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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
700,654 | KOU-10.39 | Kourovka Notebook Problem 10.39 | a) Is the membership problem soluble for the
subgroup $E_n(R)$ of $SL_n(R)$ where $R$ is a commutative ring?
b) Is the word problem soluble for the groups
$K_i(R)$ where
$K_i$ are the Quillen $K $-functors and $R$ is a commutative
ring? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.39.
Proposed by: G. A. Noskov | 2 | open | G. A. Noskov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,655 | KOU-10.40 | Kourovka Notebook Problem 10.40 | (H. Bass). Let $G$ be a group, $e$ an
idempotent matrix over ${\mathbb{Z}G$ and let $\mathrm{tr}
e=\sum\limits_{g\in G}e_gg, e_g \in {\mathbb{Z}.$ *Strong
conjecture:* for any non-trivial $x \in G$, the equation
$\sum\limits_{g\sim x} e_g=0$ holds where $\sim $ denotes
conjugacy in $G$. *Weak conjecture:* $\sum\limits_... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.40.
Proposed by: G. A. Noskov | 2 | open | G. A. Noskov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,656 | KOU-10.41 | Kourovka Notebook Problem 10.41 | (Well-known problem). Let $\Gamma$ be an almost polycyclic group
with no non-trivial finite normal subgroups, and let $k$ be a
field. The complete ring of quotients $Q(k\Gamma )$ is a matrix ring
$M_n(D)$ over a skew field. Conjecture: $n$ is the least common
multiple of the orders of the finite subgroups of $\Gamma$. ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.41.
Proposed by: G. A. Noskov
Published resolution:
The strong conjecture $G_0(k\Gamma )=G_0(k\Gamma )_{{\mathcal{F}}}$ has been
proved (J. A. Moody, *Bull. Amer. Math. Soc.*, ... | 2 | solved | G. A. Noskov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 strong conjecture $G_0(k\\Gamma )=G_0(k\\Gamma )_{{\\mathcal{F}}}$ has been\nproved (J. A. Moody, *Bull. Amer. Math. Soc.*, **17**\n(1987), 113--116)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,657 | KOU-10.42 | Kourovka Notebook Problem 10.42 | (J. T. Stafford). Let $G$ be a poly-${\mathbb{Z}
$-group and let $k$ be a field. Is it true that every finitely
generated projective $kG $-module is either a free module or an
ideal? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.42.
Proposed by: G. A. Noskov | 2 | open | G. A. Noskov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,658 | KOU-10.43 | Kourovka Notebook Problem 10.43 | Let $R$ and $S$ be associative rings with
identity such that 2 is invertible in $ S$. Let $\Lambda _I \! :
GL_n(R) \rightarrow GL_n(R/I)$ be the homomorphism corresponding
to an ideal $I$ of $ R$ and let $ E_n(R)$ be the subgroup of $
GL_n(R)$ generated by elementary transvections $t_{ij}(x)$. Let
$a_{ij}=t_{ij}(1)t_{j... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.43.
Proposed by: V. M. Petechuk
Discussion and literature:
[Unverified AI claim] Such homomorphisms are found (unverified AI solution: Achyuth Jayadevan, *Pr... | 2 | open | V. M. Petechuk | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] Such homomorphisms are found (unverified AI solution: Achyuth Jayadevan, *Preprin... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,659 | KOU-10.44 | Kourovka Notebook Problem 10.44 | Let $R$ and $S$ be associative rings with
identity such that 2 is invertible in $S$. Prove that every standard homomorphism $\Lambda$
of $ E_n(R) \subset GL_n(R)=GL(V)$ into $E_m(S) \subset
GL_m(S)=GL(W)$ originates from a collineation and a correlation,
that is, has the form $\Lambda x=(g^{-1}xg)f + (h^{-1}xh)(1 - f),... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.44.
Proposed by: V. M. Petechuk
Discussion and literature:
[Unverified AI claim] A counterexample is constructed (unverified AI solution: Achyuth Jayadevan, ... | 2 | open | V. M. Petechuk | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] A counterexample is constructed (unverified AI solution: Achyuth Jayadevan, *Prep... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,660 | KOU-10.45 | Kourovka Notebook Problem 10.45 | Let $R$ be an associative ring with
identity. Let $n \geq 3$ and suppose that $N$ is a
subgroup of $ GL_n(R)$ which is normalized by $E_n(R)$. Prove that
either $ N$ contains $E_n(R)$ or $\Lambda _I[N,E_n(R)]=1$ for a
suitable ideal $I \ne R$ of $R$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.45.
Proposed by: V. M. Petechuk | 2 | open | V. M. Petechuk | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,661 | KOU-10.46 | Kourovka Notebook Problem 10.46 | Let $R$ be an associative ring with
identity.
Prove that for every element $\sigma
\in GL_n(R)$, $n \geq 3$, there exist transvections $\tau _1, \tau _2$ such that
$[[\sigma ,\tau _1], \tau _2]$ is a unipotent element. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.46.
Proposed by: V. M. Petechuk
Discussion and literature:
[Solved] This is not always true (K. Muliarchyk, *Preprint*, 2026,
https://kourovkanotebookorg.wo... | 2 | solved | V. M. Petechuk | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] This is not always true (K. Muliarchyk, *Preprint*, 2026,\n\nhttps://kourovkanotebookorg.wordpress.com/w... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,662 | KOU-10.47 | Kourovka Notebook Problem 10.47 | Describe the automorphisms of $PE_2(R)$ in the
case where $R$ is an associative commutative ring and 2 and 3 have inverses
in it. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.47.
Proposed by: V. M. Petechuk | 2 | open | V. M. Petechuk | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,663 | KOU-10.48 | Kourovka Notebook Problem 10.48 | Let $V$ be a
vector space of finite dimension over a field of prime order.
A subset $R$ of $GL(V) \cup \{ 0\}$ is called *regular* if
$|R|=|V|$, $0,1 \in R$ and $vx \ne vy$ for any non-trivial vector
$v \in V$ and any distinct elements $x,y
\in
R$. It is obvious that $\tau ,\varepsilon ,\mu _g$ transform a
regular set ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.48.
Proposed by: N. D. Podufalov
Published resolution:
a) They were studied (N. D. Podufalov, *Algebra and Logic*,
**30**, no. 1 (1991), 62--69).
b) No. A regular set is close... | 2 | solved | N. D. Podufalov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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) They were studied (N. D. Podufalov, *Algebra and Logic*,\n**30**, no. 1 (1991), 62--69).\n\nb) No. A regular set is closed with respect to multiplication if\n... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,664 | KOU-10.49 | Kourovka Notebook Problem 10.49 | Does there exist a group $G$ satisfying the following
four conditions:
1) $G$ is simple, moreover, there is an
integer $n$
such
that $G=C^n$ for any conjugacy class $C$,
2) all maximal abelian subgroups of $G$ are
conjugate in $
G$,
3) every maximal abelian subgroup of $G$ is
self-normalizing and it is the centraliz... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.49.
Proposed by: B. Poizat | 2 | open | B. Poizat | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,665 | KOU-10.50 | Kourovka Notebook Problem 10.50 | Complex characters of a finite group $G$ induced
by linear characters of cyclic subgroups are called *induced
cyclic characters* of $G$. Computer-aided computations
(A. V. Rukolaine, *Abstracts of the 10th All--Union Sympos.
on Group Theory*, Gomel', 1986, p. 199 (Russian)) show that there
exist groups (for example, ${... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.50.
Proposed by: A. V. Rukolaine | 2 | open | A. V. Rukolaine | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,666 | KOU-10.51 | Kourovka Notebook Problem 10.51 | Describe the structure of thin abelian
$p
$-groups. Such a description is known in the class
of separable $ p
$-groups (C. Megibben, *Mich.
Math. J.*, **13**, no. 2 (1966), 153--160). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.51.
Proposed by: S. V. Rychkov | 2 | open | S. V. Rychkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,667 | KOU-10.52 | Kourovka Notebook Problem 10.52 | (R. Mines). It is well known that the topology
on the completion of an abelian group under the $ p
$-adic topology is the $p
$-adic topology. R. Warfield
has shown that this is also true in the category of nilpotent
groups. Find a categorical setting for this theorem which includes
the case of nilpotent groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.52.
Proposed by: S. V. Rychkov
Discussion and literature:
[Unverified AI claim] Such a setting is found (unverified AI solution: Achyuth Jayadevan, *Preprint... | 2 | open | S. V. Rychkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] Such a setting is found (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,668 | KOU-10.53 | Kourovka Notebook Problem 10.53 | (M. Dugas). Let ${\mathfrak{R}}$ be the *Reid
class*, i. e. the smallest containing ${\mathbb{Z}$ and closed under
direct sums and direct products. Is ${\mathfrak{R}}$ closed under direct
summands? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.53.
Proposed by: S. V. Rychkov | 2 | open | S. V. Rychkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,669 | KOU-10.54 | Kourovka Notebook Problem 10.54 | (R. G\"obel). For a cardinal number $\mu$, let
$${\mathbb{Z}^{<\mu}=\{ f \in {\mathbb{Z}^{\mu}\mid |\mathrm{supp} (f)| < \mu \} \; \; \; \; \mbox{ and} \; \; \; \;
G_{\mu}={\mathbb{Z}^{\mu}/{\mathbb{Z}^{<\mu}.$$
a) Find a non-zero direct summand $D$ of
$G_{\omega
_1}$ such
that $D\not\cong G_{\omega _1}$.
b) Investiga... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.54.
Proposed by: S. V. Rychkov | 2 | open | S. V. Rychkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,670 | KOU-10.55 | Kourovka Notebook Problem 10.55 | (A. Mader).
``Standard $B$'', that is, $B={\mathbb{Z}(p) \oplus {\mathbb{Z}(p^2 ) \oplus
\cdots$, is slender as a module over its endomorphism ring
(A. Mader, in: *Abelian Groups and Modules,
Proc., Udine, 1984*, Springer, 1984, 315--327). Which
abelian $p $-groups are slender as modules over their endomorphism
rings? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.55.
Proposed by: S. V. Rychkov | 2 | open | S. V. Rychkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,671 | KOU-10.56 | Kourovka Notebook Problem 10.56 | Is the lattice of formations of finite nilpotent groups of
class $\leq 4$ distributive? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.56.
Proposed by: A. N. Skiba
Published resolution:
No. The lattice $L$ of all varieties of nilpotent 2-groups of
class $\leq 4$ is non-distributive (Yu. A. Belov, *Algebra
and ... | 2 | solved | A. N. Skiba | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. The lattice $L$ of all varieties of nilpotent 2-groups of\nclass $\\leq 4$ is non-distributive (Yu. A. Belov, *Algebra\nand Logic*, **9**, no. 6 (1970), 371-... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,672 | KOU-10.57 | Kourovka Notebook Problem 10.57 | What are the minimal non-$A $-formations? An
*$A $-formation* is, by definition, the formation of the
finite groups all of whose Sylow subgroups are abelian. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.57.
Proposed by: A. N. Skiba | 2 | open | A. N. Skiba | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,673 | KOU-10.58 | Kourovka Notebook Problem 10.58 | Is the subsemigroup generated by the undecomposable
formations in the semigroup of formations of finite groups free? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.58.
Proposed by: A. N. Skiba | 2 | open | A. N. Skiba | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,674 | KOU-10.59 | Kourovka Notebook Problem 10.59 | Is a $p' $-group $G$ locally nilpotent if it
admits a splitting automorphism $\varphi$ of prime order $p$ such
that all subgroups of the form $\langle g, g^{\varphi}, \ldots ,
g^{\varphi ^{p-1}}\rangle $ are nilpotent? An automorphism
$\varphi$ of order $p$ is called *splitting* if $gg^{\varphi
}g^{\varphi ^{2}}\cdots ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.59.
Proposed by: A. I. Sozutov | 2 | open | A. I. Sozutov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,675 | KOU-10.60 | Kourovka Notebook Problem 10.60 | Does every periodic group ($p $-group) $A$ of
regular automorphisms of an abelian group have non-trivial
centre?
*Editors' comment (2001):* The answer is
affirmative if $A$ contains an element of order 2 or 3
(A. Kh. Zhurtov, *Siberian Math. J.*, **41**, no. 2
(2000), 268--275). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.60.
Proposed by: A. I. Sozutov | 2 | open | A. I. Sozutov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,676 | KOU-10.61 | Kourovka Notebook Problem 10.61 | Suppose that $H$ is a proper subgroup of a group
$G$, $a \in H$, $a^2 \ne 1$ and for every $g \in G\setminus H$ the
subgroup $\left< a, a^g\right>$ is a Frobenius group whose
complement contains $a$. Does the set-theoretic union of the
kernels of all Frobenius subgroups of $G$ with complement $\left<
a\right>$ constitu... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.61.
Proposed by: A. I. Sozutov | 2 | open | A. I. Sozutov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,677 | KOU-10.62 | Kourovka Notebook Problem 10.62 | Construct an example of a (periodic) group
without subgroups of index 2 which is generated by a conjugacy
class of involutions $X$ such that the order of the product of any
two involutions from $X$ is odd. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.62.
Proposed by: A. I. Sozutov
Discussion and literature:
[Solved] Example constructed (AI solution confirmed by M. Amelio: \ M. Kinyon, J. Urban, *Preprint*... | 2 | solved | A. I. Sozutov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] Example constructed (AI solution confirmed by M. Amelio: \\ 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 |
700,678 | KOU-10.63 | Kourovka Notebook Problem 10.63 | Is there a doubly transitive permutation group in which the
stabilizer of a point is infinite cyclic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.63.
Proposed by: Ya. P. Sysak
Published resolution:
No (V. D. Mazurov, *Siberian Math. J.*, **31** (1990),
615--617; \ Gy. K\'arolyi, S. J. Kov\'acs, P. P. P\'alfy,
*Aequatione... | 2 | solved | Ya. P. Sysak | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (V. D. Mazurov, *Siberian Math. J.*, **31** (1990),\n615--617; \\ Gy. K\\'arolyi, S. J. Kov\\'acs, P. P. P\\'alfy,\n*Aequationes Math.*, **39**, no. 1 (1990),... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,679 | KOU-10.64 | Kourovka Notebook Problem 10.64 | Does there exist a non-periodic doubly
transitive permutation group with a periodic stabilizer of a
point? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.64.
Proposed by: Ya. P. Sysak
Discussion and literature:
[Solved] Yes, it does exist (M. Amelio, *Preprint*, 2025, https://arxiv.org/abs/2509.11958). | 2 | solved | Ya. P. Sysak | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it does exist (M. Amelio, *Preprint*, 2025, https://arxiv.org/abs/2509.11958)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,680 | KOU-10.65 | Kourovka Notebook Problem 10.65 | Determine the structure of infinite 2-transitive
permutation groups $(G,\Omega )$ in which the stabilizer of a
point $\alpha \in \Omega$ has the form $G_{\alpha}=A\cdot
G_{\alpha \beta}$ where $G _{\alpha \beta}$ is the stabilizer of
two points $\alpha , \beta $, $\alpha \ne \beta $, such that $G
_{\alpha \beta}$ conta... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.65.
Proposed by: A. N. Fomin | 2 | open | A. N. Fomin | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,681 | KOU-10.66 | Kourovka Notebook Problem 10.66 | Is a group $G$ non-simple if it contains two
non-trivial subgroups $A$ and $B$ such that $AB \ne G$ and
$AB^g=B^gA$ for any $g \in G$? This is true if $G$ is finite
(O. H. Kegel, *Arch. Math.*, **12**, no. 2 (1961),
90--93). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.66.
Proposed by: A. N. Fomin, V. P. Shunkov
Published resolution:
No; for example, for $G$ being any simple
linearly ordered group and $A$ and $B$ arbitrary proper
convex subgr... | 2 | solved | A. N. Fomin, V. P. Shunkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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; for example, for $G$ being any simple\nlinearly ordered group and $A$ and $B$ arbitrary proper\nconvex subgroups of $G$ \\ (V. V. Bludov, *Letter of February... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,682 | KOU-10.67 | Kourovka Notebook Problem 10.67 | The class $LN{\mathfrak{M}}_p$ of locally nilpotent
groups admitting a splitting automorphism of prime order $p$ (for
definition see 10.59) is a variety of groups with operators
(E. I. Khukhro, *Math. USSR Sbornik*, **58** (1987), 119--126). Is it true that
$$LN{\mathfrak{M}}_p=({\mathfrak{N}}_{c(p)} \cap LN{\mathfrak{... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.67.
Proposed by: E. I. Khukhro | 2 | open | E. I. Khukhro | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,683 | KOU-10.68 | Kourovka Notebook Problem 10.68 | Suppose that a finite $ p
$-group $G$
admits an automorphism of order $p$ with exactly $p^m$ fixed points.
Is it true that $G$ has a subgroup whose index is bounded by a
function of $p$ and $m$ and whose nilpotency class is bounded by a
function of $m$ only? The results on $p
$-groups of
maximal class give an affirmati... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.68.
Proposed by: E. I. Khukhro
Published resolution:
Yes, it is (Yu. Medvedev, *J. London Math. Soc. (2)*, **59**, no. 3 (1999), 787--798). | 2 | solved | E. I. Khukhro | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (Yu. Medvedev, *J. London Math. Soc. (2)*, **59**, no. 3 (1999), 787--798)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,684 | KOU-10.69 | Kourovka Notebook Problem 10.69 | Suppose that $S$ is a closed oriented surface of genus $g
> 1$ and $G=\pi _1S=G_1*_A G_2$, where $G_1 \ne A \ne G_2$ and
the subgroup $A$ is finitely generated (and hence free). Is this
decomposition geometrical, that is, do there exist connected
surfaces $S_1, S_2, T$ such that $S=S_1 \cup S_2$, $S_1 \cap
S_2=T$, $G_i... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.69.
Proposed by: H. Zieschang
Published resolution:
No, not always (O. V. Bogopolski, *Geom. Dedicata*,
**94** (2002), 63--89). | 2 | solved | H. Zieschang | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (O. V. Bogopolski, *Geom. Dedicata*,\n**94** (2002), 63--89)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,685 | KOU-10.70 | Kourovka Notebook Problem 10.70 | Find a geometrical justification for the
Whitehead method for free products similar to the substantiation
given for free groups by Whitehead himself and more visual than
given in (D. J. Collins, H. Zieschang, *Math. Z.*, **185**, no. 4 (1984), 487--504; **186**, no. 3 (1984),
335--361).
*D. J. Collins' comment:* there... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.70.
Proposed by: H. Zieschang | 2 | open | H. Zieschang | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,686 | KOU-10.71 | Kourovka Notebook Problem 10.71 | Is it true that the centralizer of any
automorphism (any finite set of automorphisms) in the automorphism
group of a free group of finite rank is a finitely presented
group? This is true in the case of rank 2 and in the case of inner
automorphisms for any finite rank. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.71.
Proposed by: V. A. Churkin | 2 | open | V. A. Churkin | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,687 | KOU-10.72 | Kourovka Notebook Problem 10.72 | Prove that the formation of all finite $p $-groups does not
decompose into a product of two non-trivial subformations. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.72.
Proposed by: L. A. Shemetkov
Published resolution:
This has been proved (A. N. Skiba, L. A. Shemetkov, *Dokl. Akad.
Nauk BSSR*, **33** (1989), 581--582 (Russian)). | 2 | solved | L. A. Shemetkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 has been proved (A. N. Skiba, L. A. Shemetkov, *Dokl. Akad.\nNauk BSSR*, **33** (1989), 581--582 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,688 | KOU-10.73 | Kourovka Notebook Problem 10.73 | Enumerate all formations of finite groups all of
whose subformations are $S_n $-closed. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.73.
Proposed by: L. A. Shemetkov | 2 | open | L. A. Shemetkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,689 | KOU-10.74 | Kourovka Notebook Problem 10.74 | Suppose that a group $G$ contains an element $a$
of prime order such that its centralizer $ C_G(a)$ is finite and
all subgroups $\left< a, a^g\right> $, $g \in G$, are finite and
almost all of them are soluble. Is $G$ locally finite? This
problem is closely connected with 6.56. The question was solved in
the positive f... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.74.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,690 | KOU-10.75 | Kourovka Notebook Problem 10.75 | Suppose that a group $G$ contains an element $a$
of prime order $p$ such that the normalizer of every finite
subgroup containing $a$ has finite periodic part and all subgroups
$\left< a, a^g\right> $, $g \in G$, are finite and almost all of
them are soluble. Does $G$ possess a periodic part if $p > 2$? It
was proved in... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.75.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,691 | KOU-10.76 | Kourovka Notebook Problem 10.76 | Suppose that $G$ is a periodic group having an
infinite Sylow 2-subgroup $S$ which is either elementary abelian or
a Suzuki 2-group, and suppose that the normalizer $
N_G(S)$ is strongly embedded in $G$ and is a Frobenius group (see
6.55) with locally cyclic complement. Must $G$ be locally finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.76.
Proposed by: V. P. Shunkov
Published resolution:
Yes, it must (A. I. Sozutov, *Algebra and Logic*, **39**,
no. 5 (2000),
345--353; \
A. I. Sozutov, N. M. Suchkov, *Math. No... | 2 | solved | V. P. Shunkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it must (A. I. Sozutov, *Algebra and Logic*, **39**,\nno. 5 (2000),\n345--353; \\\nA. I. Sozutov, N. M. Suchkov, *Math. Notes*, **68**, no. 2 (2000), 237--2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,692 | KOU-10.77 | Kourovka Notebook Problem 10.77 | Suppose that $G$ is a periodic group containing
an elementary abelian subgroup $R$ of order 4. Must $G$ be locally
finite
a) if $C_G(R)$ is finite?
b) if the centralizer of every involution of
$R$ in $G$ is a
Chernikov group?
*Remark of 1999:* P. V. Shumyatsky
(*Quart. J. Math. Oxford (2)*, **49**, no. 196 (1998),
4... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.77.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,693 | KOU-10.78 | Kourovka Notebook Problem 10.78 | Does there exist a non-Chernikov group which is
a product of two Chernikov subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.78.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,986 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,694 | KOU-11.1 | Kourovka Notebook Problem 11.1 | (Well-known problem).
Describe the structure of the centralizers of unipotent elements
in almost simple groups of Lie type. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.1.
Proposed by: R. Zh. Aleev | 2 | open | R. Zh. Aleev | 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 |
700,695 | KOU-11.2 | Kourovka Notebook Problem 11.2 | Classify the simple groups
that are isomorphic to the multiplicative groups of finite rings,
in particular, of the group rings of finite groups over finite
fields and over ${\mathbb{Z}/n{\mathbb{Z}$, $n \in {\mathbb{Z}$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.2.
Proposed by: R. Zh. Aleev
Published resolution:
Such a classification is obtained in (C. Davis, T. Occhipinti, *J. Pure Appl. Algebra*, **218**, no. 4 (2014), 743--744). | 2 | solved | R. Zh. Aleev | 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: Such a classification is obtained in (C. Davis, T. Occhipinti, *J. Pure Appl. Algebra*, **218**, no. 4 (2014), 743--744)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,696 | KOU-11.3 | Kourovka Notebook Problem 11.3 | (M. Aschbacher). A
$p
$-local subgroup $H$ in a group $G$ is said to be a
*superlocal* if $H= N_G(O_p(H))$. Describe the superlocals
in alternating groups and in groups of Lie type. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.3.
Proposed by: R. Zh. Aleev | 2 | open | R. Zh. Aleev | 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 |
700,697 | KOU-11.4 | Kourovka Notebook Problem 11.4 | Is it true that the lattice of
centralizers in a group is modular if it is a sublattice of the
lattice of all subgroups? This is true for finite groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.4.
Proposed by: V. A. Antonov
Published resolution:
No, not always
(V. N. Obraztsov, *J. Algebra*, **199**, no. 1 (1998),
337--343). | 2 | solved | V. A. Antonov | 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\n(V. N. Obraztsov, *J. Algebra*, **199**, no. 1 (1998),\n337--343)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,698 | KOU-11.5 | Kourovka Notebook Problem 11.5 | Let $k$ be a commutative
ring and let $G$ be a torsion-free almost polycyclic group.
Suppose that $ P$ is a finitely generated projective module over
the group ring $kG$ and $P$ contains two elements independent over
$kG$. Is $P$ a free module? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.5.
Proposed by: V. A. Artamonov
Discussion and literature:
[Solved] No, not always (AI solution confirmed by D. Segal: \ Achyuth Jayadevan, *Preprint*, 2026,... | 2 | solved | V. A. Artamonov | 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] No, not always (AI solution confirmed by D. Segal: \\ Achyuth Jayadevan, *Preprint*, 2026, https://zenod... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,699 | KOU-11.6 | Kourovka Notebook Problem 11.6 | Let $p$ be an odd prime.
Is it true that every finite $p
$-group possesses a set
of generators of equal orders? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.6.
Proposed by: C. Bagi\'nski
Published resolution:
No, it is not true
(E. A. O'Brien, C. M. Scoppola, M. R. Vaughan-Lee, *Proc. Amer. Math. Soc.*, **134**, no. 12 (2006), 3457... | 2 | solved | C. Bagi\'nski | 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 true\n(E. A. O'Brien, C. M. Scoppola, M. R. Vaughan-Lee, *Proc. Amer. Math. Soc.*, **134**, no. 12 (2006), 3457--3464)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,700 | KOU-11.7 | Kourovka Notebook Problem 11.7 | a) Find conditions for a
group $ G$, given by its presentation, under which the
property *some term of the lower central series of $G$
is a free group* $(*)$ implies the residual
finiteness of $G$.
b) Find conditions on the structure of a
residually finite group $G$ which ensure property $(*)$ for $G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.7.
Proposed by: K. Bencs\'ath | 2 | open | K. Bencs\'ath | 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 |
700,701 | KOU-11.8 | Kourovka Notebook Problem 11.8 | b) For a finite group $X$,
let $\chi _1(X)$ denote the totality of the degrees of all
irreducible complex characters of $X$ with allowance for their
multiplicities. Suppose that $\chi _1(G)=\chi _1(H)$ for groups
$G$ and $H$. Clearly, then $|G|=|H|$. Is it true that
$H$ is soluble if $G$ is soluble?
It is known that $... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 11.8.
Proposed by: Ya. G. Berkovich
Archive of Solved Problems entry:
a) Yes, it is (H. P. Tong-Viet, *J. Algebra*, **357** (2012), 61--68;
*Monatsh. Math.*,... | 2 | partially_solved | Ya. G. Berkovich | 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 (a) Yes, it is (H. P. Tong-Viet, *J. Algebra*, **357** (2012), 61--68;\n*Monatsh. Math.*, **166**, no. 3--4 (2012), 559--577;\n*Algebr. Represent. Theory*, **15**... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,702 | KOU-11.9 | Kourovka Notebook Problem 11.9 | (I. I. Pyatetskiĭ-Shapiro). Does there exist a finite
non-soluble group $G$ such that the set of characters induced by
the trivial characters of representatives of all conjugacy classes
of subgroups of $G$ is linearly independent? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.9.
Proposed by: Ya. G. Berkovich
Discussion and literature:
[Solved] No, there are no such groups (M. Brescia, E. Ingrosso, M. Trombetti, *Preprint*, 2026, h... | 2 | solved | Ya. G. Berkovich | 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] No, there are no such groups (M. Brescia, E. Ingrosso, M. Trombetti, *Preprint*, 2026, https://arxiv.org... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,703 | KOU-11.10 | Kourovka Notebook Problem 11.10 | (R. C. Lyndon). a) Does
there exist an algorithm that, given a group word $ w(a,x)$,
recognizes whether $a$ is equal to the identity element in the
group $\left< a,x\mid a^n=1,\right.$ $\left. w(a,x)=1\right>$?
[Archived solved part(s) of 11.10:]
(R. C. Lyndon). b) Is it true
that $a \ne 1$ in $G=\langle a,x\mid a^5=1... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 11.10.
Proposed by: V. V. Bludov
Archive of Solved Problems entry:
Yes, it is true, since the mapping $a\rightarrow (1 3 5 2
4)$, $x\rightarrow (1 2 4 3 5)$ ... | 2 | partially_solved | V. V. Bludov | 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 (Yes, it is true, since the mapping $a\\rightarrow (1 3 5 2\n4)$, $x\\rightarrow (1 2 4 3 5)$ can be extended to a\nhomomorphism of the group $G$ onto the alterna... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,704 | KOU-11.11 | Kourovka Notebook Problem 11.11 | The well-known
Baer--Suzuki theorem
states that if every two conjugates of an element $a$ of
a finite group $G$ generate a finite $p
$-subgroup,
then $a$ is contained in a normal $p
$-subgroup.
a) Does
such a theorem hold in the class of periodic groups?
The case $p=2$ is of particular interest.
b) Does such a theore... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.11.
Proposed by: A. V. Borovik
Published resolution:
a) No, it does not hold for $p=2$ (V. D. Mazurov, A. Yu. Olshanskii, A. I. Sozutov, *Algebra and Logic*,
**54**, no. 2 (201... | 2 | solved | A. V. Borovik | 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) No, it does not hold for $p=2$ (V. D. Mazurov, A. Yu. Olshanskii, A. I. Sozutov, *Algebra and Logic*,\n**54**, no. 2 (2015), 161--166).\n\nb) Yes, such a theo... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,705 | KOU-11.12 | Kourovka Notebook Problem 11.12 | a) Suppose that $G$ is a
simple locally finite group in which the centralizer of some
element is a linear group, that is, a group admitting a faithful
matrix representation over a field. Is $G$ itself a linear group?
b) The same question with replacement of the
word ``linear'' by ``finitary linear''. A group $H$ is *f... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.12.
Proposed by: A. V. Borovik | 2 | open | A. V. Borovik | 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 |
700,706 | KOU-11.13 | Kourovka Notebook Problem 11.13 | Suppose that $G$ is a
periodic group with an involution $ i$ such that $i^g \cdot i$
has odd order for any $g \in G$. Is it true that the image of
$i$ in $G/O(G)$ belongs to the centre of $G/O(G)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.13.
Proposed by: A. V. Borovik
Published resolution:
No, not always (E. B. Durakov, A. I. Sozutov, *Algebra Logic*, **52**, no. 5 (2013), 422--425). | 2 | solved | A. V. Borovik | 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 (E. B. Durakov, A. I. Sozutov, *Algebra Logic*, **52**, no. 5 (2013), 422--425)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,707 | KOU-11.14 | Kourovka Notebook Problem 11.14 | Is every finite simple
group characterized by its Cartan matrix over an algebraically
closed field of characteristic 2? In (R. Brandl, *Arch.
Math.*, **38** (1982), 322--323) it is shown that for any given
finite group there exist only finitely many finite groups with
the same Cartan matrix. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.14.
Proposed by: R. Brandl | 2 | open | R. Brandl | 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 |
700,708 | KOU-11.15 | Kourovka Notebook Problem 11.15 | It is known that for each
prime number $p$ there exists a series $a_1, a_2, \ldots $ of
words in two variables such that the finite group $G$ has abelian
Sylow $p
$-subgroups if and only if $ a_k(G)=1$ for
almost all $k$. For $p=2$ such a series is known explicitly
(R. Brandl, *J. Austral. Math. Soc.*, **31** (1981),
4... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.15.
Proposed by: R. Brandl
Discussion and literature:
[Unverified AI claim] A computable construction for such a series is found (unverified AI solution: Ach... | 2 | open | R. Brandl | 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] A computable construction for such a series is found (unverified AI solution: Ach... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,709 | KOU-11.16 | Kourovka Notebook Problem 11.16 | Let $V_r$ be the class of
all finite groups $G$ satisfying a law $[x, _r y]=[x, _s y]$
for some $s=s(G) > r$. Here $[x, _1 y] = [x,y]$ and $[x,
_{i+1} y]=[[x, _i y],y]$.
a) Is there a function $f$ such that every
soluble group in $V_r$ has Fitting length $< f(r)$? For $r < 3$ see
(R. Brandl, *Bull. Austral. Math. Soc.... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.16.
Proposed by: R. Brandl | 2 | open | R. Brandl | 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 |
700,710 | KOU-11.17 | Kourovka Notebook Problem 11.17 | Let $G$ be a finite group
and let $d=d(G)$ be the least positive integer such that $G$
satisfies a law $[x, _ry]=[x, _{r+d} y]$ for some nonnegative
integer $r=r(G)$.
a) Let $e=1$ if $d(G)$ is even and $e=2$
otherwise. Is it true that the exponent of $G/F(G)$ divides $e\cdot
d(G)$?
b) If $G$ is a nonabelian simple gr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.17.
Proposed by: R. Brandl | 2 | open | R. Brandl | 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 |
700,711 | KOU-11.18 | Kourovka Notebook Problem 11.18 | Let $G(a,b)=\left<
x,y\mid x=[x, _a y], \; y=[y, _b x]\right> $. Is $G(a,b)$
finite?
It is easy to show that $G(1,b)=1$ and one can show that
$G(2,2)=1$. Nothing is known about $G(2,3)$. If one could show
that every minimal simple group is a quotient of some $ G(a,b)$,
then this would yield a very nice sequence of wor... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.18.
Proposed by: R. Brandl | 2 | open | R. Brandl | 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 |
700,712 | KOU-11.19 | Kourovka Notebook Problem 11.19 | (C. Sims). Is the $n$th
term of the lower central series of an absolutely free group the
normal closure of the set of basic commutators (in some fixed free
generators) of weight exactly $n$?
D. Jackson announced a
positive answer for $n\leq 5$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.19.
Proposed by: A. Gaglione, D. Spellman | 2 | open | A. Gaglione, D. Spellman | 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 |
700,713 | KOU-11.20 | Kourovka Notebook Problem 11.20 | Suppose we have $[a,b]=[c,d]$ in an absolutely free group, where
$a, b, [a,b]$ are basic commutators (in some fixed free
generators). If $c$ and $d$ are arbitrary (proper) commutators,
does it follow that $a=c$ and $b=d$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.20.
Proposed by: A. Gaglione, D. Spellman
Published resolution:
No, not always.
For example, if $x_1, x_2, x_3$ are free generators, \ $
x_1<x_2< x_3$, $a=[[x_2,x_1],x_3]$, \ $... | 2 | solved | A. Gaglione, D. Spellman | 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.\nFor example, if $x_1, x_2, x_3$ are free generators, \\ $\nx_1<x_2< x_3$, $a=[[x_2,x_1],x_3]$, \\ $b=[x_2,x_1]$,\n\\ $c=b^{-1}$, \\ $d=b^{x_3b}$... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,714 | KOU-11.21 | Kourovka Notebook Problem 11.21 | Let ${\mathfrak{N}}_p$ denote the
formation of all finite $p
$-groups, for a given prime
number $p$. Is it true that, for every subformation ${\mathfrak{F}}$ of
${\mathfrak{N}}_p$, there exists a variety ${\mathfrak{M}}$ such that ${\mathfrak{F}}={\mathfrak{N}}_p \cap {\mathfrak{M}}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.21.
Proposed by: A. F. Vasil'yev
Published resolution:
No, it is not true.
There is a natural one-to-one correspondence between formations of
finite $p $-groups and varieties o... | 2 | solved | A. F. Vasil'yev | 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 true.\nThere is a natural one-to-one correspondence between formations of\nfinite $p $-groups and varieties of pro-$p $-groups: for every\nvariety ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,715 | KOU-11.22 | Kourovka Notebook Problem 11.22 | Characterize all
$p
$-groups, $p$ a prime, that can be faithfully
represented as $n\times n$ triangular matrices over a division
ring of characteristic $p$.
If $p \geq n$ the solution is given in
(B. A. F. Wehrfritz, *Bull. London Math.
Soc.*, **19** (1987), 320--324). The corresponding question with
$p=0$ was solved ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.22.
Proposed by: B. A. F. Wehrfritz | 2 | open | B. A. F. Wehrfritz | 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 |
700,716 | KOU-11.23 | Kourovka Notebook Problem 11.23 | An automorphism $\varphi$
of a group $G$ is called a *nil-automorphism* if, for every
$a \in G$, there exists $n$ such that $[a, _n \varphi ] = 1$.
Here $[x, _1 y]=[x,y]$ and $[x, _{i+1} y]=[[x, _i y],y]$. An
automorphism $\varphi$ is called an *$e$-automorphism*
if, for any two $\varphi
$-invariant subgroups $A$
and $... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.23.
Proposed by: V. G. Vilyatser
Discussion and literature:
[Solved] No, not every (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.co... | 2 | solved | V. G. Vilyatser | 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] No, not every (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/wp-content/upl... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,717 | KOU-11.24 | Kourovka Notebook Problem 11.24 | A Fitting class ${\mathfrak{F}}$ is
said to be *local* if there exists a group function $f$ (for
definition see (L. A. Shemetkov, *Formations of Finite Groups*,
Moscow, Nauka, 1978 (Russian)) such that $f(p)$ is a Fitting class
for every prime number $p$ and ${\mathfrak{F}}={\mathfrak{G}}_{\pi ( {\mathfrak{F}})} \bigca... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.24.
Proposed by: N. T. Vorob'\"ev
Published resolution:
No, not every
(L. A. Shemetkov, A. F. Vasil'yev, *Abstracts of the Conf.
of Mathematicians of Belarus', Part 1*, Grodno,... | 2 | solved | N. T. Vorob'\"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: No, not every\n(L. A. Shemetkov, A. F. Vasil'yev, *Abstracts of the Conf.\nof Mathematicians of Belarus', Part 1*, Grodno, 1992, p. 56\n(Russian); \\ S. F. Kamor... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,718 | KOU-11.25 | Kourovka Notebook Problem 11.25 | b) Do there exist local
Fitting classes which are decomposable into a non-trivial product
of Fitting classes and in every such a decomposition all factors
are non-local? \ For the definition of the *product* of
Fitting classes see (N. T. Vorob'\"ev, *Math. Notes*, **43**, no. 2 (1988), 91--94).
[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: 11.25.
Proposed by: N. T. Vorob'\"ev
Archive of Solved Problems entry:
Yes, there does (N. T. Vorob'ev, A. N. Skiba, *Problems in Algebra*, **8**, Gomel', 19... | 2 | partially_solved | N. T. Vorob'\"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": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (Yes, there does (N. T. Vorob'ev, A. N. Skiba, *Problems in Algebra*, **8**, Gomel', 1995, 55--58 (Russian)).); remaining part(s) in main section."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,719 | KOU-11.26 | Kourovka Notebook Problem 11.26 | Does there exist a group which is not isomorphic to
outer automorphism group of a metabelian group with trivial
center? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.26.
Proposed by: R. G\"obel
Published resolution:
No, given any group $G$ there is a metabelian group $M$ with
trivial center such that $\mathrm{Out} M \cong G$ (R. G\"obel,
A.... | 2 | solved | R. G\"obel | 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, given any group $G$ there is a metabelian group $M$ with\ntrivial center such that $\\mathrm{Out} M \\cong G$ (R. G\\\"obel,\nA. Paras, *J. Pure Appl. Algebr... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,720 | KOU-11.27 | Kourovka Notebook Problem 11.27 | What are the minimum numbers of generators for
groups $G$ satisfying $S \leq G \leq \mathrm{Aut} S$ where $S$ is a
finite simple non-abelian group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.27.
Proposed by: K. Gruenberg
Published resolution:
The number $d(G)$
is found (mod CFSG) for every such a group
$G$. In particular, $d(G)=\max
\{ 2, d(G/S)\}$ and $d(G)\leq 3$... | 2 | solved | K. Gruenberg | 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: The number $d(G)$\nis found (mod CFSG) for every such a group\n$G$. In particular, $d(G)=\\max\n\\{ 2, d(G/S)\\}$ and $d(G)\\leq 3$ (F. Dalla Volta,\nA. Lucchini... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,721 | KOU-11.28 | Kourovka Notebook Problem 11.28 | Suppose the prime graph
of the finite group $G$ is disconnected. (This means that the set
of prime divisors of the order of $G$ is the disjoint union of
non-empty subsets $\pi$ and $\pi '$ such that $G$ contains no
element of order $pq$ where $p \in \pi$, $q \in \pi '$.) Then
P. A. Linnell (*Proc. London Math. Soc.*, *... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.28.
Proposed by: K. Gruenberg | 2 | open | K. Gruenberg | 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 |
700,722 | KOU-11.29 | Kourovka Notebook Problem 11.29 | Let $F$ be a free group
and ${\mathfrak{f}}={\mathbb{Z}F(F - 1)$ the augmentation ideal of the
integral group ring ${\mathbb{Z}F$. For any normal subgroup $R$ of
$F$ define the corresponding ideal ${\mathfrak{r}}={\mathbb{Z}F(R - 1)=
_{\mathrm{id}} (r - 1\mid r \in R)$. One may identify, for instance,
$F \cap (1 + {\ma... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 11.29.
Proposed by: N. D. Gupta
Archive of Solved Problems entry:
Yes, it is (N. D. Gupta, Yu. V. Kuz'min, *J. Pure Appl. Algebra*,
**78**, no. 1 (1992), 165... | 2 | partially_solved | N. D. Gupta | 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 (Yes, it is (N. D. Gupta, Yu. V. Kuz'min, *J. Pure Appl. Algebra*,\n**78**, no. 1 (1992), 165--172).); remaining part(s) in main section."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,723 | KOU-11.30 | Kourovka Notebook Problem 11.30 | Is it true that the rank of a torsion-free soluble group
is equal to the rank of any of its subgroups of finite index? The
answer is affirmative for groups having a rational series
(D. I. Zaitsev, in: *Groups with restrictions on subgroups*,
Naukova dumka, Kiev, 1971, 115--130 (Russian)). We note also that
every torsio... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.30.
Proposed by: D. I. Zaitsev | 2 | open | D. I. Zaitsev | 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 |
700,724 | KOU-11.31 | Kourovka Notebook Problem 11.31 | Is a radical group
polycyclic if it is a product of two polycyclic subgroups? The
answer is affirmative for soluble and for hyperabelian groups
(D. I. Zaitsev, *Math. Notes*, **29**, no. 4 (1981),
247--252; \ J. C. Lennox, J. E. Roseblade, *Math. Z.*,
**170** (1980), 153--154). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.31.
Proposed by: D. I. Zaitsev | 2 | open | D. I. Zaitsev | 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 |
700,725 | KOU-11.32 | Kourovka Notebook Problem 11.32 | Describe the primitive
finite linear groups that contain a matrix with *simple spectrum*,
that is, a matrix all of whose eigenvalues are of multiplicity 1.
See partial results in (A. Zalesskii, I. D. Suprunenko, *Commun. Algebra*, **26**, no. 3 (1998), 863--888; **28**, no. 4
(2000), 1789--1833; \ Ch. Rudloff, A. Zales... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.32.
Proposed by: A. E. Zalesski\u{\i
Discussion and literature:
[Solved] The project has been completed in (L. Di Martino, M. Pellegrini, A. Zalesski,
*J. Gr... | 2 | solved | A. E. Zalesski\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": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] The project has been completed in (L. Di Martino, M. Pellegrini, A. Zalesski,\n*J. Group Theory*, **23**... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,726 | KOU-11.33 | Kourovka Notebook Problem 11.33 | b) Let $G(q)$ be a simple Chevalley group over a
field of order $q$. Prove that there exists $m$ such that the
restriction of every non-one-dimensional representation of
$G(q^m)$ over a field of prime characteristic not dividing $q$ to
$G(q)$ contains all irreducible representations of $G(q)$ as
composition factors.
[... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 11.33.
Proposed by: A. E. Zalesski\u{\i
Archive of Solved Problems entry:
This is proved in (D. Gluck, *J. Algebra*, **155**, no. 2
(1993), 221--237). | 2 | partially_solved | A. E. Zalesski\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": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (This is proved in (D. Gluck, *J. Algebra*, **155**, no. 2\n(1993), 221--237).); remaining part(s) in main section."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,727 | KOU-11.34 | Kourovka Notebook Problem 11.34 | Describe the complex representations of quasisimple finite groups
which remain irreducible after reduction modulo any prime number
$q$. An important example: representations of degree $(p^k - 1)/2$
of the symplectic group $Sp(2k,p)$ where $k \in {\mathbb{N}$ and $p$
is an odd prime. See partial results in
(P. H. Tiep, ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.34.
Proposed by: A. E. Zalesski\u{\i | 2 | open | A. E. Zalesski\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 |
700,728 | KOU-11.35 | Kourovka Notebook Problem 11.35 | Suppose that $H$ is a finite linear group over
${\mathbb{C}$ and $h$ is an element of $H$ of prime order $p$ which
is not contained in any abelian normal subgroup. Is it true that
$h$ has at least $(p - 1)/2$ different eigenvalues? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.35.
Proposed by: A. E. Zalesskii
Published resolution:
Yes, it is
(G. R. Robinson, *J. Algebra*, **178**, no. 2 (1995),
635--642). | 2 | solved | A. E. Zalesskii | 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(G. R. Robinson, *J. Algebra*, **178**, no. 2 (1995),\n635--642)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,729 | KOU-11.36 | Kourovka Notebook Problem 11.36 | Let $G=B(m,n)$ be the
free Burnside group of rank $m$ and of odd exponent $n \gg 1$. Are
the following statements true?
a) Every 2-generated subgroup of $G$ is
isomorphic to the Burnside $n$-product of two cyclic groups.
b)
Every automorphism $\varphi$ of $G$ such that
$\varphi ^n=1$ and $b^{\varphi}\cdot b^{\varphi ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 11.36.
Proposed by: S. V. Ivanov
Archive of Solved Problems entry:
e) No, there are nontrivial zero divisors (S. V. Ivanov, R. Mikhailov, *Canad. Math. Bull.... | 2 | partially_solved | S. V. Ivanov | 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 (e) No, there are nontrivial zero divisors (S. V. Ivanov, R. Mikhailov, *Canad. Math. Bull.*, **57** (2014), 326--334).); remaining part(s) in main section."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,730 | KOU-11.37 | Kourovka Notebook Problem 11.37 | a) Can the free Burnside
group $ B(m,n)$, for any $m$ and $ n$, be given by defining relations
of the form $v^n=1$ such that for any natural divisor $d$ of $n$
distinct from $n$ the element $v^d$ is not trivial in $B(m,n)$? This
is true for odd $n \geq 665$, and for all
$n\geq 2^{48}$ divisible by
$2^9$.
[Archived sol... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 11.37.
Proposed by: S. V. Ivanov
Archive of Solved Problems entry:
Yes, it can (S. V. Ivanov, *Int. J. Algebra Comput.*, **4**, no. 1--2 (1994), 1--308). | 2 | partially_solved | S. V. Ivanov | 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 (Yes, it can (S. V. Ivanov, *Int. J. Algebra Comput.*, **4**, no. 1--2 (1994), 1--308).); remaining part(s) in main section."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,731 | KOU-11.38 | Kourovka Notebook Problem 11.38 | Does there exist a
finitely presented Noetherian group which is not almost
polycyclic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.38.
Proposed by: S. V. Ivanov | 2 | open | S. V. Ivanov | 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 |
700,732 | KOU-11.39 | Kourovka Notebook Problem 11.39 | (Well-known problem).
Does there exist a group which is not almost polycyclic and
whose integral group ring is Noetherian? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.39.
Proposed by: S. V. Ivanov | 2 | open | S. V. Ivanov | 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 |
700,733 | KOU-11.40 | Kourovka Notebook Problem 11.40 | Prove or disprove that a
torsion-free group $G$ with the small cancellation condition
$C'(\lambda )$ where $\lambda \ll 1$ necessarily has the ${\mathcal{U}P}$-property (and therefore $KG$ has no zero divisors). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.40.
Proposed by: S. V. Ivanov | 2 | open | S. V. Ivanov | 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 |
700,734 | KOU-11.42 | Kourovka Notebook Problem 11.42 | Does there exist
a torsion-free group having exactly three conjugacy classes and
containing a subgroup of index 2? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.42.
Proposed by: A. V. Izosov
Published resolution:
Yes, there does
(A. Minasyan, *Comment. Math. Helv.*, **84** (2009),
259--296). | 2 | solved | A. V. Izosov | 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 does\n(A. Minasyan, *Comment. Math. Helv.*, **84** (2009),\n259--296)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,735 | KOU-11.43 | Kourovka Notebook Problem 11.43 | For a finite group $X$, we denote by $k(X)$ the
number of its conjugacy classes. Is it true that $k(AB) \leq
k(A)k(B)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.43.
Proposed by: L. S. Kazarin
Published resolution:
No, it is not true in general:
let $G=\langle a,b\mid a^{30}=b^2=1, a^b=a^{-1}\rangle\cong
D_{60}$ be the dihedral group of... | 2 | solved | L. S. Kazarin | 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 true in general:\nlet $G=\\langle a,b\\mid a^{30}=b^2=1, a^b=a^{-1}\\rangle\\cong\nD_{60}$ be the dihedral group of order $60$, and let $A=\\langle... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,736 | KOU-11.44 | Kourovka Notebook Problem 11.44 | For a finite group $X$,
we denote by $r(X)$ its sectional rank. Is it true that the
sectional rank of a finite $p $-group, which is a product $AB$
of its subgroups $A$ and $ B$, is bounded by some linear function
of $r(A)$ and $r(B)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.44.
Proposed by: L. S. Kazarin | 2 | open | L. S. Kazarin | 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 |
700,737 | KOU-11.45 | Kourovka Notebook Problem 11.45 | A $t$-$(v,k,\lambda )$
*design* ${\mathcal{D}}=(X,{\mathcal{B}})$ contains a set $X$ of $v$
points and a set ${\mathcal{B}}$ of $k$-element subsets of $X$ called
*blocks* such that each $t$-element subset of $X$ is
contained in $\lambda$ blocks. Prove that there are no nontrivial
block-transitive 6-designs. (We have sh... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.45.
Proposed by: P. J. Cameron,
C. E. Praeger | 2 | open | P. J. Cameron,
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 |
700,738 | KOU-11.46 | Kourovka Notebook Problem 11.46 | a) Does
there exist a finite 3-group $G$ of nilpotency class 3 with the
property $[a, a^{\varphi}]=1$ for all $a \in G$ and all
endomorphisms $\varphi$ of $G$? (See A. Caranti, *J. Algebra*, **97**, no. 1 (1985), 1--13.)
[Archived solved part(s) of 11.46:]
b) Does there exist a finite
$p
$-group of nilpotency class gr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 11.46.
Proposed by: A. Caranti
Archive of Solved Problems entry:
b) Yes, there does (I. Malinowska, *Rend. Sem. Mat.
Padova*, **91** (1994), 265--271).
c) Y... | 2 | partially_solved | A. Caranti | 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) Yes, there does (I. Malinowska, *Rend. Sem. Mat.\nPadova*, **91** (1994), 265--271).\n\nc) Yes, it does (A. Abdollahi, A. Faghihi, S. A. Linton,\nE. A. O'Brie... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,739 | KOU-11.47 | Kourovka Notebook Problem 11.47 | Let ${\mathcal{L}}_d$ be the
homogeneous component of degree $d$ in a free Lie algebra ${\mathcal{L}}$
of rank 2 over the field of order 2. What is the dimension of the
fixed point space in ${\mathcal{L}}_d$ for the automorphism of ${\mathcal{L}}$
which interchanges two elements of a free generating set of ${\mathcal{L... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.47.
Proposed by: L. G. Kov\'acs
Published resolution:
It is found; R. M. Bryant and R. St\"ohr (*Arch. Math.*, **67**, no. 4 (1996), 281--289) confirmed the conjecture from
(M.... | 2 | solved | L. G. Kov\'acs | 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: It is found; R. M. Bryant and R. St\\\"ohr (*Arch. Math.*, **67**, no. 4 (1996), 281--289) confirmed the conjecture from\n(M. W. Short, *Commun. Algebra*, **23**... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,740 | KOU-11.48 | Kourovka Notebook Problem 11.48 | Is the commutator $[x,
y, y, y, y, y, y]$ a product of fifth powers in the free
group $\left< x,y\right>$? If not, then the Burnside group
$B(2,5)$ is infinite. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.48.
Proposed by: A. I. Kostrikin | 2 | open | A. I. Kostrikin | 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 |
700,741 | KOU-11.49 | Kourovka Notebook Problem 11.49 | B. Hartley (*Proc.
London Math. Soc.*, **35**, no. 1 (1977), 55--75) constructed
an example of a non-countable Artinian ${\mathbb{Z}G$-module where
$G$ is a metabelian group with the minimum condition for
normal subgroups. It follows that there exists a non-countable
soluble group (of derived length 3) satisfying Min-$... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.49.
Proposed by: L. A. Kurdachenko | 2 | open | L. A. Kurdachenko | 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 |
700,742 | KOU-11.50 | Kourovka Notebook Problem 11.50 | Let $A, C$ be abelian
groups. If $A[n]=
0$, i. e. for $a \in A$, $na=0$ implies $a=0$, then the
sequence \ $\displaystyle{
\frac{\mathrm{Hom} (C,A)}{n\mathrm{Hom} (C,A)}\; \;
\rightarrowtail \mathrm{Hom} \left(\frac{C}{nC},\; \frac{A}{nA}\right)
\twoheadrightarrow \mathrm{Ext} (C,A)[n]}
$ \
is exact. Given \ $ \display... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.50.
Proposed by: A. Mader | 2 | open | A. Mader | 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 |
700,743 | KOU-11.51 | Kourovka Notebook Problem 11.51 | Are there (large,
non-trivial) classes ${\mathfrak{X}}$ of torsion-free abelian groups
such that for $A, C \in {\mathfrak{X}}$ the group $\mathrm{Ext} (C,A)$
is torsion-free? It is a fact (E. L. Lady, A. Mader, *J. Algebra*, **140** (1991), 36--64) that two groups of such a class are
nearly isomorphic if and only if th... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.51.
Proposed by: A. Mader | 2 | open | A. Mader | 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 |
700,744 | KOU-11.52 | Kourovka Notebook Problem 11.52 | (Well-known problem). A
permutation group on a set $\Omega $ is called *sharply doubly
transitive* if for any two pairs $(\alpha , \beta )$ and
$(\gamma , \delta )$ of elements of $\Omega$ such that $\alpha \ne
\beta$ and $\gamma \ne \delta $, there is exactly one element of
the group taking $\alpha$ to $\gamma$ and $\... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.52.
Proposed by: V. D. Mazurov
Published resolution:
No, not every (E. Rips, Y. Segev, K. Tent, *J. Europ. Math. Soc.*, **19**, no. 10 (2017), 2895--2910). | 2 | solved | V. D. Mazurov | 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 (E. Rips, Y. Segev, K. Tent, *J. Europ. Math. Soc.*, **19**, no. 10 (2017), 2895--2910)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,745 | KOU-11.53 | Kourovka Notebook Problem 11.53 | (P. Kleidman). Do the sporadic
simple groups of Rudvalis $ Ru$, Mathieu $M_{22}$, and Higman--Sims
$HS$ embed into the simple group $E_7(5)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.53.
Proposed by: V. D. Mazurov
Published resolution:
Yes, they do
(P. B. Kleidman, R. A. Wilson, *J. Algebra*, **157**,
no. 2 (1993), 316--330). | 2 | solved | V. D. Mazurov | 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, they do\n(P. B. Kleidman, R. A. Wilson, *J. Algebra*, **157**,\nno. 2 (1993), 316--330)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,746 | KOU-11.54 | Kourovka Notebook Problem 11.54 | Is it true that in the group
of coloured braids only the identity braid is a conjugate to its
inverse? (For definition see Archive, 10.24.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.54.
Proposed by: G. S. Makanin
Published resolution:
The answers to both 11.54 and 11.55 are affirmative,
since the group $K_n$ of coloured braids is
embeddable into the group ... | 2 | solved | G. S. Makanin | 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: The answers to both 11.54 and 11.55 are affirmative,\nsince the group $K_n$ of coloured braids is\nembeddable into the group of those automorphisms of the free g... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,747 | KOU-11.55 | Kourovka Notebook Problem 11.55 | Is it true that extraction of roots in the
group of coloured braids is uniquely determined? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.55.
Proposed by: G. S. Makanin
Published resolution:
The answers to both 11.54 and 11.55 are affirmative,
since the group $K_n$ of coloured braids is
embeddable into the group ... | 2 | solved | G. S. Makanin | 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: The answers to both 11.54 and 11.55 are affirmative,\nsince the group $K_n$ of coloured braids is\nembeddable into the group of those automorphisms of the free g... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,748 | KOU-11.56 | Kourovka Notebook Problem 11.56 | a) Does every infinite residually finite group
contain an infinite abelian subgroup? This is equivalent to
the following: does every infinite residually finite group
contain a non-identity element with an infinite centralizer?
By a famous theorem of Shunkov
a torsion group with an involution having a
finite centralize... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.56.
Proposed by: A. Mann | 2 | open | A. Mann | 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 |
700,749 | KOU-11.57 | Kourovka Notebook Problem 11.57 | An *upper composition
factor* of a group $G$ is a composition factor of some finite
quotient of $ G$. Is there any restriction on the set of non-abelian
upper composition factors of a finitely generated group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 11th issue (1990). Original problem number: 11.57.
Proposed by: A. Mann, D. Segal
Published resolution:
There are no restrictions: any set of non-abelian finite simple
groups can be the set of upper composition factors of a... | 2 | solved | A. Mann, D. Segal | 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: There are no restrictions: any set of non-abelian finite simple\ngroups can be the set of upper composition factors of a 63-generator\ngroup; if we allow the gro... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,750 | KOU-11.58 | Kourovka Notebook Problem 11.58 | Describe the finite
groups which contain a tightly embedded subgroup $H$ such that a
Sylow 2-subgroup of $H$ is a direct product of a quaternion
group of order 8 and a non-trivial elementary group. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.58.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"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 |
700,751 | KOU-11.59 | Kourovka Notebook Problem 11.59 | A $TI$-subgroup $A$ of a
group $G$ is called a *subgroup of root type* if $[A,
A^g]=1$ whenever $N_A(A^g) \ne 1$. Describe the finite groups
containing a cyclic subgroup of order 4 as a subgroup of root
type. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.59.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"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 |
700,752 | KOU-11.60 | Kourovka Notebook Problem 11.60 | Is it true that the
hypothetical Moore graph with 3250 vertices of valence 57 has no
automorphisms of order 2? M. Aschbacher (*J. Algebra*, **19**, no. 4 (1971), 538--540) proved that this graph is not a
graph of rank 3. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.60.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"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 |
700,753 | KOU-11.61 | Kourovka Notebook Problem 11.61 | Let $F$ be a non-abelian
free pro-$p
$-group. Is it true that the subset $\{ r
\in F\mid \mathrm{cd} (F/(r)) \leq 2\} $ is dense in $F$? Here $
(r)$ denotes the closed normal subgroup of $F$ generated by $r$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 11.61.
Proposed by: O. V. Mel'nikov | 2 | open | O. V. Mel'nikov | 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 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.