Let $A_n$ be the alternating group of degree $n$, let $S=\{(123), (124),\d... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.50.
Proposed by: D. V. Lytkina
Published resolution:
a) Yes, it is true (D. O. Revin, *Letter of 21 April 2018 *; see also the reference for part (b) below; \ A. Abdollahi, *Le... | 2 | solved | D. V. Lytkina | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) Yes, it is true (D. O. Revin, *Letter of 21 April 2018 *; see also the reference for part (b) below; \\ A. Abdollahi, *Letter of 3 May 2018* ). Both proofs su... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,599 | KOU-19.51 | Kourovka Notebook Problem 19.51 | A finite group is *monomial* if each irreducible character of it is induced from a linear character of some subgroup. Monomial groups are soluble (Taketa). The group is *normally* (*subnormally*) *monomial* if each irreducible character is induced from a linear character of some normal (respectively, subnormal) subgrou... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.51.
Proposed by: A. Mann | 2 | open | A. Mann | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,600 | KOU-19.52 | Kourovka Notebook Problem 19.52 | The Gruenberg--Kegel graph (or the prime graph) $GK(G)$ of a finite group $G$ has vertex set consisting of all prime divisors of the order of $G$, and different vertices $p$ and $q$ are adjacent in $GK(G)$ if and only if the number $pq$ is an element order in $G$. Is there a finite non-solvable group $G$ such that $GK(... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.52.
Proposed by: N. V. Maslova | 2 | open | N. V. Maslova | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,601 | KOU-19.53 | Kourovka Notebook Problem 19.53 | Let $G$ be a group generated by elements $x,y,z$ such that
$$x^3=y^2=z^2=(xy)^3=(yz)^3=1\quad \text{and}\ \quad g^{12}=1 \text{ for all }g\in G.$$
Is it true that $|G|\leq 12$?
This is true if $G$ is finite (D. V. Lytkina, V. D. Mazurov, *Siberian Math. J.*, **56**, no. 3 (2015), 471--475). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.53.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,602 | KOU-19.54 | Kourovka Notebook Problem 19.54 | What are the chief factors of a finite group in which every
2-maximal subgroup is not $m$-maximal for any $m\ge 3$?
A subgroup $H$ of a group $G$ is said to be $m$-maximal if there is a chain of subgroups $ H=H_0<H_1< \ldots < H_{m-1}< H_m=G$ in which
$H_i$ is maximal in $H_{i+1}$ for every $i$. Note that for every $m... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.54.
Proposed by: V. S. Monakhov | 2 | open | V. S. Monakhov | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,603 | KOU-19.55 | Kourovka Notebook Problem 19.55 | Suppose that in a finite group $G$ every maximal subgroup $M$ is supersoluble whenever $\pi (M)=\pi (G)$, where $\pi (G)$ is the set of all prime divisors of the order of $G$.
a) What are the non-abelian composition factors of $G$?
b) Determine the exact upper bounds for the nilpotency length, the rank, and the $p$-l... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.55.
Proposed by: V. S. Monakhov
Published resolution:
a) Every nonabelian finite simple group can occur as a composition factor of $G$ (A. Moret\'o, *Monatsh. Math.*, **195**, ... | 2 | solved | V. S. Monakhov | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) Every nonabelian finite simple group can occur as a composition factor of $G$ (A. Moret\\'o, *Monatsh. Math.*, **195**, no. 3 (2021), 497--500).\n\nb) There i... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,604 | KOU-19.56 | Kourovka Notebook Problem 19.56 | A $\star$-commutator is a commutator $[x,y]$ of two elements $x,y$ of coprime prime-power orders. Is a finite group $G$ soluble if $|ab|\ge |a||b|$ for any $\star$\nobreakdash- commutators $a$ and $b$ of coprime orders? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.56.
Proposed by: V. S. Monakhov
Discussion and literature:
[Unverified AI claim] Yes, it is (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,
h... | 2 | open | V. S. Monakhov | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
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"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: M. Kinyon, J. Urban, *Preprint*, 2026,\nhttps... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,605 | KOU-19.57 | Kourovka Notebook Problem 19.57 | What are the non-abelian composition factors of a finite group $G$
in which every maximal subgroup is simple or $p$\nobreakdash- nilpotent for some fixed odd prime $p\in \pi (G)$?
*Comment of 2026*: These factors are determined in the cases where $G$ is non-simple, or alternating, or sporadic simple (A. Beltr\'an, F. ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.57.
Proposed by: V. S. Monakhov, V. N. Tyutyanov
Discussion and literature:
[Solved] These factors are described (AI solution confirmed by V. N. Tyutyanov: \... | 2 | solved | V. S. Monakhov, V. N. Tyutyanov | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
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"id": 2,
"level": 2,
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"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
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"display_name": "Kourovka Notebook (Issues 1–21)",
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"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
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"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] These factors are described (AI solution confirmed by V. N. Tyutyanov: \\ R. Sater, *Preprint*, 2026, ht... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,606 | KOU-19.58 | Kourovka Notebook Problem 19.58 | What are the non-abelian composition factors of a finite group
in which every maximal subgroup is simple or $p$-decomposable
for some fixed odd prime $p\in \pi (G)$?
*Comment of 2026*: These factors are determined in the cases where $G$ is non-simple, or alternating, or sporadic simple (A. Beltr\'an, F. Viudez, *Prepr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.58.
Proposed by: V. S. Monakhov, V. N. Tyutyanov
Discussion and literature:
[Solved] These factors are described (AI solution confirmed by V. N. Tyutyanov: \... | 2 | solved | V. S. Monakhov, V. N. Tyutyanov | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
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"slug": "group-theory",
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} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
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"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] These factors are described (AI solution confirmed by V. N. Tyutyanov: \\ R. Sater, *Preprint*, 2026, ht... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,607 | KOU-19.59 | Kourovka Notebook Problem 19.59 | Does the group of isometries of $\mathbb{Q}^3$ have a free non-abelian subgroup such that every non-trivial element acts without nontrivial fixed points in $\mathbb{Q}^3$?
The answer is yes if the field of rational numbers $\mathbb{Q}$ is replaced by the field $\mathbb{R}$ of real numbers (see G. Tomkowicz, S. Wagon, ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.59.
Proposed by: J. Mycielski | 2 | open | J. Mycielski | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,608 | KOU-19.60 | Kourovka Notebook Problem 19.60 | Let $R$ be a principal ideal domain, let $d$ be a positive integer, and let $X_1, \ldots, X_m$ be members of $\mathrm{SL}(d,R)$ (or of $\mathrm{GL}(d,R)$). Is it decidable whether or not these $m$ matrices freely generate a free group? If it is, design an effective algorithm for computing the answer. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.60.
Proposed by: P. M. Neumann | 2 | open | P. M. Neumann | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,609 | KOU-19.61 | Kourovka Notebook Problem 19.61 | Let $\mathfrak{A}=\{\mathfrak{A}_r\ |\ r\in \Phi\}$ be an elementary carpet of type $\Phi$ over a commutative ring $K$
(see 7.28), and let $\Phi(\mathfrak{A})=\langle x_r(\mathfrak{A}_r)\ |\ r\in\Phi\rangle$ be its carpet subgroup. Define the *closure* of the carpet $\mathfrak{A}$ to be the set of additive subgroups
$\... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.61.
Proposed by: Ya. N. Nuzhin
Discussion and literature:
[Solved] Yes, it is (AI solution confirmed by Ya. N. Nuzhin: \ M. Kinyon, J. Urban, *Preprint*, 202... | 2 | solved | Ya. N. Nuzhin | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "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 (AI solution confirmed by Ya. N. Nuzhin: \\ M. Kinyon, J. Urban, *Preprint*, 2026,\nhttps://k... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,610 | KOU-19.62 | Kourovka Notebook Problem 19.62 | Let $\mathfrak{A}=\{\mathfrak{A}_r\ |\ r\in \Phi\}$ be an elementary carpet of type $\Phi$ of rank $l\geq 2$ (see 7.28). For $p\in \Phi$, define a set of additive subgroups
$ \mathfrak{B}_p=\sum C_{ij,rs}\mathfrak{A}_r^i\mathfrak{A}_s^j$, where the sum is taken over all natural numbers $i$, $j$ and roots $r,s\in \Phi$ ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.62.
Proposed by: Ya. N. Nuzhin
Discussion and literature:
[Solved] Yes, it is (AI solution confirmed by Ya. N. Nuzhin: \ M. Kinyon, J. Urban, *Preprint*, 202... | 2 | solved | Ya. N. Nuzhin | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "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 (AI solution confirmed by Ya. N. Nuzhin: \\ M. Kinyon, J. Urban, *Preprint*, 2026,\nhttps://k... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,611 | KOU-19.63 | Kourovka Notebook Problem 19.63 | Let $\mathfrak{A}=\{\mathfrak{A}_r\ |\ r\in \Phi\}$ be an elementary carpet of type $\Phi$ over a commutative ring $K$
(see 7.28) and let $\mathfrak{A}_r^2=\{t^2\mid t\in\mathfrak{A}_r\}$. Are the inclusions
$\mathfrak{A}_r^2\mathfrak{A}_{-r}\subseteq\mathfrak{A}_r$, $r\in\Phi$, sufficient for the carpet $\mathfrak{A}$... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.63.
Proposed by: Ya. N. Nuzhin
Discussion and literature:
[Solved] Yes, they are sufficient (Ya. N. Nuzhin, *J. Siberian Fed. Univ. Ser. Math. Phys.*, **16**... | 2 | solved | Ya. N. Nuzhin | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
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"id": 2,
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"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
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"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
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"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, they are sufficient (Ya. N. Nuzhin, *J. Siberian Fed. Univ. Ser. Math. Phys.*, **16**, no. 6 (2023)... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,612 | KOU-19.64 | Kourovka Notebook Problem 19.64 | Let $G$ be a group, and $(g_1, \ldots , g_n)$ a tuple of its elements. The *type* of this tuple in $G$, denoted $Tp^G(g_1, \ldots , g_n)$, is the set of all first order formulas in free variables $x_1, \ldots ,x_n$ in the standard group theory language which are true on $(g_1, \ldots , g_n)$ in $G$.
Two groups $G$ and ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.64.
Proposed by: B. I. Plotkin | 2 | open | B. I. Plotkin | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
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} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,613 | KOU-19.65 | Kourovka Notebook Problem 19.65 | It is well known that varieties of groups form a free semigroup $N$ (A. L. Shmel'kin, *DAN SSSR*, **149** (1963), 543--545 (Russian)); \ B. H. Neuman, H. Neumann, P. M. Neumann, *Math. Z.*, **80** (1962), 44--62). Varieties of linear representations over an infinite field of zero characteristic also form a free semigro... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.65.
Proposed by: B. I. Plotkin | 2 | open | B. I. Plotkin | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,614 | KOU-19.66 | Kourovka Notebook Problem 19.66 | We say that a variety $\Theta$ is of *Tarski type* if any two non-abelian $\Theta$-free groups of finite rank are elementarily equivalent.
a) Find examples of Tarski type varieties distinct from the variety of all groups.
b) Is it true that the {Burnside variety $B_n$} of all groups of exponent $n$, where $n$ is big ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.66.
Proposed by: \smash{B. I. Plotkin, E. B. Plotkin | 2 | open | \smash{B. I. Plotkin, E. B. Plotkin | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,615 | KOU-19.67 | Kourovka Notebook Problem 19.67 | Let $G\le \mathrm{Sym}(\Omega)$, where $\Omega$ is finite. The $2$-closure $G^{(2)}$ of the group $G$ is defined to be the largest subgroup of $\mathrm{Sym}(\Omega)$ containing $G$ which has the same orbits as $G$ in the induced action on $\Omega\times\Omega$. Is it true that if $G$ is solvable, then every composition ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.67.
Proposed by: I. Ponomarenko
Published resolution:
No, it is not true (S. V. Skresanov,
*Algebra Logic*, **58**, no. 3 (2019), 249--253). | 2 | solved | I. Ponomarenko | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it is not true (S. V. Skresanov,\n*Algebra Logic*, **58**, no. 3 (2019), 249--253)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,616 | KOU-19.68 | Kourovka Notebook Problem 19.68 | For a finite group $G$ and a permutation group $K$, let $b_G(K)$ denote the number of conjugacy classes of regular subgroups of $K$ isomorphic to $G$. Does there exist a function $f$ such that
$b_G(K)\le n^{f(r)}$ for every abelian group $G$ of order $n$ and rank $r$, and every group $K$ such that $K^{(2)}=K$? (See Arc... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.68.
Proposed by: I. Ponomarenko | 2 | open | I. Ponomarenko | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
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"display_name": "Group Theory",
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,617 | KOU-19.69 | Kourovka Notebook Problem 19.69 | (P. Wesolek).
The acronym tdlc stands for totally disconnected and locally compact, and tdlcsc for tdlc and second countable. Let $\rm{Res}(G)$ denote the intersection of all open normal subgroups of a topological group $G$. The class of elementary tdlcsc groups is defined as the smallest class $\mathscr{E}$ of tdlcsc ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.69.
Proposed by: C. Reid | 2 | open | C. Reid | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,618 | KOU-19.70 | Kourovka Notebook Problem 19.70 | (P. Wesolek and P.-E. Caprace).
Let $\mathscr{S}$ denote the class of nondiscrete compactly generated, topologically simple tdlc groups. Let $G$ be a non-elementary tdlcsc group. Is there a compactly generated closed subgroup $H \leq G$ such that $H$ has a continuous quotient in $\mathscr{S}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.70.
Proposed by: C. Reid | 2 | open | C. Reid | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,619 | KOU-19.71 | Kourovka Notebook Problem 19.71 | (G. Willis).
Can a group $G$ in $\mathscr{S}$ be such that every element of $G$ normalizes a compact open subgroup of $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.71.
Proposed by: C. Reid | 2 | open | C. Reid | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,620 | KOU-19.72 | Kourovka Notebook Problem 19.72 | Let $G$ be a group in $\mathscr{S}$. Can every element of $G$ have trivial contraction group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.72.
Proposed by: C. Reid | 2 | open | C. Reid | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,621 | KOU-19.73 | Kourovka Notebook Problem 19.73 | (P.-E. Caprace and N. Monod).
Is there a compactly generated, locally compact group that is topologically simple, but not abstractly simple? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.73.
Proposed by: C. Reid, S. M. Smith | 2 | open | C. Reid, S. M. Smith | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,622 | KOU-19.74 | Kourovka Notebook Problem 19.74 | A subgroup $H$ of a group $G$ is called *pronormal* if $H$
and $H^g$ are conjugate in $\langle H,H^g\rangle$ for every $g\in
G$. A subgroup $H$ of a group $G$ is called *abnormal* if $g\in \langle H,H^g\rangle$ for every $g\in G$.
Does there exist an infinite group that does not contain nontrivial proper pronormal sub... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.74.
Proposed by: D. O. Revin
Discussion and literature:
[Solved] Yes, it does (S. Corson, *Monatsh. Math.* (2025) https://doi.org/10.1007/s00605-025-02116-8)... | 2 | solved | D. O. Revin | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "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 (S. Corson, *Monatsh. Math.* (2025) https://doi.org/10.1007/s00605-025-02116-8)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,623 | KOU-19.75 | Kourovka Notebook Problem 19.75 | Let $P$ be a finite 2-group of exponent $2^e$ such that the rank of every abelian subgroup is at most $r$. Is it true that $|P|\le 2^{r(e+1)}$?
This bound would be sharp (for a direct product of quaternion groups). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.75.
Proposed by: B. Sambale
Published resolution:
No, it is not true, as follows from the examples constructed in (A. Yu. Olshanskii, *Math. Notes*, **23** (1978), 183--185) (A... | 2 | solved | B. Sambale | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it is not true, as follows from the examples constructed in (A. Yu. Olshanskii, *Math. Notes*, **23** (1978), 183--185) (A. Mann, *Letter of 23 April 2018*).... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,624 | KOU-19.76 | Kourovka Notebook Problem 19.76 | A semigroup presentation is called *tree-like* if all
relations have the form $a=bc$ where $a, b, c$ are letters and no two
relations share the left-hand side or the right-hand side.
Is it decidable whether the semigroup given by a finite tree-like
presentation contains an idempotent?
This is equivalent to the questio... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.76.
Proposed by: M. Sapir | 2 | open | M. Sapir | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,625 | KOU-19.77 | Kourovka Notebook Problem 19.77 | *The Stable Small Cancellation Conjecture*: If $ F$ is a free group of rank $r\geq 2$, then, for any fixed $n > 0$, there exists a generic subset $S$ of $F^n$ such that,
for any automorphism $\phi$ of $F$, the set $\phi(S)$ satisfies the small cancellation condition $C'(1/6)$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.77.
Proposed by: P. E. Schupp | 2 | open | P. E. Schupp | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,626 | KOU-19.78 | Kourovka Notebook Problem 19.78 | Does there exist an infinite simple subgroup of $SL(2,{\mathbb{Q})$?
This is a special case of a question which Serge Cantat asked me about ``non-algebraic'' simple subgroups of $GL(n,{\mathbb{C})$. It is also a special case of 15.57. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.78.
Proposed by: J.-P. Serre | 2 | open | J.-P. Serre | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,627 | KOU-19.79 | Kourovka Notebook Problem 19.79 | (T. Springer). Let $G$ be a group. Suppose that for every integer $n > 0$ the group $G$ has a unique (up to isomorphism) irreducible complex linear representation of dimension $n$. (Note that $G = SL(2,{\mathbb{Q})$ has these properties, by a theorem of Borel--Tits (*Ann. Math.*, **97** (1973), 499--571).) Is it true t... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.79.
Proposed by: J.-P. Serre | 2 | open | J.-P. Serre | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,628 | KOU-19.80 | Kourovka Notebook Problem 19.80 | For a periodic group $G$, let $\pi _e(G)$ denote the set of orders of elements of $G$. A periodic group $G$ is said to be an *$OC_n $-group* if $\pi _e(G)=\{ 1, 2, \ldots , n\}$.
Is it true that every $OC_7 $-group is isomorphic to the alternating group $A_7$?
For finite groups, the answer is affirmative. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.80.
Proposed by: W. J. Shi
Published resolution:
Yes, it is true (E. Jabara, A. S. Mamontov, *Siberian Math. J.*, **62**, no. 1 (2021), 94--104). | 2 | solved | W. J. Shi | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is true (E. Jabara, A. S. Mamontov, *Siberian Math. J.*, **62**, no. 1 (2021), 94--104)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,629 | KOU-19.81 | Kourovka Notebook Problem 19.81 | (Well-known problem). Is the conjugacy problem in the braid group $B_n$ in the class NP (that is, decidable in nondeterministic polynomial time with respect to the maximum of the lengths $|u|$, $|v|$, where $u, v$ are the input braid words)? A stronger question: given two conjugate elements of $B_n$ represented by brai... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.81.
Proposed by: V. Shpilrain
Published resolution:
Yes, there is such a conjugator, even for any mapping class groups (J. Tao,
*Geom. Funct. Anal.*, **23**, no. 1 (2013), 415-... | 2 | solved | V. Shpilrain | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there is such a conjugator, even for any mapping class groups (J. Tao,\n*Geom. Funct. Anal.*, **23**, no. 1 (2013), 415--466)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,630 | KOU-19.82 | Kourovka Notebook Problem 19.82 | Let $h^*(G)$ denote the generalized Fitting height of a finite group $G$ defined as the minimum number $k$ such that $F^*_k(G)=G$, where $F^*_1(G)=F^*(G)$ is the generalized Fitting subgroup of $G$, and by induction $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_i(G))$. If $G$ is soluble, then $h^*(G)=h(G)$ is the F... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.82.
Proposed by: P. Shumyatsky | 2 | open | P. Shumyatsky | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
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"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,631 | KOU-19.83 | Kourovka Notebook Problem 19.83 | An element $g$ of a group $G$ is almost Engel if there is a finite set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[x, {}_n g]$ belong to ${\mathscr E}(g)$, that is, for every $x\in G$ there is a positive integer $n(x,g)$ such that $[x, {}_n g]\in {\mathscr E}(g)$ whenever $n\geq n... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.83.
Proposed by: P. Shumyatsky | 2 | open | P. Shumyatsky | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,632 | KOU-19.84 | Kourovka Notebook Problem 19.84 | Let $\mathbb{P}$ be the set of all primes, and let
$\sigma =\{\sigma_{i} \mid i\in I\}$ be some partition of
$\mathbb{P}$ into disjoint subsets. A finite group $G$ is said to be *$\sigma$-primary* if $G$ is a $\sigma_{i}$-group for some $i$; *$\sigma$-nilpotent* if $G$ is a direct product of $\sigma$-primary groups;
*$... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.84.
Proposed by: A. N. Skiba
Published resolution:
No, not always. For example, in $G = S_5$ with partition $\sigma = \{2, 3\}\cup \{5\}$ the subgroup $A =\langle (12) \rangle$... | 2 | solved | A. N. Skiba | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always. For example, in $G = S_5$ with partition $\\sigma = \\{2, 3\\}\\cup \\{5\\}$ the subgroup $A =\\langle (12) \\rangle$ is $\\sigma$-subnormal in $... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,633 | KOU-19.85 | Kourovka Notebook Problem 19.85 | Suppose that every Schmidt subgroup of a finite group $G$ is ${\sigma}$-subnormal in $ G$ (see Archive 19.84). Is it true that then there is a normal $\sigma$-nilpotent subgroup $N\leq G$ such that $G/N$ is cyclic?
An affirmative answer is known if $\sigma =\{\{2\}, \{3 \}, \ldots \}$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.85.
Proposed by: A. N. Skiba
Published resolution:
Yes, it is true (X. Yi, S. F. Kamornikov, *J. Algebra*, **560** (2020), 181--191). | 2 | solved | A. N. Skiba | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
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"slug": "group-theory",
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"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (X. Yi, S. F. Kamornikov, *J. Algebra*, **560** (2020), 181--191)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,634 | KOU-19.86 | Kourovka Notebook Problem 19.86 | Suppose that a finite group $G$ has a
$\sigma_{i}$-Hall subgroup for every $i \in I$. Suppose that a subgroup $A\leq G$ is such that $A\cap H$ is a
$\sigma_{i}$-Hall subgroup of $A$ for every $i\in I$ and every $\sigma_{i}$-Hall subgroup $H$ of $G$ (see 19.84 in Archive). Is it true that then $A$ is ${\sigma}$-subnorma... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.86.
Proposed by: A. N. Skiba | 2 | open | A. N. Skiba | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,635 | KOU-19.87 | Kourovka Notebook Problem 19.87 | Suppose that for every Sylow subgroup $P$ of a finite group $G$ and every maximal subgroup $V$ of $P$ there is a
${\sigma}$-soluble subgroup $T$ such that $VT=G$. Is it true that then $G$ is ${\sigma}$-soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.87.
Proposed by: A. N. Skiba
Published resolution:
Yes, it is true (A.-M. Liu, W. Guo, I. N. Safonova, A. N. Skiba, *J. Algebra*, **585** (2021), 280--293); another solution us... | 2 | solved | A. N. Skiba | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is true (A.-M. Liu, W. Guo, I. N. Safonova, A. N. Skiba, *J. Algebra*, **585** (2021), 280--293); another solution using CFSG is in (S. F. Kamornikov, V.... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,636 | KOU-19.88 | Kourovka Notebook Problem 19.88 | Suppose that for every Sylow subgroup $P$ of a finite group $G$ and every maximal subgroup $V$ of $P$ there is a
${\sigma}$-nilpotent subgroup $T$ such that $VT=G$. Is it true that then $G$ is ${\sigma}$-nilpotent? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.88.
Proposed by: A. N. Skiba
Published resolution:
Yes, it is true (A.-M. Liu, W. Guo, I. N. Safonova, A. N. Skiba, *J. Algebra*, **585** (2021), 280--293); another solution us... | 2 | solved | A. N. Skiba | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is true (A.-M. Liu, W. Guo, I. N. Safonova, A. N. Skiba, *J. Algebra*, **585** (2021), 280--293); another solution using CFSG is in (S. F. Kamornikov, V.... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,637 | KOU-19.89 | Kourovka Notebook Problem 19.89 | A permutation group is *subdegree-finite* if every orbit of every point stabiliser is finite. Let $\Omega$ be countably infinite, and let $G$ be a closed and subdegree-finite subgroup of $\mathrm{Sym}(\Omega)$ regarded as a topological group under the topology of pointwise convergence.
*Conjecture*: If the minimal deg... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.89.
Proposed by: S. M. Smith
Discussion and literature:
[Solved] The conjecture is proved (F. Lehner, *Bull. London Math. Soc.*, **55** (2023), 1874--1879). | 2 | solved | S. M. Smith | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "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 proved (F. Lehner, *Bull. London Math. Soc.*, **55** (2023), 1874--1879)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,638 | KOU-19.90 | Kourovka Notebook Problem 19.90 | A *skew brace* is a set $B$ equipped with two operations $+$ and $\cdot$ such that $(B,+)$ is an additively written (but not necessarily abelian) group, $(B,\cdot )$ is a multiplicatively written group, and $a\cdot (b+c)=ab-a+ac$ for any $a,b,c\in B$.
b) Is there a skew brace with non-soluble additive group but nilpot... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 19.90.
Proposed by: A. Smoktunowicz, L. Vendramin
Archive of Solved Problems entry:
a) Yes, there is (T. Nasybullov, *J. Algebra*, **540** (2019), 156--167).... | 2 | partially_solved | A. Smoktunowicz, L. Vendramin | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (a) Yes, there is (T. Nasybullov, *J. Algebra*, **540** (2019), 156--167).\n\nd) No, there is not (C. Tsang, Q. Chao, * Int. J. Algebra Comput.*, **30**, no. 2 (2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,639 | KOU-19.91 | Kourovka Notebook Problem 19.91 | Let $G$ be a finite group with an abelian Sylow $p$-subgroup $A$. Suppose that
$B$ is a strongly closed elementary abelian subgroup of $A$. Without invoking the
Classification Theorem for Finite Simple Groups (CFSG), prove that $G$ has a normal
subgroup $N$ such that $B =\Omega _1(A\cap N)$.
For $p = 2$, this is a cor... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.91.
Proposed by: R. Solomon | 2 | open | R. Solomon | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,640 | KOU-19.92 | Kourovka Notebook Problem 19.92 | Let $A$ be the algebra of $3\times 3$ skew-Hermitian matrices over the real octonions, where multiplication is given by bracket product. Determine $\mathrm{Aut}(A)$.
The question might be of interest for octonion algebras over a different base field or ring. The question was inspired by an observation of John Faulkner... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.92.
Proposed by: R. Solomon
Discussion and literature:
[Solved] It is determined: $Aut(A)\cong G_2 \times SO(3)$, which follows from the result about derivat... | 2 | solved | R. Solomon | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] It is determined: $Aut(A)\\cong G_2 \\times SO(3)$, which follows from the result about derivations in (... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,641 | KOU-19.93 | Kourovka Notebook Problem 19.93 | *Conjecture*:
There exists a function $f:\mathbb{N}\times\mathbb{N}\to \mathbb{N}$ such that, if $G$ is a finite $p$-group with $d$ generators and $G$ has no epimorphic images isomorphic to the wreath product $C_p\wr C_p$, then each factor of the $p$-lower central series of $G$ has order bounded above by $f(p,d)$
a) f... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.93.
Proposed by: P. Spiga
Discussion and literature:
[Solved] a) The conjecture is refuted for $p\geq 7$ (AI solution confirmed by P. Spiga: \ M. Kinyon, J. ... | 2 | partially_solved | P. Spiga | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
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"id": 2,
"level": 2,
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"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
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"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] a) The conjecture is refuted for $p\\geq 7$ (AI solution confirmed by P. Spiga: \\ M. Kinyon, ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,642 | KOU-19.94 | Kourovka Notebook Problem 19.94 | By definition two elements $g_1$ and $g_2$ of a free metabelian group $F$ with basis $\{x_1,\ldots, x_n\}$ have the same distribution if the equations $g_1(x_1,\ldots,
x_n) = g$ and $g_2(x_1,\ldots, x_n) = g$ have the same number of solutions in any finite metabelian group $G$ for every $g \in G$. Is it true that eleme... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.94.
Proposed by: E. I. Timoshenko
Discussion and literature:
[Solved] No, it is not true (V. Ionin, A. Semidetnov, *Preprint*, 2026, https://arxiv.org/pdf/26... | 2 | solved | E. I. Timoshenko | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it is not true (V. Ionin, A. Semidetnov, *Preprint*, 2026, https://arxiv.org/pdf/2608.29219)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,643 | KOU-19.95 | Kourovka Notebook Problem 19.95 | Let $G_\Gamma$ be a partially commutative soluble group of derived length $n \geq 3$ with defining graph $\Gamma$ (the definition is similar to the case of $n=2$, see 17.104). Is it true that the centralizer of any vertex of the graph is generated by its adjacent vertices? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.95.
Proposed by: E. I. Timoshenko | 2 | open | E. I. Timoshenko | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,644 | KOU-19.96 | Kourovka Notebook Problem 19.96 | Let $M$ be a free metabelian group of finite rank $r \geq 2$, and let $P$ be the set of its primitive elements. (An element of a relatively free group is said to be primitive if it can be included in a basis of the group.)
a) Is $P$ a first-order definable set?
b) Is the set of bases
of the group
$M$ a first-order de... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.96.
Proposed by: E. I. Timoshenko | 2 | open | E. I. Timoshenko | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,645 | KOU-19.97 | Kourovka Notebook Problem 19.97 | Let $G$ be an $m$-generated group the elementary theory of which $Th(G)$ coincides with the elementary theory
$Th(F)$ of a free soluble group $F$ of derived length $n
\geq 3$ of finite rank $r \geq 2$. Must the groups $G$ and
$F$ be isomorphic?
The answer is affirmative if $m \leq r$ or $n=2$ | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.97.
Proposed by: E. I. Timoshenko | 2 | open | E. I. Timoshenko | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,646 | KOU-19.98 | Kourovka Notebook Problem 19.98 | A connected graph $\Sigma$ is a *symmetrical extension* of a graph
$\Gamma$ by a graph $\Delta$ if there exist a vertex-transitive group $G$ of automorphisms of $\Sigma$ and an imprimitivity system $\sigma$ of $G$ on the set of vertices of $\Sigma$ such that the quotient graph $\Sigma/\sigma$ is
isomorphic to $\Gamma$ ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.98.
Proposed by: V. I. Trofimov
Published resolution:
(a) No, not always, as follows from the construction in the proof of Theorem H in (M. de la Salle, R. Tessera, *J. Topolog... | 2 | solved | V. I. Trofimov | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: (a) No, not always, as follows from the construction in the proof of Theorem H in (M. de la Salle, R. Tessera, *J. Topology*, **12** (2019), 705--743).\n\n(b) No... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,647 | KOU-19.99 | Kourovka Notebook Problem 19.99 | Is it true that for any positive integer $t$ there is a positive integer $n(t)$ such that any finite group with at least $n(t)$ conjugate classes of soluble maximal subgroups of Fitting height at most $t$ is itself a soluble group of Fitting height at most $t$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.99.
Proposed by: A. F. Vasil'ev, T. I. Vasil'eva | 2 | open | A. F. Vasil'ev, T. I. Vasil'eva | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,648 | KOU-19.100 | Kourovka Notebook Problem 19.100 | Suppose that a finite group $G$ admits a factorization $G=AB=BC=CA$, where $A, B, C$ are abnormal supersoluble subgroups. Is $G$ supersoluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.100.
Proposed by: A. F. Vasil'ev, T. I. Vasil'eva
Discussion and literature:
[Solved] Yes, it is (L. S. Kazarin, V. N. Tyutyanov, *Preprint*, 2023 (Russian),... | 2 | solved | A. F. Vasil'ev, T. I. Vasil'eva | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "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 (L. S. Kazarin, V. N. Tyutyanov, *Preprint*, 2023 (Russian),\n\nhttps://kourovkanotebookorg.w... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,649 | KOU-19.101 | Kourovka Notebook Problem 19.101 | The maximum length of a chain of nested centralizers of a group is called its $c$-dimension. Let $G$ be a locally finite group of finite $c$-dimension $k$, and let $S$ be the preimage in $G$ of the socle of $G/R$, where $R$ is the locally solvable radical of $G$. Is it true that the factor group $G/S$ contains an abeli... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 19th issue (2018). Original problem number: 19.101.
Proposed by: A. V. Vasil'ev
Published resolution:
Yes, it is true (A. A. Buturlakin,
*J. Algebra Appl.*, **18**, no. 12 (2019), 1950223, 12 pp). | 2 | solved | A. V. Vasil'ev | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is true (A. A. Buturlakin,\n*J. Algebra Appl.*, **18**, no. 12 (2019), 1950223, 12 pp)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,650 | KOU-19.102 | Kourovka Notebook Problem 19.102 | A subgroup $H$ of a free group $F$ is called *inert* if $r(H\cap K)\leq r(K)$ for every $K\leq F$; and *compressed* if $r(H)\leq r(K)$ for every $H\leq K\leq F$.
Is it true that compressed subgroups are inert? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.102.
Proposed by: E. Ventura
Discussion and literature:
[Solved] Yes, it is true (A. Jaikin-Zapirain, *Preprint*, 2024, https://arxiv.org/pdf/2403.09515 ). | 2 | solved | E. Ventura | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (A. Jaikin-Zapirain, *Preprint*, 2024, https://arxiv.org/pdf/2403.09515 )."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,651 | KOU-19.103 | Kourovka Notebook Problem 19.103 | Is it true that an intersection of compressed subgroups is compressed?
It is known that arbitrary intersections of inert subgroups are inert. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.103.
Proposed by: E. Ventura
Discussion and literature:
[Solved] Yes, it is true (A. Jaikin-Zapirain, *Preprint*, 2024, https://arxiv.org/pdf/2403.09515 ). | 2 | solved | E. Ventura | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (A. Jaikin-Zapirain, *Preprint*, 2024, https://arxiv.org/pdf/2403.09515 )."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,652 | KOU-19.104 | Kourovka Notebook Problem 19.104 | Is there an algorithm which decides whether a given subgroup of $F$ is inert?
An algorithm to decide whether a given $H$ is compressed is known. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.104.
Proposed by: E. Ventura
Discussion and literature:
[Solved] Yes, there is (A. Jaikin-Zapirain, *Preprint*, 2024, https://arxiv.org/pdf/2403.09515 ). | 2 | solved | E. Ventura | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, there is (A. Jaikin-Zapirain, *Preprint*, 2024, https://arxiv.org/pdf/2403.09515 )."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,653 | KOU-19.105 | Kourovka Notebook Problem 19.105 | Is it true that the fixed subgroups of endomorphisms of $F$ are inert? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 19.105.
Proposed by: E. Ventura
Discussion and literature:
[Solved] Yes, it is true (A. Jaikin-Zapirain, *Preprint*, 2024, https://arxiv.org/pdf/2403.09515 ). | 2 | solved | E. Ventura | 2,018 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (A. Jaikin-Zapirain, *Preprint*, 2024, https://arxiv.org/pdf/2403.09515 )."
} | 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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