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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
700,754 | KOU-11.62 | Kourovka Notebook Problem 11.62 | Describe the groups over
which any equation is soluble. In particular, is it true that this
class of groups coincides with the class of torsion-free groups?
It is easy to see that a group, over which every equation is
soluble, is torsion-free. On the other hand, S. D. Brodskiĭ
(*Siberian Math. J.*, **25**, no. 2 (1984)... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.62.
Proposed by: D. I. Moldavanski\u{\i | 2 | open | D. I. Moldavanski\u{\i | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,755 | KOU-11.63 | Kourovka Notebook Problem 11.63 | Suppose that $G$ is a
one-relator group containing non-trivial elements of finite order
and $N$ is a subgroup of $G$ generated by all elements of finite
order. Is it true that any subgroup of $G$ that intersects $N$
trivially is a free group? One can show that the answer is
affirmative in the cases where $ G/N$ has non... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.63.
Proposed by: D. I. Moldavanski\u{\i | 2 | open | D. I. Moldavanski\u{\i | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,756 | KOU-11.64 | Kourovka Notebook Problem 11.64 | Let $\pi (G)$ denote the set of
prime divisors of the order of a finite group $G$. Are there only
finitely many finite simple groups $G$, different from alternating
groups, which have a proper subgroup $H$ such that $\pi (H)=\pi (G)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.64.
Proposed by: V. S. Monakhov
Published resolution:
No, there are
infinitely many such groups. If
$G=S_{4k}(2^s)$ and $H\cong \Omega ^{-}_{4k}(2^s)$, then $\pi
(G)=\pi (H)$ (... | 2 | solved | V. S. Monakhov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, there are\ninfinitely many such groups. If\n$G=S_{4k}(2^s)$ and $H\\cong \\Omega ^{-}_{4k}(2^s)$, then $\\pi\n(G)=\\pi (H)$ (V. I. Zenkov, *Letter of March, ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,757 | KOU-11.65 | Kourovka Notebook Problem 11.65 | *Conjecture:* any
finitely generated soluble torsion-free pro-$p
$-group
with decidable elementary theory is an analytic
pro-$p
$-group. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.65.
Proposed by: A. G. Myasnikov,
V. N. Remeslennikov | 2 | open | A. G. Myasnikov,
V. N. Remeslennikov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,758 | KOU-11.66 | Kourovka Notebook Problem 11.66 | (Yu. L. Ershov). Is the
elementary theory of a free pro-$p
$-group decidable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.66.
Proposed by: A. G. Myasnikov, V. N. Remeslennikov | 2 | open | A. G. Myasnikov, V. N. Remeslennikov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,759 | KOU-11.67 | Kourovka Notebook Problem 11.67 | Does there exist a
torsion-free group
with exactly 3 classes of conjugate elements such that no
non-trivial conjugate class contains a pair of inverse elements? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.67.
Proposed by: B. Neumann
Discussion and literature:
[Solved] Yes, there does (V. Bagayoko, https://arxiv.org/abs/2509.09186). | 2 | solved | B. Neumann | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 does (V. Bagayoko, https://arxiv.org/abs/2509.09186)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,760 | KOU-11.68 | Kourovka Notebook Problem 11.68 | Can every fully ordered
group be embedded in a fully ordered group (continuing the given
order) with only 3 classes of conjugate elements? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.68.
Proposed by: B. Neumann
Published resolution:
No, not every
(V. V. Bludov, *Algebra and Logic*, **44**, no. 6
(2005), 370--380). | 2 | solved | B. Neumann | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 every\n(V. V. Bludov, *Algebra and Logic*, **44**, no. 6\n(2005), 370--380)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,761 | KOU-11.69 | Kourovka Notebook Problem 11.69 | A group $G$ acting on a
set $\Omega$ will be said to be {\it \mathrm{1}-$\{
2\}$-transitive} if it acts transitively on the set $ \Omega
^{1,\{ 2\}}=\{ (\alpha , \{ \beta ,\gamma \} )\mid \alpha ,\beta
,\gamma \mbox{ distinct}\} .$ Thus $G$ is 1-$\{ 2\}$-transitive
if and only if it is transitive and a stabilizer $G_{\... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.69.
Proposed by: P. M. Neumann | 2 | open | P. M. Neumann | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,762 | KOU-11.70 | Kourovka Notebook Problem 11.70 | Let $F$ be an infinite
field or a skew-field.
a) Find all transitive subgroups of $ PGL(2,F)$
acting on the projective line $F \cup \{ \infty \} $.
c) What are the flag-transitive subgroups of
$PGL(d + 1, F)$?
d) What subgroups of $PGL(d + 1, F)$ are
2-transitive on the points of $PG(d,F)$?
[Archived solved part(s)... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 11.70.
Proposed by: P. M. Neumann, C. E. Praeger
Archive of Solved Problems entry:
b) Such necessary and sufficient conditions are given in (S. A. Zyubin, *S... | 2 | partially_solved | P. M. Neumann, C. E. Praeger | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (b) Such necessary and sufficient conditions are given in (S. A. Zyubin, *Siberian Electron. Math. Rep.*, **11** (2014), 64--69 (Russian)).); remaining part(s) in... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,763 | KOU-11.71 | Kourovka Notebook Problem 11.71 | Let A be a finite group
with a normal subgroup $H$. A subgroup $U$ of $H$ is called an *$A$-covering subgroup* of $H$ if $\bigcup\limits_{a\in A} U^a=H$.
Is there a function $f\!: {\mathbb{N} \rightarrow {\mathbb{N}$ such that
whenever
$U< H < A$, where $A$ is a finite group, $H$ is a normal
subgroup of $ A$ of index $... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.71.
Proposed by: P. M. Neumann, C. E. Praeger | 2 | open | P. M. Neumann, C. E. Praeger | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,764 | KOU-11.72 | Kourovka Notebook Problem 11.72 | Suppose that a variety of
groups ${\mathfrak{V}}$ is *non-regular*, that is, the free group
$F_{n+1}({\mathfrak{V}})$ is embeddable in $F_n({\mathfrak{V}})$ for some $
n$. Is it true that then every countable group in ${\mathfrak{V}}$ is
embeddable in an $n$-generated group in ${\mathfrak{V}}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.72.
Proposed by: A. Yu. Olshanskii | 2 | open | A. Yu. Olshanskii | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,765 | KOU-11.73 | Kourovka Notebook Problem 11.73 | If a relatively free
group is finitely presented, is it virtually nilpotent? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.73.
Proposed by: A. Yu. Olshanskii | 2 | open | A. Yu. Olshanskii | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,766 | KOU-11.74 | Kourovka Notebook Problem 11.74 | Let $G$ be a non-elementary
hyperbolic group and let $G^n$ be the subgroup generated by the $n$th
powers of the elements of $G$.
a) (M. Gromov). Is it true that $G/G^n$ is
infinite for some $n = n(G)$?
b) Is it true that $\bigcap\limits_{n=1}^{\infty} G^n=\{ 1\}
$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.74.
Proposed by: A. Yu. Olshanskii
Published resolution:
a) Yes, it is; b) Yes, it is (S. V. Ivanov,
A. Yu. Olshanskii, *Trans. Amer. Math. Soc.*, **348**, N
6 (1996), 2091--21... | 2 | solved | A. Yu. Olshanskii | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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; b) Yes, it is (S. V. Ivanov,\nA. Yu. Olshanskii, *Trans. Amer. Math. Soc.*, **348**, N\n6 (1996), 2091--2138)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,767 | KOU-11.75 | Kourovka Notebook Problem 11.75 | Let us consider the class of groups with $n$
generators and $ m$ relators. A subclass of this class is called
*dense* if the ratio of the number of presentations of the
form $\left< a_1,\ldots ,a_n\mid R_1,\ldots ,R_m\right>$ (where
$|R_i|=d_i$) for groups from this subclass to the number of all
such presentations conv... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.75.
Proposed by: A. Yu. Olshanskii
Published resolution:
This is proved (G. N. Arzhantseva, A. Yu. Olshanskii,
*Math. Notes*, **59**, no. 4 (1996), 350--355). | 2 | solved | A. Yu. Olshanskii | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This is proved (G. N. Arzhantseva, A. Yu. Olshanskii,\n*Math. Notes*, **59**, no. 4 (1996), 350--355)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,768 | KOU-11.76 | Kourovka Notebook Problem 11.76 | (Well-known problem). Is
the group of collineations of a finite non-Desarguasian
projective plane defined over a semi-field soluble? (The
hypothesis on the plane means that the corresponding regular set,
see Archive, 10.48, is closed under addition.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.76.
Proposed by: N. D. Podufalov | 2 | open | N. D. Podufalov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,769 | KOU-11.77 | Kourovka Notebook Problem 11.77 | (Well-known problem). Describe the finite translation planes whose
collineation groups act doubly transitively on the set of
points of the line at infinity. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.77.
Proposed by: N. D. Podufalov | 2 | open | N. D. Podufalov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,770 | KOU-11.78 | Kourovka Notebook Problem 11.78 | An isomorphism of groups
of points of algebraic groups is called *semialgebraic* if
it can be represented as a composition of an isomorphism of
translation of the field of definition and a rational morphism.
a) Is it true that the existence of an
isomorphism of groups of points of two directly undecomposable
algebraic... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.78.
Proposed by: K. N. Ponomar\"ev | 2 | open | K. N. Ponomar\"ev | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,771 | KOU-11.79 | Kourovka Notebook Problem 11.79 | Let $G$ be a finite group of automorphisms of an
infinite field $F$ of characteristic $p$. Taking integral powers of
the elements of $F$ and the action of $G$ define the action of the
group ring ${\mathbb{Z}G$ of $G$ on the multiplicative group of $F$. Is
it true that any subfield of $F$ that contains the images of all... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.79.
Proposed by: K. N. Ponomar\"ev
Published resolution:
Yes
(K. N. Ponomar\"ev, *Siber. Math. J.*, **33** (1992),
1094--1099). | 2 | solved | K. N. Ponomar\"ev | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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\n(K. N. Ponomar\\\"ev, *Siber. Math. J.*, **33** (1992),\n1094--1099)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,772 | KOU-11.80 | Kourovka Notebook Problem 11.80 | Let $G$ be a primitive
permutation group on a finite set $\Omega$ and suppose that, for
$\alpha \in \Omega $, $G_{\alpha}$ acts 2-transitively on one of
its orbits in $\Omega \setminus \{ \alpha \} $. By
(C. E. Praeger, *J. Austral. Math. Soc. (A)*, **45**,
1988, 66--77) either
(a) $T \leq G \leq \mathrm{Aut} T$ for s... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.80.
Proposed by: C. E. Praeger | 2 | open | C. E. Praeger | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,773 | KOU-11.81 | Kourovka Notebook Problem 11.81 | A topological group is
said to be *$F$-balanced* if for any subset $X$ and any
neighborhood of the identity $U$ there is a neighborhood of the
identity $V$ such that $VX \subseteq XU$. Is every $ F$-balanced group
*balanced*, that is, does it have a basis of neighborhoods of
the identity consisting of invariant sets? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.81.
Proposed by: I. V. Protasov | 2 | open | I. V. Protasov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,774 | KOU-11.82 | Kourovka Notebook Problem 11.82 | Let $R$ be the normal
closure of an element $r$ in a free group $F$ with the natural length
function and suppose that $s$ is an element of minimal length in $R$.
Is it true that $s$ is conjugate to one of the following elements:
$r$, $r^{-1}$, $[r,f]$, $[r^{-1},f]$ for some $f \in F$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.82.
Proposed by: V. N. Remeslennikov
Published resolution:
No, not always (J. McCool, *Glasgow Math. J.*, **43**,
no. 1 (2001), 123--124). | 2 | solved | V. N. Remeslennikov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always (J. McCool, *Glasgow Math. J.*, **43**,\nno. 1 (2001), 123--124)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,775 | KOU-11.83 | Kourovka Notebook Problem 11.83 | Is the conjugacy problem
soluble for finitely generated abelian-by-polycyclic groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.83.
Proposed by: V. N. Remeslennikov | 2 | open | V. N. Remeslennikov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,776 | KOU-11.84 | Kourovka Notebook Problem 11.84 | Is the isomorphism
problem soluble
a) for finitely generated metabelian groups?
b) for finitely generated soluble groups of
finite rank? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.84.
Proposed by: V. N. Remeslennikov | 2 | open | V. N. Remeslennikov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,777 | KOU-11.85 | Kourovka Notebook Problem 11.85 | Let $F$ be a free
pro-$p
$-group with a basis $X$ and let $ R=r^F$ be the
closed normal subgroup generated by an element $ r$. We say that
an element $s$ of $F$ is *associated to $r$* if
$s^F=R$.
a) Suppose that none of the elements
associated to $r$ is a $p$th power of an element in
$F$. Is it true that $F/R$ is tors... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.85.
Proposed by: N. S. Romanovski\u{\i | 2 | open | N. S. Romanovski\u{\i | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,778 | KOU-11.86 | Kourovka Notebook Problem 11.86 | Does every group
$G=\left< x_1,\ldots ,x_n\mid r_1=\cdots =r_m=1\right> $ possess,
in a natural way, a homomorphic image $H=\left< x_1,\ldots
,x_n\mid s_1=\cdots =s_m=1\right> $
a) such that $H$ is a torsion-free group?
b) such that the integral group ring of $H$ is embeddable
in a skew field?
If the stronger assert... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.86.
Proposed by: V. A. Roman'kov | 2 | open | V. A. Roman'kov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,779 | KOU-11.87 | Kourovka Notebook Problem 11.87 | (Well-known problem). Is
the automorphism group of a free metabelian group of rank $n
\geq 4$ finitely presented? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.87.
Proposed by: V. A. Roman'kov | 2 | open | V. A. Roman'kov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,780 | KOU-11.88 | Kourovka Notebook Problem 11.88 | We define the *length* $l(g)$ of an Engel element $g$ of a
group $G$ to be the smallest number $l$ such that $[h,g; l]=1$
for all $h \in G$. Here $[h,g; 1]=[h,g]$ and $[h,g; i +
1]=[[h,g; i],g]$. Does there exist a polynomial
function $\varphi (x,y)$ such that $l(uv) \leq \varphi
(l(u),l(v))$? Up to now, it is unknown ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.88.
Proposed by: V. A. Roman'kov
Published resolution:
No, it does not (L. V. Dolbak, *Siberian Math. J.*, **47**, no. 1 (2006), 55--57). | 2 | solved | V. A. Roman'kov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
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"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"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: No, it does not (L. V. Dolbak, *Siberian Math. J.*, **47**, no. 1 (2006), 55--57)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,781 | KOU-11.89 | Kourovka Notebook Problem 11.89 | Let $k$ be an infinite
cardinal number. Describe the epimorphic images of the Cartesian
power $\prod\limits_{k}{\mathbb{Z}$ of the group ${\mathbb{Z}$ of
integers. Such a description is known for $k=\omega _0$
(R. Nunke, *Acta Sci. Math. (Szeged)*, **23** (1963),
67--73). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.89.
Proposed by: S. V. Rychkov | 2 | open | S. V. Rychkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,782 | KOU-11.90 | Kourovka Notebook Problem 11.90 | Let ${\mathfrak{V}}$ be a
variety of groups and let $k$ be an infinite cardinal number.
A group $G \in {\mathfrak{V}}$ of rank $k$ is called {\it almost free
in
${\mathfrak{V}}$} if each of its subgroups of rank less than $k$ is
contained in a subgroup of $G$ which is free in ${\mathfrak{V}}$. For
what $k$ do there exi... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.90.
Proposed by: S. V. Rychkov | 2 | open | S. V. Rychkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,783 | KOU-11.91 | Kourovka Notebook Problem 11.91 | Prove that a hereditary formation ${\mathfrak{F}}$ of finite soluble
groups is local if every finite soluble non-simple minimal
non-${\mathfrak{F}}$-group is a Shmidt group (that is, a non-nilpotent
finite group all of whose proper subgroups are nilpotent). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.91.
Proposed by: V. N. Semenchuk
Published resolution:
This is proved (A. N. Skiba, *Dokl. Akad. Nauk Belorus.
SSR*, **34**, no. 11 (1990), 982--985 (Russian)). | 2 | solved | V. N. Semenchuk | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This is proved (A. N. Skiba, *Dokl. Akad. Nauk Belorus.\nSSR*, **34**, no. 11 (1990), 982--985 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,784 | KOU-11.92 | Kourovka Notebook Problem 11.92 | What are the soluble
hereditary non-one-generator formations of finite groups all of
whose proper hereditary subformations are one-generated? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.92.
Proposed by: A. N. Skiba | 2 | open | A. N. Skiba | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,785 | KOU-11.93 | Kourovka Notebook Problem 11.93 | Is the variety of all lattices generated by the
block lattices (see Archive, 9.62) of finite groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.93.
Proposed by: D. M. Smirnov
Published resolution:
No, it is not
(D. M. Smirnov, *Siberian Math. J.*, **33**, no. 4
(1992), 663--668). | 2 | solved | D. M. Smirnov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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\n(D. M. Smirnov, *Siberian Math. J.*, **33**, no. 4\n(1992), 663--668)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,786 | KOU-11.94 | Kourovka Notebook Problem 11.94 | Describe all *simply
reducible* groups, that is, groups such that all their
characters are real and the tensor product of any two irreducible
representations contains no multiple components. This question is
interesting for physicists. Is every finite simply reducible group
soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.94.
Proposed by: S. P. Strunkov
Published resolution:
Yes, it is
(L. S. Kazarin, E. I. Chankov,
*Sb. Math.*, **201**, no. 5 (2010), 655--668). | 2 | solved | S. P. Strunkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is\n(L. S. Kazarin, E. I. Chankov,\n*Sb. Math.*, **201**, no. 5 (2010), 655--668)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,787 | KOU-11.95 | Kourovka Notebook Problem 11.95 | Suppose that $G$ is a $p
$-group
$G$ containing an element $a$ of order $p$
such that the subgroup $\left< a, a^g\right> $ is finite for any
$g$ and the set $C_G(a) \cap a^G$ is finite. Is it true that $G$
has non-trivial
centre? This is true for 2-groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.95.
Proposed by: S. P. Strunkov | 2 | open | S. P. Strunkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,788 | KOU-11.96 | Kourovka Notebook Problem 11.96 | (a) Is it true that, for a
given number $n$, there exist only finitely many finite simple
groups each of which contains an involution which commutes with at
most $n$ involutions of the group?
(b) Is it true that there are no
infinite simple groups satisfying this condition? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.96.
Proposed by: S. P. Strunkov
Discussion and literature:
[Solved] (a) Yes, it is true (mod CFSG); moreover, if a locally finite group has such an involutio... | 2 | solved | S. P. Strunkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] (a) Yes, it is true (mod CFSG); moreover, if a locally finite group has such an involution, then it is f... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,789 | KOU-11.97 | Kourovka Notebook Problem 11.97 | Are there only finitely many
finite simple groups with a given set of all different values of
irreducible characters on a single element? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.97.
Proposed by: S. P. Strunkov
Published resolution:
No; all
complex irreducible characters of the groups $L_2(2^m)$, $m\geq
2$, take the values $0, \pm1$ on involutions (V. D... | 2 | solved | S. P. Strunkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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; all\ncomplex irreducible characters of the groups $L_2(2^m)$, $m\\geq\n2$, take the values $0, \\pm1$ on involutions (V. D. Mazurov)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,790 | KOU-11.98 | Kourovka Notebook Problem 11.98 | a) (R. Brauer). Find the
best-possible estimate of the form $|G| \leq f(r)$ where $r$ is the
number of conjugacy classes of elements in a finite (simple)
group $G$.
[Archived solved part(s) of 11.98:]
b) Let $r$ be the number of conjugacy classes of
elements in a finite (simple) group $G$. Is it true that $|G|\leq
\ma... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 11.98.
Proposed by: S. P. Strunkov
Archive of Solved Problems entry:
No, this is not true: the group $M_{22}$ has order 443520 and
contains 12 conjugacy clas... | 2 | partially_solved | S. P. Strunkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (No, this is not true: the group $M_{22}$ has order 443520 and\ncontains 12 conjugacy classes (T. Plunkett, *Letter of May, 9,\n2000*).); remaining part(s) in mai... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,791 | KOU-11.99 | Kourovka Notebook Problem 11.99 | Find (in group-theoretic
terms) necessary and sufficient conditions for a finite group to
have complex irreducible characters having defect 0 for more than
one prime number dividing the order of the group. Express the
number of such characters in the same terms. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.99.
Proposed by: S. P. Strunkov | 2 | open | S. P. Strunkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,792 | KOU-11.100 | Kourovka Notebook Problem 11.100 | Is it true that every
periodic conjugacy biprimitively finite group (see 6.59) can be
obtained from
2-groups and binary finite groups by taking extensions? This
question is of independent interest for $p $-groups, $p$ an odd
prime. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.100.
Proposed by: S. P. Strunkov | 2 | open | S. P. Strunkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,793 | KOU-11.101 | Kourovka Notebook Problem 11.101 | Does there
exist a Golod group (see 9.76) with finite centre? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.101.
Proposed by: A. V. Timofeenko
Published resolution:
Yes, there is a Golod group with trivial centre
(V. A. Sereda, A. I. Sozutov, *Algebra and Logic*, **45**, no. 2 (2006)... | 2 | solved | A. V. Timofeenko | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there is a Golod group with trivial centre\n(V. A. Sereda, A. I. Sozutov, *Algebra and Logic*, **45**, no. 2 (2006), 134--138)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,794 | KOU-11.102 | Kourovka Notebook Problem 11.102 | Does there exist a
residually soluble but insoluble group satisfying the maximum
condition on subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.102.
Proposed by: J. Wiegold | 2 | open | J. Wiegold | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,795 | KOU-11.103 | Kourovka Notebook Problem 11.103 | Is a
2-group satisfying the minimum condition for centralizers necessarily
locally finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.103.
Proposed by: John S. Wilson
Published resolution:
Yes, it is (F. O. Wagner, *J. Algebra*, **217**, no. 2
(1999),
448--460). | 2 | solved | John S. Wilson | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (F. O. Wagner, *J. Algebra*, **217**, no. 2\n(1999),\n448--460)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,796 | KOU-11.104 | Kourovka Notebook Problem 11.104 | Let $G$ be a finite
group of order $p^a\cdot q^b\cdots $, where $p, q, \ldots $ are
distinct primes. Introduce distinct variables $x_p, x_q, \ldots $
corresponding to $p, q, \ldots $ . Define functions $f, \varphi$
from the lattice of subgroups of $G$ to the polynomial ring ${\mathbb{Z}[x_p,x_q,\ldots ]$ as follows:
(... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.104.
Proposed by: G. E. Wall
Published resolution:
Yes, the converses are
true:
a) for solvable groups (E. Detomi, A. Lucchini, *J. London
Math. Soc. (2)*, **70** (2004), 165--... | 2 | solved | G. E. Wall | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, the converses are\ntrue:\na) for solvable groups (E. Detomi, A. Lucchini, *J. London\nMath. Soc. (2)*, **70** (2004), 165--181); \\ b) for\n$p\n$-solvable g... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,797 | KOU-11.105 | Kourovka Notebook Problem 11.105 | b) Let ${\mathfrak{V}}$ be a
variety of groups. Its relatively free group of given rank has a
presentation $F/N$, where $F$ is absolutely free of the same rank
and $N$ fully invariant in $F$. The associated Lie ring ${\mathcal{L}}(F/N)$ has a presentation ${ L}/J$, where ${ L}$ is the free Lie
ring of the same rank and... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 11.105.
Proposed by: G. E. Wall
Archive of Solved Problems entry:
No, not always; the ideal $J$ is not fully invariant for
$F/(F^2)^4$, that is, for ${\mathf... | 2 | partially_solved | G. E. Wall | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (No, not always; the ideal $J$ is not fully invariant for\n$F/(F^2)^4$, that is, for ${\\mathfrak{V}}={\\mathfrak{B}}_4{\\mathfrak{B}}_2$\n(D. Groves, *J. Algebra... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,798 | KOU-11.106 | Kourovka Notebook Problem 11.106 | Can every periodic group be embedded in a simple
periodic group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.106.
Proposed by: R. Phillips
Published resolution:
Yes (A. Yu. Olshanskii, *Ukrain. Math. J.*, **44** (1992),
761--763). | 2 | solved | R. Phillips | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (A. Yu. Olshanskii, *Ukrain. Math. J.*, **44** (1992),\n761--763)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,799 | KOU-11.107 | Kourovka Notebook Problem 11.107 | Is there a non-linear
locally finite simple group each of whose proper subgroups is
residually finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.107.
Proposed by: R. Phillips | 2 | open | R. Phillips | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,800 | KOU-11.108 | Kourovka Notebook Problem 11.108 | Is every locally finite simple group absolutely
simple? A group $G$ is said to be *absolutely simple* if the only
composition series of $G$ is $\{ 1, G\} $. For equivalent
formulations see (R. E. Phillips, *Rend. Sem. Mat. Univ.
Padova*, **79** (1988), 213--220). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.108.
Proposed by: R. Phillips
Published resolution:
No, not every (U. Meierfrankenfeld, in: *Proc. Int. Conf. Finite
and Locally Finite Groups, Istanbul, 1994*, Kl\"uwer, 1995,... | 2 | solved | R. Phillips | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 every (U. Meierfrankenfeld, in: *Proc. Int. Conf. Finite\nand Locally Finite Groups, Istanbul, 1994*, Kl\\\"uwer, 1995,\n189--212)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,801 | KOU-11.109 | Kourovka Notebook Problem 11.109 | Is it true that the
union of an ascending series of groups of Lie type of equal ranks
over some fields should be also a group of Lie type of the same
rank over a field? (For locally finite fields the answer is
``yes'', the theorem in this case is due to V. Belyaev,
A. Borovik, B. Hartley \& G. Shute, and S. Thomas.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.109.
Proposed by: R. Phillips | 2 | open | R. Phillips | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,802 | KOU-11.110 | Kourovka Notebook Problem 11.110 | Is it possible to embed $SL(2,{\mathbb{Q})$ in the
multiplicative group of some division ring? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.110.
Proposed by: B. Hartley
Published resolution:
No, it is not (W. Dicks, B. Hartley, *Commun. Algebra*,
**19** (1991), 1919--1943). | 2 | solved | B. Hartley | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (W. Dicks, B. Hartley, *Commun. Algebra*,\n**19** (1991), 1919--1943)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,803 | KOU-11.111 | Kourovka Notebook Problem 11.111 | Does every infinite
locally finite simple group $G$ contain a nonabelian finite simple
subgroup? Is there an infinite tower of such subgroups in $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.111.
Proposed by: B. Hartley | 2 | open | B. Hartley | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,804 | KOU-11.112 | Kourovka Notebook Problem 11.112 | Let $L=L(K(p))$ be the
associated Lie ring of a free countably generated group $K(p)$ of
the Kostrikin variety of locally finite groups of a given prime
exponent $p$. Is it true that
a) $L$ is a relatively free Lie ring?
c) all identities of $L$ follow from a finite
number of identities of $L$?
[Archived solved part... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 11.112.
Proposed by: E. I. Khukhro
Archive of Solved Problems entry:
No, there are two relations of weight 29 which hold in $L(K(7))$
(in two generators, of ... | 2 | partially_solved | E. I. Khukhro | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (No, there are two relations of weight 29 which hold in $L(K(7))$\n(in two generators, of multiweights $(14,15)$ and $(15,14)$) that\nare not consequences of mult... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,805 | KOU-11.113 | Kourovka Notebook Problem 11.113 | (B. Hartley). Is it
true that the derived length of a nilpotent periodic group
admitting a regular automorphism of prime-power order $p^n$ is
bounded by a function of $p$ and $n$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.113.
Proposed by: E. I. Khukhro | 2 | open | E. I. Khukhro | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,806 | KOU-11.114 | Kourovka Notebook Problem 11.114 | Is every locally graded
group of finite special rank almost hyperabelian? This is true in
the class of periodic locally graded groups (*Ukrain.
Math. J.*, **42**, no. 7 (1990), 855--861). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.114.
Proposed by: N. S. Chernikov | 2 | open | N. S. Chernikov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,807 | KOU-11.115 | Kourovka Notebook Problem 11.115 | Suppose that $X$ is a
free non-cyclic group, $N$ is a non-trivial normal subgroup of $X$
and $T$ is a proper subgroup of $ X$ containing $N$. Is it true
that $[T,N] < [X,N]$ if $N$ is not maximal in $X$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.115.
Proposed by: V. P. Shaptala
Discussion and literature:
[Unverified AI claim] No, not always (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 20... | 2 | open | V. P. Shaptala | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] No, not always (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,\n\... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,808 | KOU-11.116 | Kourovka Notebook Problem 11.116 | The *dimension* of a
partially ordered set $\left< P, \leq \right> $ is, by
definition, the least cardinal number $\delta$ such that the
relation $\leq$ is an intersection of $\delta$ relations of linear
order on $P$. Is it true that, for any Chernikov group which does
not contain a direct product of two quasicyclic gr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.116.
Proposed by: L. N. Shevrin
Discussion and literature:
[Solved] Yes, it is true (AI solution confirmed by P. Shumyatsky: \ M. Kinyon, J. Urban, *Preprint... | 2 | solved | L. N. Shevrin | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (AI solution confirmed by P. Shumyatsky: \\ M. Kinyon, J. Urban, *Preprint*, 2026,\nhttp... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,809 | KOU-11.117 | Kourovka Notebook Problem 11.117 | Let ${\mathfrak{X}}$ be a
soluble non-empty Fitting class. Is it true that every finite
non-soluble group possesses an ${\mathfrak{X}}$-injector? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.117.
Proposed by: L. A. Shemetkov | 2 | open | L. A. Shemetkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,810 | KOU-11.118 | Kourovka Notebook Problem 11.118 | What are the hereditary
soluble local formations ${\mathfrak{F}}$ of finite groups such that every
finite group has an ${\mathfrak{F}}$-covering subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.118.
Proposed by: L. A. Shemetkov | 2 | open | L. A. Shemetkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,811 | KOU-11.119 | Kourovka Notebook Problem 11.119 | Is it true that, for any
non-empty set of primes $\pi$, the $\pi $-length of any finite
$\pi $-soluble group does not exceed the derived length of its
Hall $\pi $-subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.119.
Proposed by: L. A. Shemetkov | 2 | open | L. A. Shemetkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,812 | KOU-11.120 | Kourovka Notebook Problem 11.120 | Let ${\mathfrak{F}}$ be a
soluble saturated Fitting formation. Is it true that $l_{\mathfrak{F}}(G)
\leq f(c_{\mathfrak{F}}(G))$, where $c_{\mathfrak{F}}(G)$ is the length of a
composition series of some ${\mathfrak{F}}$-covering subgroup of
a finite soluble group $G$? This is Problem 17 in
(L. A. Shemetkov, *Formation... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.120.
Proposed by: L. A. Shemetkov | 2 | open | L. A. Shemetkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,813 | KOU-11.121 | Kourovka Notebook Problem 11.121 | Does there exist a local
formation of finite groups which has a non-trivial decomposition
into a product of two formations and in any such a decomposition
both factors are non-local? Local products of non-local formations
do exist. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.121.
Proposed by: L. A. Shemetkov | 2 | open | L. A. Shemetkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,814 | KOU-11.122 | Kourovka Notebook Problem 11.122 | Does every non-zero
submodule of a free module over the group ring of a torsion-free
group contain a free cyclic submodule? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.122.
Proposed by: A. L. Shmel'kin | 2 | open | A. L. Shmel'kin | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,815 | KOU-11.123 | Kourovka Notebook Problem 11.123 | (Well-known problem).
For a given group $G$, define the following sequence of groups:
$A_1(G)=G$, $A_{i+1}(G)=\mathrm{Aut} (A_i(G))$. Does there exist a
finite group $G$ for which this sequence contains infinitely many
non-isomorphic groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.123.
Proposed by: M. Short | 2 | open | M. Short | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,816 | KOU-11.124 | Kourovka Notebook Problem 11.124 | Let $F$ be a non-cyclic
free group and
$R$ a non-cyclic subgroup of $F$. Is it true that if $[R,R]$
is a normal subgroup of $F$ then $R$ is also a normal subgroup of
$F$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.124.
Proposed by: V. E. Shpilrain
Discussion and literature:
[Solved] Yes, it is true (A. Tursunbaev, *Preprint*, 2026,
https://arxiv.org/abs/2609.06755). | 2 | solved | V. E. Shpilrain | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. Tursunbaev, *Preprint*, 2026,\nhttps://arxiv.org/abs/2609.06755)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,817 | KOU-11.125 | Kourovka Notebook Problem 11.125 | Let $G$ be a finite
group admitting a regular elementary abelian group of automorphisms
$V$ of order $p^n$. Is it true that the subgroup
$H=\bigcap\limits_{v\in V\setminus \{ 1\}} [G,v]$ is nilpotent? In
the case of an affirmative answer, does there exist a function
depending only on $p$, $n$, and the derived length of... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.125.
Proposed by: P. V. Shumyatski\u{\i | 2 | open | P. V. Shumyatski\u{\i | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,818 | KOU-11.126 | Kourovka Notebook Problem 11.126 | Do there exist a constant $h$ and a function $f$ with the
following property: if a finite soluble group $G$ admits an
automorphism $\varphi$ of order 4 such that $|C_G(\varphi )| \leq
m$, then $G$ has a normal series
$1\leq M\leq N\leq G$ such that the index
$|G:N|$ does not exceed $f(m)$, the group $N/M$ is nilpotent ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.126.
Proposed by: P. V. Shumyatski\u{\i
Published resolution:
Yes, there do (N. Yu. Makarenko, E. I. Khukhro, *Algebra and Logic*,
**45** (2006), 326--343). | 2 | solved | P. V. Shumyatski\u{\i | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there do (N. Yu. Makarenko, E. I. Khukhro, *Algebra and Logic*,\n**45** (2006), 326--343)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,819 | KOU-11.127 | Kourovka Notebook Problem 11.127 | Is every group of
exponent 12 locally finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.127.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,820 | KOU-11.128 | Kourovka Notebook Problem 11.128 | A group $G$ is said to be a *$K$-group* if
for every subgroup $A \leq G$ there exists a subgroup $B \leq G$
such that $A \cap B=1$ and $\left< A,B\right> =G$. Is it true that
normal subgroups of $K$-groups are also $K$-groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.128.
Proposed by: M. Emaldi
Published resolution:
No, it is not
(V. N. Obraztsov, *J. Austral. Math. Soc. (Ser. A)*, **61**, no. 2 (1996), 267--288). | 2 | solved | M. Emaldi | 1,990 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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\n(V. N. Obraztsov, *J. Austral. Math. Soc. (Ser. A)*, **61**, no. 2 (1996), 267--288)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,821 | KOU-12.1 | Kourovka Notebook Problem 12.1 | a) H. Bass (*Topology*, **4**, no. 4
(1966), 391--400) has constructed explicitly a proper subgroup of
finite index in the group of units of the integer group ring of a
finite cyclic group. Calculate the index of Bass' subgroup.
[Archived solved part(s) of 12.1:]
b) (A. A. Bovdi). H. Bass (*Topology*,
**4**, no. 4 (19... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 12.1.
Proposed by: R. Zh. Aleev
Archive of Solved Problems entry:
They are constructed: for $p $-groups,
$p\ne 2$, in
(K. Hoechsmann, J. Ritter, *J. Pure App... | 2 | partially_solved | R. Zh. Aleev | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (They are constructed: for $p $-groups,\n$p\\ne 2$, in\n(K. Hoechsmann, J. Ritter, *J. Pure Appl. Algebra*, **68**, no. 3 (1990), 325--333); for the general case ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,822 | KOU-12.2 | Kourovka Notebook Problem 12.2 | Is it true that every non-discrete group topology
(in an abelian group) can be strengthened up to a maximal complete
group topology? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.2.
Proposed by: V. I. Arnautov
Published resolution:
No, not always, since
under the assumption of the Continuum Hypothesis there exist
non-complete maximal topologies (V. I. A... | 2 | solved | V. I. Arnautov | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, since\nunder the assumption of the Continuum Hypothesis there exist\nnon-complete maximal topologies (V. I. Arnautov,\nE. G. Zelenyuk, *Ukrain. M... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,823 | KOU-12.3 | Kourovka Notebook Problem 12.3 | A $p
$-group is called *thin* if
every set of pairwise incomparable (by inclusion) normal subgroups
contains $\leq p + 1$ elements. Is the number of thin
pro-$p
$-groups finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.3.
Proposed by: R. Brandl | 2 | open | R. Brandl | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,824 | KOU-12.4 | Kourovka Notebook Problem 12.4 | Let $G$ be a group and assume that we have
$[x,y]^3=1$ for all $x,y \in G$. Is $G'$ of finite exponent? Is
$G$ soluble? By a result of N. D. Gupta and N. S. Mendelssohn,
1967, we know that $G'$ is a 3-group. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.4.
Proposed by: R. Brandl | 2 | open | R. Brandl | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,825 | KOU-12.5 | Kourovka Notebook Problem 12.5 | Does there exist a countable
non-trivial filter in the lattice of quasivarieties of metabelian
torsion-free groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.5.
Proposed by: A. I. Budkin
Published resolution:
No, there is no such filter
(S. V. Lenyuk, *Siberian Math. J.*, **39**, no. 1 (1998),
57--62). | 2 | solved | A. I. Budkin | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, there is no such filter\n(S. V. Lenyuk, *Siberian Math. J.*, **39**, no. 1 (1998),\n57--62)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,826 | KOU-12.6 | Kourovka Notebook Problem 12.6 | Let $G$ be a finitely generated group and suppose
that $H$ is a $p
$-subgroup of $G$ such that $H$
contains no non-trivial normal subgroups of $G$ and $HX=XH$ for
any subgroup $ X$ of $G$. Is then $G/C_G(H^G)$ a $p
$-group where $H^G$ is the normal closure of $H$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.6.
Proposed by: G. Busetto | 2 | open | G. Busetto | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,827 | KOU-12.7 | Kourovka Notebook Problem 12.7 | Is it true that every radical hereditary
formation of finite groups is a composition one? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.7.
Proposed by: A. F. Vasil'ev
Published resolution:
No, it is not
(S. F. Kamornikov, *Math.
Notes*, **55**, no. 6 (1994), 586--588). | 2 | solved | A. F. Vasil'ev | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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\n(S. F. Kamornikov, *Math.\nNotes*, **55**, no. 6 (1994), 586--588)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,828 | KOU-12.8 | Kourovka Notebook Problem 12.8 | Let ${\mathcal{V}}$ be a non-trivial variety of groups
and let $a_1,\ldots ,a_r$ freely generate a free group $F_r(V)$ in
${\mathcal{V}}$. We say $F_r({\mathcal{V}})$ *strongly discriminates*
${\mathcal{V}}$ just in case every finite system of inequalities
$w_i(a_1,\ldots ,a_r,x_1,\ldots ,x_k) \ne 1$ for $1 \leq i \leq... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.8.
Proposed by: A. M. Gaglione,
D. Spellman | 2 | open | A. M. Gaglione,
D. Spellman | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,829 | KOU-12.9 | Kourovka Notebook Problem 12.9 | Following Bass, call a group *tree-free* if
there is an ordered Abelian group $\Lambda$ and a $\Lambda $-tree
$X$ such that $G$ acts freely without inversion on $X$.
a) Must every finitely generated tree-free
group satisfy the
maximal condition for Abelian subgroups?
b) The same question for finitely
presented tree-f... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.9.
Proposed by: A. M. Gaglione,
D. Spellman | 2 | open | A. M. Gaglione,
D. Spellman | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,830 | KOU-12.10 | Kourovka Notebook Problem 12.10 | (P. Neumann). Can the free group on two
generators be embedded in $\mathrm{Sym} ({\mathbb{N})$ so that the
image of every non-identity element has only a finite number of
orbits? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.10.
Proposed by: A. M. W. Glass
Published resolution:
Yes, it can
(H. D. Macpherson, in: *Ordered groups and infinite permutation
groups, Partially based on Conf. Luminy, Franc... | 2 | solved | A. M. W. Glass | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it can\n(H. D. Macpherson, in: *Ordered groups and infinite permutation\ngroups, Partially based on Conf. Luminy, France, 1993* (*Mathematics and its Applic... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,831 | KOU-12.11 | Kourovka Notebook Problem 12.11 | Suppose that $G, H$ are countable or finite
groups and $A$ is a proper subgroup of $G$ and $H$ containing no
non-trivial subgroup normal in both $G$ and $H$. Can $G*_A H$ be
embedded in $\mathrm{Sym} ({\mathbb{N})$ so that the image is highly
transitive?
*Comment of 2013:*
for partial results, see (S. V. Gunhouse, *Hi... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.11.
Proposed by: A. M. W. Glass | 2 | open | A. M. W. Glass | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,832 | KOU-12.12 | Kourovka Notebook Problem 12.12 | Is the conjugacy problem for nilpotent finitely
generated lattice-ordered groups soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.12.
Proposed by: A. M. W. Glass | 2 | open | A. M. W. Glass | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,833 | KOU-12.13 | Kourovka Notebook Problem 12.13 | If $\left< \Omega , \leq \right>$ is the
countable universal poset, then $G=\mathrm{Aut} (\left< \Omega ,
\leq \right> )$ is simple (A. M. W. Glass, S. H. McCleary,
M. Rubin, *Math. Z.*, **214**, no. 1 (1993), 55--66). If
$H$ is a subgroup of $G$ such that $|G:H| < 2^{\aleph _0}$ and $H$
is transitive on $\Omega$, does... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.13.
Proposed by: A. M. W. Glass | 2 | open | A. M. W. Glass | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,834 | KOU-12.14 | Kourovka Notebook Problem 12.14 | If $T$ is a
countable theory, does there exist a model ${\mathfrak{A}}$ of $T$ such
that the theory of $\mathrm{Aut} ({\mathfrak{A}})$ is undecidable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.14.
Proposed by: M. Giraudet, A. M. W. Glass
Published resolution:
Yes, moreover, every first
order theory having infinite models has a model whose automorphism
group has undec... | 2 | solved | M. Giraudet, A. M. W. Glass | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, moreover, every first\norder theory having infinite models has a model whose automorphism\ngroup has undecidable existential theory (V. V. Bludov,\nM. Girau... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,835 | KOU-12.15 | Kourovka Notebook Problem 12.15 | Suppose that, in a finite 2-group $G$, any two
elements are conjugate whenever their normal closures coincide. Is
it true that the derived subgroup of $G$ is abelian? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.15.
Proposed by: E. A. Golikova, A. I. Starostin
Discussion and literature:
[Solved] No, not always (K. Muliarchyk, *Preprint*, 2026,
https://kourovkanotebo... | 2 | solved | E. A. Golikova, A. I. Starostin | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (K. Muliarchyk, *Preprint*, 2026,\n\nhttps://kourovkanotebookorg.wordpress.com/wp-content... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,836 | KOU-12.16 | Kourovka Notebook Problem 12.16 | Is the class of groups of recursive
automorphisms of arbitrary models closed with respect to taking
free products? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.16.
Proposed by: S. S. Goncharov | 2 | open | S. S. Goncharov | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,837 | KOU-12.17 | Kourovka Notebook Problem 12.17 | Find a description of autostable periodic
abelian groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.17.
Proposed by: S. S. Goncharov | 2 | open | S. S. Goncharov | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,838 | KOU-12.19 | Kourovka Notebook Problem 12.19 | Is it true that, for every $n \geq 2$ and every
two epimorphisms $\varphi$ and $\psi$ of a free group $F_{2n}$ of
rank $2n$ onto $F_n \times F_n$, there exists an automorphism
$\alpha$ of $F_{2n}$ such that $\alpha \varphi =\psi $? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.19.
Proposed by: R. I. Grigorchuk | 2 | open | R. I. Grigorchuk | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,839 | KOU-12.20 | Kourovka Notebook Problem 12.20 | (Well-known problem). Is
R. Thompson's group $$F=\left< x_0, x_1,\ldots \mid
x_n^{x_i}= x_{n+1}, \; \; i < n,\;\;n=1, 2, \ldots \right>
=$$
$$= \left< x_0,\ldots ,x_4\mid x_1^{x_0}=x_2, \; \;
x_2^{x_0}=x_3, \; \; x_2^{x_1}=x_3, \; \; x_3^{x_1}=x_4, \; \;
x_3^{x_2}=x_4\right> $$
amenable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.20.
Proposed by: R. I. Grigorchuk | 2 | open | R. I. Grigorchuk | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,840 | KOU-12.21 | Kourovka Notebook Problem 12.21 | Let $R$ be a commutative ring with identity and
let $G$ be a finite group. Prove that the ring $a(RG)$ of $R
$-representations of $G$ has no non-trivial idempotents. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.21.
Proposed by: P. M. Gudivok,
V. P. Rud'ko | 2 | open | P. M. Gudivok,
V. P. Rud'ko | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,841 | KOU-12.22 | Kourovka Notebook Problem 12.22 | Let $\Delta (G)$ be the augmentation ideal of
the integer group ring of an arbitrary group $G$. Then $D_n(G)=G \cap
(1 + \Delta ^n(G))$ contains the $n $th lower central subgroup
$\gamma _n(G)$ of $G$.
a) Is it true that $D_n(G)/\gamma_n(G)$ is
central in $G/\gamma _n(G)$?
b) Is it true that $D_n(G)/\gamma _n(G)$ has... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.22.
Proposed by: N. D. Gupta, Yu. V. Kuz'min
Published resolution:
a) No, not always; moreover,
$D_n(G)/\gamma_n(G)$ need not be contained in any term of the
upper central seri... | 2 | solved | N. D. Gupta, Yu. V. Kuz'min | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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; moreover,\n$D_n(G)/\\gamma_n(G)$ need not be contained in any term of the\nupper central series of $G/\\gamma _n(G)$ with fixed number\n(N. D.... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,842 | KOU-12.23 | Kourovka Notebook Problem 12.23 | The *index of permutability* of a group
$G$ is defined to be the minimal integer $k \geq 2$ such that for
each $k $-tuple $x_1,\ldots ,x_k$ of elements in $G$ there is a
non-identity permutation $\sigma$ on $k$ symbols such that
$x_1\cdots x_k=x_{\sigma (1)}\cdots x_{\sigma (k)}$. Determine the
index of permutability o... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.23.
Proposed by: M. Gutsan | 2 | open | M. Gutsan | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,843 | KOU-12.24 | Kourovka Notebook Problem 12.24 | Given a ring $R$ with identity, the automorphisms
of $R[[x]]$ sending $x$ to $x\left( 1 + \sum_{i=1}^{\infty}
a_ix^i\right) $, $a_i\in R$, form a group $N(R)$. We know that
$N({\mathbb{Z})$ contains a copy of the free group $F_2$ of rank 2 and,
from work of A. Weiss, that $N({\mathbb{Z}/p{\mathbb{Z})$ contains a copy
o... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.24.
Proposed by: D. L. Johnson
Published resolution:
Yes, it does (R. Camina,
*J. Algebra*, **196**, no. 1 (1997), 101--113).
I. B. Fesenko noted that this fact might have been... | 2 | solved | D. L. Johnson | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does (R. Camina,\n*J. Algebra*, **196**, no. 1 (1997), 101--113).\nI. B. Fesenko noted that this fact might have been known to\nspecialists in the theory... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,844 | KOU-12.25 | Kourovka Notebook Problem 12.25 | Let $G$ be a finite group acting irreducibly on a
vector space $V$. An orbit $\alpha ^G$ for $\alpha \in V$ is said to
be *$p$-regular* if the stabilizer of $\alpha$ in $G$ is a
$p' $-subgroup. Does $G$ have a regular orbit on $V$ if it has a
$p
$-regular orbit for every prime $p$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.25.
Proposed by: Jiping Zhang
Published resolution:
No, not always.
For example, let $H={\mathbb{A}_4\wr {\mathbb{Z}_5$, \ $V=O_2(H)$, and $G$
be a complement to $V$ in $H$, $G... | 2 | solved | Jiping Zhang | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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.\nFor example, let $H={\\mathbb{A}_4\\wr {\\mathbb{Z}_5$, \\ $V=O_2(H)$, and $G$\nbe a complement to $V$ in $H$, $G$ acting on\n$V$ by conjugation... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,845 | KOU-12.26 | Kourovka Notebook Problem 12.26 | (Shi Shengming). Is it true that a finite
$p
$-soluble group $G$ has a $p
$-block of
defect zero if and only if there exists an element $x \in O_{p'}(G)$
such that $C_G(x)$ is a $p' $-subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.26.
Proposed by: Jiping Zhang
Published resolution:
No, it is not. The
group $3^2 . GL_2(3)$ has a 2-block of defect $0$, but the
centralizer of every element from $O(G)$ has e... | 2 | solved | Jiping Zhang | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. The\ngroup $3^2 . GL_2(3)$ has a 2-block of defect $0$, but the\ncentralizer of every element from $O(G)$ has even order\n(V. I. Zenkov, in: *Trud... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,846 | KOU-12.27 | Kourovka Notebook Problem 12.27 | Let $G$ be a simple locally finite group. We say
that $G$ is a *group of finite type* if there is a
non-trivial permutational representation of $G$ such that some
finite subgroup of $G$ has no regular orbits. Investigate and,
perhaps, classify
the simple locally finite groups of finite type. Partial results
see in (B. ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.27.
Proposed by: A. E. Zalesski\u{\i | 2 | open | A. E. Zalesski\u{\i | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,847 | KOU-12.28 | Kourovka Notebook Problem 12.28 | Let $G$ be a group. A function $f\! : G
\rightarrow {\mathbb{C}$ is called
\begin{enumerate}
\item[1)] *normed* if $f(1)=1$;
\item[2)] *central* if $f(gh)=f(hg)$ for all
$g,h \in G$;
\item[3)] *positive-definite* if $\sum\limits_{k,
l}f(g_k^{-1}g_l) \overline{\! c}_kc_l \geq 0$ for any $g_1,\ldots
,g_n \in G$ and an... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.28.
Proposed by: A. E. Zalesski\u{\i | 2 | open | A. E. Zalesski\u{\i | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,848 | KOU-12.29 | Kourovka Notebook Problem 12.29 | Classify the locally finite groups for which the
augmentation ideal of the complex group algebra is a simple ring.
The problem goes back to I. Kaplansky (1965). For partial
results, see (K. Bonvallet, B. Hartley, D. S. Passman,
M. K. Smith, *Proc. Amer. Math. Soc.*, **56**, no. 1
(1976), 79--82; \ A. E. Zalesskiĭ, *Alg... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.29.
Proposed by: A. E. Zalesski\u{\i | 2 | open | A. E. Zalesski\u{\i | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,849 | KOU-12.30 | Kourovka Notebook Problem 12.30 | (O. N. Golovin). On the class of all groups,
do there exist associative operations which satisfy the postulates
of MacLane and Mal'cev (that is, which are free functorial and
hereditary) and which are different from taking free and direct
products? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.30.
Proposed by: S. V. Ivanov | 2 | open | S. V. Ivanov | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,850 | KOU-12.31 | Kourovka Notebook Problem 12.31 | For relatively free groups $G$, prove or disprove
the following conjecture of P. Hall: if a word $v$ takes only
finitely many values on $G$ then the verbal subgroup $vG$ is finite. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.31.
Proposed by: S. V. Ivanov
Published resolution:
The conjecture is
disproved (A. Storozhev, *Proc. Amer. Math. Soc.*, **124**, no. 10 (1996), 2953--2954). | 2 | solved | S. V. Ivanov | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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\ndisproved (A. Storozhev, *Proc. Amer. Math. Soc.*, **124**, no. 10 (1996), 2953--2954)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,851 | KOU-12.32 | Kourovka Notebook Problem 12.32 | Prove an analogue of Higman's theorem for the
Burnside variety ${\mathfrak{B}}_n$ of groups of odd exponent $n \gg
1$, that is, prove that every recursively presented group of
exponent $n$ can be embedded in a finitely presented (in ${\mathfrak{B}}_n$) group of exponent $n$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 12th issue (1992). Original problem number: 12.32.
Proposed by: S. V. Ivanov
Published resolution:
It is proved (A. Yu. Olshanskii, *J. Algebra*, **560** (2020), 960--1052). | 2 | solved | S. V. Ivanov | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: It is proved (A. Yu. Olshanskii, *J. Algebra*, **560** (2020), 960--1052)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,852 | KOU-12.33 | Kourovka Notebook Problem 12.33 | Suppose that $G$ is a finite group and $x$ is an
element of $G$ such that the subgroup $\left< x,y\right>$ has odd
order for any $y$ conjugate to $x$ in $G$. Prove, without using
CFSG, that the normal closure of $x$ in $G$ is a group of odd
order. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.33.
Proposed by: L. S. Kazarin | 2 | open | L. S. Kazarin | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,853 | KOU-12.34 | Kourovka Notebook Problem 12.34 | Describe the finite groups $G$ such that the sum
of the cubes of the degrees of all irreducible complex characters
is at most $|G|\cdot \log _2|G|$. The question is interesting for
applications in the theory of signal processing. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 12.34.
Proposed by: L. S. Kazarin | 2 | open | L. S. Kazarin | 1,992 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
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