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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
700,554 | KOU-9.19 | Kourovka Notebook Problem 9.19 | a) Let $n(X)$ denote the minimum of the indices of
proper subgroups of a group $X$. A subgroup $A$ of a finite group $G$
is called *wide* if $A$ is a maximal element by inclusion of
the set $\{ X\mid X \mbox{ is a proper subgroup of }G\mbox{ and
}n(X)=n(G)\} $. Find all wide subgroups in finite projective special
linea... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 9.19.
Proposed by: V. D. Mazurov
Archive of Solved Problems entry:
This was proved (S. V. Zharov, V. D. Mazurov, in: *Matematicheskoye programmirovaniye i pri... | 2 | partially_solved | V. D. Mazurov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 was proved (S. V. Zharov, V. D. Mazurov, in: *Matematicheskoye programmirovaniye i prilozheniya*,\nEkaterinburg, 1995, 96--97 (Russian)).); remaining part(s... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,555 | KOU-9.21 | Kourovka Notebook Problem 9.21 | Let $P$ be a maximal parabolic subgroup of the
smallest
index in a finite group $G$ of Lie type $ E_6$, $ E_7$, $ E_8$,
or $^2E_6$ and let $X$ be a subgroup such that $PX=G$. Is it
true that $X=G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.21.
Proposed by: V. D. Mazurov
Published resolution:
Yes, it is true (mod CFSG) (C. Hering, M. W. Liebeck, J. Saxl,
*J. Algebra*, **106**, no. 2 (1987), 517--527). | 2 | solved | V. D. Mazurov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is true (mod CFSG) (C. Hering, M. W. Liebeck, J. Saxl,\n*J. Algebra*, **106**, no. 2 (1987), 517--527)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,556 | KOU-9.23 | Kourovka Notebook Problem 9.23 | Let $G$ be a finite group, $B$ a block of
characters of $ G$, $D(B)$ its defect group and $k(B)$
(respectively, $k_0(B)$) the number of all irreducible complex
characters (of height 0) lying in $B$. *Conjectures:*
a) (R. Brauer) \ $k(B) \leq |D(B)|$; this has
been proved for $p $-soluble groups (D. Gluck, K. Magaard,
... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.23.
Proposed by: V. D. Mazurov
Discussion and literature:
[Solved] c) The conjecture is proved (G. Malle, G. Navarro, A. A. Schaeffer Fry, P. H. Tiep, *Ann. Ma... | 2 | partially_solved | V. D. Mazurov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] c) The conjecture is proved (G. Malle, G. Navarro, A. A. Schaeffer Fry, P. H. Tiep, *Ann. Math... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,557 | KOU-9.24 | Kourovka Notebook Problem 9.24 | (J. G. Thompson). *Conjecture:* every
finite simple non-abelian group $G$ can be represented in the form
$ G=CC$, where $C$ is some conjugacy class of $G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.24.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,558 | KOU-9.25 | Kourovka Notebook Problem 9.25 | Find an algorithm which recognizes, by an
equation $w(x_1,\ldots ,x_n)=1$ in a free group $F$ and by a list of
finitely generated subgroups $H_1,\ldots ,H_n$ of $ F$, whether there is a
solution of this equation satisfying the condition $x_1 \in
H_1$, $\ldots$ , $x_n \in H_n$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.25.
Proposed by: G. S. Makanin
Published resolution:
Such an algorithm is found in (V. Diekert, C. Guti{\'e}rrez, C. Hagenah,
*Inf. Comput.*, **202**, no. {2} (2005), 105--140). | 2 | solved | G. S. Makanin | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 an algorithm is found in (V. Diekert, C. Guti{\\'e}rrez, C. Hagenah,\n*Inf. Comput.*, **202**, no. {2} (2005), 105--140)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,559 | KOU-9.26 | Kourovka Notebook Problem 9.26 | b) Describe the finite groups of 2-local 3-rank 1
which have non-cyclic Sylow 3-subgroups.
[Archived solved part(s) of 9.26:]
a) Describe the finite groups of 2-local 3-rank 1
which have 3-rank at least 3. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 9.26.
Proposed by: A. A. Makhn\"ev
Archive of Solved Problems entry:
Described in (A. A. Makhn\"ev, *Siber. Math. J.*, **29**
(1988), 951--959). | 2 | partially_solved | A. A. Makhn\"ev | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (Described in (A. A. Makhn\\\"ev, *Siber. Math. J.*, **29**\n(1988), 951--959).); 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 |
700,560 | KOU-9.27 | Kourovka Notebook Problem 9.27 | Let $M$ be a subgroup of a finite group $G$, \ $A$ an
abelian 2-subgroup of $M$, and suppose that $A^g$ is not
contained in $M$ for some $g$ from $G$. Determine the structure
of $G$ under the hypothesis that $\left< A,A^x\right> =G$
whenever the subgroup $A^x$, $x \in G$, is not contained in $M$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.27.
Proposed by: A. A. Makhn\"ev
Published resolution:
It is described mod CFSG (V. I. Zenkov, *Algebra and Logic*,
**35**, no. 3 (1996), 160--163). | 2 | solved | A. A. Makhn\"ev | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 described mod CFSG (V. I. Zenkov, *Algebra and Logic*,\n**35**, no. 3 (1996), 160--163)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,561 | KOU-9.28 | Kourovka Notebook Problem 9.28 | Suppose that a finite group $G$ is generated by a
conjugacy class $D$ of involutions and let $D_i=\{ d_1\cdots
d_i\mid d_1,\ldots ,d_i$ are different pairwise commuting elements
of $ D\} $. What is $G$, if $ D_1,\ldots ,D_n$ are all its
different conjugacy classes of involutions? For example, the
Fischer groups $F_{22}... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.28.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,562 | KOU-9.29 | Kourovka Notebook Problem 9.29 | (Well-known problem). According to a classical
theorem of Magnus, the word problem is soluble in 1-relator
groups. Do there exist 2-relator groups with insoluble word
problem? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.29.
Proposed by: Yu. I. Merzlyakov | 2 | open | Yu. I. Merzlyakov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,563 | KOU-9.30 | Kourovka Notebook Problem 9.30 | (Well-known problem). A finite set of reductions
$u_i \rightarrow v_i$ of words on a finite alphabet $\Sigma
=\Sigma ^{-1}$ is called a *group set of reductions* if
$\mbox{*length*} (u_i) > \mbox{*length*} (v_i)$ or
$\mbox{*length*} (u_i)=\mbox{*length*} (v_i)$ and $u_i
> v_i$ in the lexicographical ordering, and every... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.30.
Proposed by: Yu. I. Merzlyakov
Published resolution:
Yes, there do (J. Avenhaus, K. Madlener, F. Otto, *Trans.
Amer. Math. Soc.*, **297** (1986), 427--443). | 2 | solved | Yu. I. Merzlyakov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there do (J. Avenhaus, K. Madlener, F. Otto, *Trans.\nAmer. Math. Soc.*, **297** (1986), 427--443)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,564 | KOU-9.31 | Kourovka Notebook Problem 9.31 | Let $k$ be a field of characteristic 0. According
to (Yu. I. Merzlyakov, *Proc. Steklov Inst. Math.*, **167** (1986), 263--266), the family $\mathrm{Rep}_k(G)$ of all
canonical matrix representations of a $k$-powered group $G$ over $
k$ may be regarded as an affine $ k$-variety. Find an explicit
form of equations defin... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.31.
Proposed by: Yu. I. Merzlyakov | 2 | open | Yu. I. Merzlyakov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,565 | KOU-9.32 | Kourovka Notebook Problem 9.32 | What locally compact groups satisfy the following
condition: the product of any two closed subgroups is also a closed
subgroup? Abelian groups with this property were described in
(Yu. N. Mukhin, *Math. Notes*, **8** (1970), 755--760). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.32.
Proposed by: Yu. N. Mukhin
Published resolution:
Such groups are described (W. Herfort, K. H. Hofmann, F. G. Russo, *Adv. Math.*, **390** (2021), 107894). | 2 | solved | Yu. N. Mukhin | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 groups are described (W. Herfort, K. H. Hofmann, F. G. Russo, *Adv. Math.*, **390** (2021), 107894)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,566 | KOU-9.33 | Kourovka Notebook Problem 9.33 | (F. K\"ummich, H. Scheerer). If $H$ is a
closed
subgroup of a
connected locally-compact group $G$ such that $ \overline{\!
HX}= \overline{\! H} \overline{\! X}$ for every closed
subgroup $X$ of $G$, then is $H$ normal? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.33.
Proposed by: Yu. N. Mukhin
Published resolution:
Yes, it is (C. Scheiderer, *Monatsh. Math.*, **98** (1984), 75--81). | 2 | solved | Yu. N. Mukhin | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (C. Scheiderer, *Monatsh. Math.*, **98** (1984), 75--81)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,567 | KOU-9.34 | Kourovka Notebook Problem 9.34 | (S. K. Grosser, W. N. Herfort). Does there exist
an infinite compact $p
$-group in which the
centralizers of all elements are finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.34.
Proposed by: Yu. N. Mukhin
Published resolution:
No, it does not exist, in view of the connection of this problem
with the Restricted Burnside Problem (S. K. Grosser,
W. N. H... | 2 | solved | Yu. N. Mukhin | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it does not exist, in view of the connection of this problem\nwith the Restricted Burnside Problem (S. K. Grosser,\nW. N. Herfort, *Trans. Amer. Math. Soc.*,... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,568 | KOU-9.35 | Kourovka Notebook Problem 9.35 | A topological group is said to be *inductively compact* if any finite set of its elements is
contained in a compact subgroup. Is this property preserved under
lattice isomorphisms in the class of locally compact groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.35.
Proposed by: Yu. N. Mukhin | 2 | open | Yu. N. Mukhin | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,569 | KOU-9.36 | Kourovka Notebook Problem 9.36 | Characterize the lattices of closed subgroups in
locally compact groups. In the discrete case this was done in
(B. V. Yakovlev, *Algebra and Logic*, **13**, no. 6
(1974), 400--412). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.36.
Proposed by: Yu. N. Mukhin | 2 | open | Yu. N. Mukhin | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,570 | KOU-9.37 | Kourovka Notebook Problem 9.37 | Is a compactly generated inductively prosoluble
locally compact group prosoluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.37.
Proposed by: Yu. N. Mukhin | 2 | open | Yu. N. Mukhin | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,571 | KOU-9.38 | Kourovka Notebook Problem 9.38 | A group is said to be *compactly covered* if
it is the union of its compact subgroups. In a null-dimensional
locally compact group, are maximal compactly covered subgroups
closed? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.38.
Proposed by: Yu. N. Mukhin | 2 | open | Yu. N. Mukhin | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,572 | KOU-9.39 | Kourovka Notebook Problem 9.39 | Let $\Omega$ be a countable set and ${\mathfrak{m}}$ a
cardinal number such that $\aleph _0 \leq {\mathfrak{m}} \leq 2^{\aleph
_0}$ (we assume Axiom of Choice but not Continuum Hypothesis). Does
there exist a permutation group $G$ on $\Omega$ that has exactly
${\mathfrak{m}}$ orbits on the power set ${\mathfrak{P}}(\Om... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.39.
Proposed by: P. M. Neumann | 2 | open | P. M. Neumann | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,573 | KOU-9.40 | Kourovka Notebook Problem 9.40 | Let $\Omega$ be a countably infinite set. Define
a *moiety* of $\Omega$ to be a subset $\Sigma$ such that
both $\Sigma$ and $\Omega \setminus \Sigma$ are infinite. Which
permutation groups on $ \Omega$ are transitive on moieties? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.40.
Proposed by: P. M. Neumann | 2 | open | P. M. Neumann | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,574 | KOU-9.41 | Kourovka Notebook Problem 9.41 | Let $\Omega$ be a countably infinite set. For $k
\geq 2$ define a *$k$-section* of $\Omega$ to be a partition
of $\Omega$ as union of $k$ infinite sets.
b) Does there exist a group that is
transitive on
$k$-sections
but not on $(k + 1)$-sections?
c)
Does there exist a transitive permutation
group on $\Omega$ that is ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 9.41.
Proposed by: P. M. Neumann
Main section comment:
[Solved] c) Yes, it does exist (AI solution confirmed by S. M. Corson: \ Achyuth Jayadevan, *Preprint*,... | 2 | partially_solved | P. M. Neumann | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. Let $U$ be a non-principal ultrafilter in $\n{\\mathfrak{P}}(\\Omega )$ and let $G=\\{ g\\in \\mathrm{Sym}(\\Omega )\\mid \\mathrm{Fix}(g)\\in U... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,575 | KOU-9.42 | Kourovka Notebook Problem 9.42 | Let $\Omega$ be a countable set and let $D$ be
the set of total order relations on $ \Omega$ for which $ \Omega$
is order-isomorphic with ${\mathbb{Q}$. Does there exist a transitive
proper subgroup $G$ of $\mathrm{Sym} (\Omega )$ which is transitive
on $D$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.42.
Proposed by: P. M. Neumann
Discussion and literature:
[Solved] An affirmative answer is consistent with the ZFC axioms of set theory (S. M. Corson, S. Shel... | 2 | solved | P. M. Neumann | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] An affirmative answer is consistent with the ZFC axioms of set theory (S. M. Corson, S. Shelah, *Preprin... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,576 | KOU-9.43 | Kourovka Notebook Problem 9.43 | b) The group $G$ indicated in (N. D. Podufalov,
*Abstracts of the 9th All--Union Symp. on Group Theory*,
Moscow, 1984, 113--114 (Russian)) allows us to construct a
projective plane of order 3: one can take as lines any class of
subgroups of order $2\cdot 5\cdot 11\cdot 17$ conjugate under $ S$
and add four more lines i... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 9.43.
Proposed by: N. D. Podufalov
Archive of Solved Problems entry:
Not always. The answer is affirmative for $n=1$. But for $n>1$ the
planes in question may... | 2 | partially_solved | N. D. Podufalov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (Not always. The answer is affirmative for $n=1$. But for $n>1$ the\nplanes in question may not have the Galois property. For example, if\nthere exists a near-fie... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,577 | KOU-9.44 | Kourovka Notebook Problem 9.44 | A topological group is said to be *layer
compact* if the full inverse images of all of its compacts under
mappings $x \rightarrow x^n$, $n=1, 2, \ldots $, are
compacts. Describe the locally compact locally soluble layer
compact groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.44.
Proposed by: V. M. Poletskikh | 2 | open | V. M. Poletskikh | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,578 | KOU-9.45 | Kourovka Notebook Problem 9.45 | Let $a$ be a vector in ${\mathbb{R}^n$ with rational
coordinates and set $$S(a)=\{ ka + b\mid k \in {\mathbb{Z}, b \in {\mathbb{Z}^n\} .$$
It is obvious that $S(a)$ is a discrete subgroup in
${\mathbb{R}^n$ of rank $n$. Find necessary and sufficient conditions
in terms of the coordinates of $a$ for $S(a)$ to have an
or... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.45.
Proposed by: Yu. D. Popov,
I. V. Protasov
Discussion and literature:
[Unverified AI claim] Conditions found (unverified AI solution: M. Kinyon, J. Urban, *... | 2 | open | Yu. D. Popov,
I. V. Protasov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] Conditions found (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,\... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,579 | KOU-9.46 | Kourovka Notebook Problem 9.46 | Let $G$ be a locally compact group of countable weight and
$L(G)$ the space of all its closed subgroups equipped with the
$E$-topology. Then is $L(G)$ a $k$-space? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.46.
Proposed by: I. V. Protasov
Published resolution:
Yes (I. V. Protasov, *Dokl. Akad. Nauk Ukr. SSR Ser. A*, **10** (1986), 64--66 (Russian)). | 2 | solved | I. V. Protasov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (I. V. Protasov, *Dokl. Akad. Nauk Ukr. SSR Ser. A*, **10** (1986), 64--66 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,580 | KOU-9.47 | Kourovka Notebook Problem 9.47 | Is it true that every scattered compact can be
homeomorphically embedded into the space of all closed non-compact
subgroups with $ E$-topology of a suitable locally compact
group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.47.
Proposed by: I. V. Protasov
Discussion and literature:
[Unverified AI claim] No, not every (unverified AI solutions: M. Kinyon, J. Urban, *Preprint of 17 S... | 2 | open | I. V. Protasov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] No, not every (unverified AI solutions: M. Kinyon, J. Urban, *Preprint of 17 Sept... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,581 | KOU-9.48 | Kourovka Notebook Problem 9.48 | In a null-dimensional locally compact group, is
the set of all compact elements closed? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.48.
Proposed by: I. V. Protasov
Published resolution:
Yes, it is (G. A. Willis, *Math. Ann.*, **300** (1994),
341--363). | 2 | solved | I. V. Protasov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (G. A. Willis, *Math. Ann.*, **300** (1994),\n341--363)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,582 | KOU-9.49 | Kourovka Notebook Problem 9.49 | Let $G$ be a compact group of weight $> \omega
_2$. Is it true that the space of all closed subgroups of $G$ with
respect to $E$-topology is non-dyadic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.49.
Proposed by: I. V. Protasov,
Yu. V. Tsybenko
Published resolution:
Yes, it is true (Yu. V. Tsybenko, *Ukrain. Math. J.*, **38** (1986), 542--545). | 2 | solved | I. V. Protasov,
Yu. V. Tsybenko | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is true (Yu. V. Tsybenko, *Ukrain. Math. J.*, **38** (1986), 542--545)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,583 | KOU-9.50 | Kourovka Notebook Problem 9.50 | Is every 4-Engel group
a) without elements of order 2 and 5
necessarily soluble?
b) satisfying the identity $x^5=1$ necessarily
locally finite?
c) (R. I. Grigorchuk) satisfying the identity
$x^8=1$ necessarily locally finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.50.
Proposed by: Yu. P. Razmyslov
Published resolution:
a) Yes; this follows from the local nilpotency of 4-Engel groups
(G. Traustason, *Int. J. Algebra Comput.*, **15** (2005),... | 2 | solved | Yu. P. Razmyslov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) Yes; this follows from the local nilpotency of 4-Engel groups\n(G. Traustason, *Int. J. Algebra Comput.*, **15** (2005),\n309--316; \\ G. Havas, M. R. Vaughan... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,584 | KOU-9.51 | Kourovka Notebook Problem 9.51 | Does there exist a finitely presented soluble
group satisfying the maximum con- dition on normal subgroups which
has insoluble word problem? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.51.
Proposed by: D. J. S. Robinson | 2 | open | D. J. S. Robinson | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,585 | KOU-9.52 | Kourovka Notebook Problem 9.52 | Does a finitely presented soluble group of finite
rank have soluble conjugacy problem? Note: there is an algorithm
to decide conjugacy to a given element of the group. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.52.
Proposed by: D. J. S. Robinson | 2 | open | D. J. S. Robinson | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,586 | KOU-9.53 | Kourovka Notebook Problem 9.53 | Is the isomorphism problem soluble for finitely
presented soluble groups of finite rank? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.53.
Proposed by: D. J. S. Robinson | 2 | open | D. J. S. Robinson | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,587 | KOU-9.54 | Kourovka Notebook Problem 9.54 | If $G=HK$ is a soluble group and $ H$, $ K$ are
minimax groups, is it true that $G$ is a minimax group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.54.
Proposed by: D. J. S. Robinson
Published resolution:
Yes, it is true (J. S. Wilson, *J. Pure Appl. Algebra*, **53**, no. 3 (1988), 297--318; Ya. P. Sysak, *Radical modules
ov... | 2 | solved | D. J. S. Robinson | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is true (J. S. Wilson, *J. Pure Appl. Algebra*, **53**, no. 3 (1988), 297--318; Ya. P. Sysak, *Radical modules\nover groups of finite rank*, Inst. of Mat... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,588 | KOU-9.55 | Kourovka Notebook Problem 9.55 | Does there exist a finite $p
$-group
$G$ and a central augmented automorphism $\varphi$ of ${\mathbb{Z}G$
such that $\varphi $, if extended to $\widehat{{\mathbb{Z}}_pG$, the
$p
$-adic group ring, is *not* conjugation by unit
in $\widehat{{\mathbb{Z}}_pG$ followed by a homomorphism induced from
a group automorphism? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.55.
Proposed by: K. W. Roggenkamp | 2 | open | K. W. Roggenkamp | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,589 | KOU-9.56 | Kourovka Notebook Problem 9.56 | Find all finite groups with the property that the
tensor square of any ordinary irreducible character is
multiplicity free. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.56.
Proposed by: J. Saxl
Published resolution:
Such groups are shown to be soluble (L. S. Kazarin, E. I. Chankov,
*Sb. Math.*, **201**, no. 5 (2010), 655--668), and abundance of ... | 2 | solved | J. Saxl | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 groups are shown to be soluble (L. S. Kazarin, E. I. Chankov,\n*Sb. Math.*, **201**, no. 5 (2010), 655--668), and abundance of examples shows that a classif... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,590 | KOU-9.57 | Kourovka Notebook Problem 9.57 | The set of subformations of a given formation is
a lattice with respect to operations of intersection and
generation. What formations of finite groups have distributive
lattices of subformations? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.57.
Proposed by: A. N. Skiba | 2 | open | A. N. Skiba | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,591 | KOU-9.58 | Kourovka Notebook Problem 9.58 | Can a product of non-local formations of finite groups be
local? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.58.
Proposed by: A. N. Skiba, L. A. Shemetkov
Published resolution:
Yes, it can (V. A. Vedernikov, *Math. Notes*, **46** (1989),
910--913; \ N. T. Vorob'ev, in: *Proc. 7th Region... | 2 | solved | A. N. Skiba, L. A. Shemetkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it can (V. A. Vedernikov, *Math. Notes*, **46** (1989),\n910--913; \\ N. T. Vorob'ev, in: *Proc. 7th Regional Sci.\nSession Math.*, Zielona Goru, 1989, 79--... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,592 | KOU-9.59 | Kourovka Notebook Problem 9.59 | (W. Gasch\"utz). Prove that the formation
generated by a finite group has finite lattice of subformations. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.59.
Proposed by: A. N. Skiba, L. A. Shemetkov
Published resolution:
Counterexamples are constructed (V. P. Burichenko, *J. Algebra*, **372**
(2012), 428--458). | 2 | solved | A. N. Skiba, L. A. Shemetkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: Counterexamples are constructed (V. P. Burichenko, *J. Algebra*, **372**\n(2012), 428--458)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,593 | KOU-9.60 | Kourovka Notebook Problem 9.60 | Let ${\mathfrak{F}}$ and ${\mathfrak{H}}$ be local formations
of finite groups and suppose that ${\mathfrak{F}}$ is not contained in
${\mathfrak{H}}$. Does ${\mathfrak{F}}$ necessarily have at least one minimal
local non-${\mathfrak{H}}$-subformation? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.60.
Proposed by: \mbox{A. N. Skiba, L. A. Shemetkov
Published resolution:
No, not necessarily (V. P. Burichenko, *Trudy Inst. Math. National Akad. Sci. Belarus'*, **21**, no. 1 (... | 2 | solved | \mbox{A. N. Skiba, L. A. Shemetkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 necessarily (V. P. Burichenko, *Trudy Inst. Math. National Akad. Sci. Belarus'*, **21**, no. 1 (2013),\n15--24 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,594 | KOU-9.61 | Kourovka Notebook Problem 9.61 | Two varieties are said to be *$S$-equivalent* if they have the same Mal'cev theory
(D. M. Smirnov, *Algebra and Logic*, **22**, no. 6
(1983), 492--501). What is the cardinality of the set of
$S$-equivalent varieties of groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.61.
Proposed by: D. M. Smirnov | 2 | open | D. M. Smirnov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,595 | KOU-9.62 | Kourovka Notebook Problem 9.62 | In any group $G$, the cosets of all of its normal
subgroups together with the empty set form the *block
lattice* $C(G)$ with respect to inclusion, which, for infinite
$|G|$, is subdirectly irreducible and, for finite $|G| \geq 3$, is
even simple (D. M. Smirnov, A. V. Reibol'd, *Algebra and
Logic*, **23**, no. 6 (1984),... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.62.
Proposed by: D. M. Smirnov
Published resolution:
For each $g \in G$ the filter $\{ x \in C(G)\mid g \in x\} $ is
modular (and even arguesian); the variety generated by the bl... | 2 | solved | D. M. Smirnov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: For each $g \\in G$ the filter $\\{ x \\in C(G)\\mid g \\in x\\} $ is\nmodular (and even arguesian); the variety generated by the block\nlattices of groups does ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,596 | KOU-9.63 | Kourovka Notebook Problem 9.63 | Is a finite group of
the form $G=ABA$
soluble
if $A$ is an
abelian subgroup and $B$ is a cyclic subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.63.
Proposed by: Ya. P. Sysak
Published resolution:
Yes, it is soluble (mod CFSG) (D. L. Zagorin, L. S. Kazarin,
*Dokl. Math.*, **53**, no. 2 (1996), 237--239). | 2 | solved | Ya. P. Sysak | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 soluble (mod CFSG) (D. L. Zagorin, L. S. Kazarin,\n*Dokl. Math.*, **53**, no. 2 (1996), 237--239)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,597 | KOU-9.64 | Kourovka Notebook Problem 9.64 | Is it true that, in a group of the form $ G=AB$,
every subgroup $N$ of $A \cap B$ which is subnormal both in $A$
and in $B$ is subnormal in $G$? The answer is affirmative in the
case of finite groups (H. Wielandt). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.64.
Proposed by: Ya. P. Sysak
Published resolution:
No, not always (C. Casolo, U. Dardano, *J. Group Theory*, **7** (2004), 507--520). | 2 | solved | Ya. P. Sysak | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (C. Casolo, U. Dardano, *J. Group Theory*, **7** (2004), 507--520)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,598 | KOU-9.65 | Kourovka Notebook Problem 9.65 | Is a locally soluble group periodic if it is a
product of two periodic subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.65.
Proposed by: Ya. P. Sysak | 2 | open | Ya. P. Sysak | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,599 | KOU-9.66 | Kourovka Notebook Problem 9.66 | a) *B. Jonsson's Conjecture:* elementary
equivalence is preserved under taking free products in the class
of all groups, that is, if $Th(G_1)=Th(G_2)$ and $Th(H_1)=Th(H_2)$
for groups
$G_1$, $G_2$, $H_1$, $H_2$, then $Th(G_1*H_1)=Th(G_2*H_2)$.
b) It may be interesting to consider also the
following *weakened conjectur... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.66.
Proposed by: A. D. Taimanov
Discussion and literature:
[Solved] The conjecture (also known as R. L. Vaught's conjecture) is proved (Z. Sela, *Preprint*, 20... | 2 | solved | A. D. Taimanov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] The conjecture (also known as R. L. Vaught's conjecture) is proved (Z. Sela, *Preprint*, 2010, https://a... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,600 | KOU-9.67 | Kourovka Notebook Problem 9.67 | (A. Tarski). Let
$F_n$ be a free group of rank $ n$; is it true that $
Th(F_2)=Th(F_3)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.67.
Proposed by: A. D. Taimanov
Published resolution:
Yes, it is
(O. Kharlampovich, A. Myasnikov, *J. Algebra*, **302**,
(2006), 451--552; \ Z. Sela, *Geom. Funct. Anal.*, **16**... | 2 | solved | A. D. Taimanov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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(O. Kharlampovich, A. Myasnikov, *J. Algebra*, **302**,\n(2006), 451--552; \\ Z. Sela, *Geom. Funct. Anal.*, **16** (2006),\n707--730)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,601 | KOU-9.68 | Kourovka Notebook Problem 9.68 | Let ${\mathfrak{V}}$ be a variety of groups which is
not the variety of all groups, and let $p$ be a prime. Is there a
bound on the
$
p
$-lengths of the finite $p
$-soluble groups
whose Sylow $ p
$-subgroups are in ${\mathfrak{V}}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.68.
Proposed by: John S. Wilson | 2 | open | John S. Wilson | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,602 | KOU-9.69 | Kourovka Notebook Problem 9.69 | (P. Cameron). Let $G$ be a finite primitive
permutation group and suppose that the stabilizer $G_{\alpha}$ of
a point $\alpha$ induces on some of its orbits $\Delta \ne \{
\alpha \}$ a regular permutation group. Is it true that
$|G_{\alpha}|=|\Delta |$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.69.
Proposed by: A. N. Fomin
Discussion and literature:
[Solved] No, not always (P. Spiga, *J. Group Theory*, **25**, no. 1 (2022), 113--126). | 2 | solved | A. N. Fomin | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (P. Spiga, *J. Group Theory*, **25**, no. 1 (2022), 113--126)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,603 | KOU-9.70 | Kourovka Notebook Problem 9.70 | Prove or disprove the conjecture of
P. Cameron (*Bull.
London Math. Soc.*, **13**, no. 1 (1981), 1--22): if $G$ is a
finite primitive permutation group of subrank $m$, then either the
rank of $G$ is bounded by a function of $m$ or the order of a
point stabilizer is at most $m$. The *subrank* of a
transitive permutation... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.70.
Proposed by: A. N. Fomin | 2 | open | A. N. Fomin | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,604 | KOU-9.71 | Kourovka Notebook Problem 9.71 | Is it true that every infinite 2-transitive
permutation groups with locally soluble point stabilizer has a
non-trivial irreducible finite-dimensional representation over some
field? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.71.
Proposed by: A. N. Fomin | 2 | open | A. N. Fomin | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,605 | KOU-9.72 | Kourovka Notebook Problem 9.72 | Let $G_1$ and $G_2$ be Lie groups with the
following property: each $G_i$ contains a nilpotent simply
connected normal Lie subgroup $B_i$ such that $G_i/B_i \cong
SL_2(K)$, where $K={\mathbb{R}$ or ${\mathbb{C}$. Assume that $G_1$ and $G_2$
are contained as closed subgroups in a topological group $G$, that
$G_1 \cap G_... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.72.
Proposed by: A. L. Chermak | 2 | open | A. L. Chermak | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,606 | KOU-9.73 | Kourovka Notebook Problem 9.73 | Let ${\mathfrak{F}}$ be any local formation of finite soluble groups
containing all finite nilpotent groups. Prove that $H^{{\mathfrak{F}}}K=KH^{{\mathfrak{F}}}$ for any two
subnormal subgroups $H$ and $K$ of an arbitrary finite group
$G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.73.
Proposed by: L. A. Shemetkov
Published resolution:
This has been proved, even without the hypothesis of solubility of
${\mathfrak{F}}$ \ (S. F. Kamornikov, *Dokl. Akad. Nauk
... | 2 | solved | L. A. Shemetkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, even without the hypothesis of solubility of\n${\\mathfrak{F}}$ \\ (S. F. Kamornikov, *Dokl. Akad. Nauk\nBSSR*, **33**, no. 5 (1989), 396--... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,607 | KOU-9.74 | Kourovka Notebook Problem 9.74 | Find all local formations ${\mathfrak{F}}$ of finite
groups such that every finite minimal non-${\mathfrak{F}}$-group is
either a Shmidt group (that is, a non-nilpotent finite group all
of whose proper subgroups are nilpotent) or a group of prime
order. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.74.
Proposed by: L. A. Shemetkov
Published resolution:
They are found (S. F. Kamornikov, *Siberian Math. J.*, **35**, no. 4 (1994), 713--721). | 2 | solved | L. A. Shemetkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: They are found (S. F. Kamornikov, *Siberian Math. J.*, **35**, no. 4 (1994), 713--721)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,608 | KOU-9.75 | Kourovka Notebook Problem 9.75 | Find all local formations ${\mathfrak{F}}$ of finite
groups such that, in every finite group, the set of ${\mathfrak{F}}$-subnormal subgroups forms a lattice.
*Comment of 2015*: Subgroup-closed formations with this property are described in (Xiaolan Yi, S. F. Kamornikov, *J. Algebra*, **444** (2015),
143--151). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.75.
Proposed by: L. A. Shemetkov | 2 | open | L. A. Shemetkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,609 | KOU-9.76 | Kourovka Notebook Problem 9.76 | We define a *Golod group* to be the
$r$-generated, $r \geq 2$, subgroup $\left< 1 + x_1+I,\right.$
$\left. 1+x_2+I,\ldots ,1+x_r+I\right>$ of the adjoint group
$1 +
F/I$ of the factor-algebra $F/I$, where $F$ is a free algebra of
polynomials without constant terms in non-commuting variables $x_1,
x_2, \ldots , x_r$ ove... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.76.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,610 | KOU-9.77 | Kourovka Notebook Problem 9.77 | Does there exist an infinite finitely generated
residually finite binary finite group all of whose Sylow subgroups
are finite? A. V. Rozhkov (Dr. of Sci. Disser., 1997) showed that
such a group does exist if the condition of finiteness of the
Sylow subgroups is weakened to local finiteness. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.77.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,611 | KOU-9.78 | Kourovka Notebook Problem 9.78 | Does there exist a periodic residually finite $
F^{\displaystyle{*}}$-group (see Archive, 7.42) all of whose Sylow
subgroups are finite and which is not binary finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.78.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,612 | KOU-9.79 | Kourovka Notebook Problem 9.79 | (A. G. Kurosh). Is every group with the minimum
condition countable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.79.
Proposed by: V. P. Shunkov
Published resolution:
No (V. N. Obraztsov, *Math. USSR--Sb.*, **66** (1989), 541--553). | 2 | solved | V. P. Shunkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. N. Obraztsov, *Math. USSR--Sb.*, **66** (1989), 541--553)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,613 | KOU-9.80 | Kourovka Notebook Problem 9.80 | Are the 2-elements of a group with the minimum condition
contained in its locally finite radical? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.80.
Proposed by: V. P. Shunkov
Published resolution:
Not necessarily (A. Yu. Olshanskii, *The geometry of
defining relations in groups*, Kluwer, Dordrecht, 1991). | 2 | solved | V. P. Shunkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: Not necessarily (A. Yu. Olshanskii, *The geometry of\ndefining relations in groups*, Kluwer, Dordrecht, 1991)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,614 | KOU-9.81 | Kourovka Notebook Problem 9.81 | Does there exist a simple group with the minimum condition
possessing a non-trivial quasi-cyclic subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.81.
Proposed by: V. P. Shunkov
Published resolution:
Yes, there does (V. N. Obraztsov, *Math. USSR--Sb.*, **66** (1989), 541--553). | 2 | solved | V. P. Shunkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (V. N. Obraztsov, *Math. USSR--Sb.*, **66** (1989), 541--553)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,615 | KOU-9.82 | Kourovka Notebook Problem 9.82 | An infinite group, all of whose proper subgroups are finite, is
called *quasi-finite*. Is it true that an element of a
quasi-finite group $G$ is central if and only if it is contained
in infinitely many subgroups of $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.82.
Proposed by: V. P. Shunkov
Published resolution:
No (K. I. Lossov, Dep. no. 5529-V89, VINITI, Moscow, 1988
(Russian)). | 2 | solved | V. P. Shunkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (K. I. Lossov, Dep. no. 5529-V89, VINITI, Moscow, 1988\n(Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,616 | KOU-9.83 | Kourovka Notebook Problem 9.83 | Suppose that $G$ is a (periodic) $
p
$-conjugacy biprimitively finite group (see 6.59)
which has a finite Sylow
$
p
$-subgroup. Is it then true that all Sylow
$p
$-subgroups of $G$ are conjugate? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.83.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,617 | KOU-9.84 | Kourovka Notebook Problem 9.84 | a) Is every binary finite 2-group of finite
exponent locally finite?
b) The same question for $p
$-groups for $p
> 2$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.84.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,618 | KOU-10.1 | Kourovka Notebook Problem 10.1 | Let $p$ be a
prime number. Describe
the
groups of order
$
p^9$ of nilpotency class 2 which contain subgroups $X$ and $Y$
such
that $|X|=|Y|=p^3$ and any non-identity elements $x \in X$, $y
\in
Y$ do not commute. An answer to this question would yield a
description of the semifields of order $p^3$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.1.
Proposed by: S. N. Adamov, A. N. Fomin
Published resolution:
They are described (V. A. Antonov,
*Preprint*, Chelyabinsk, 1999 (Russian)). | 2 | solved | S. N. Adamov, 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 | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: They are described (V. A. Antonov,\n*Preprint*, Chelyabinsk, 1999 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,619 | KOU-10.2 | Kourovka Notebook Problem 10.2 | A *mixed identity* of a group $G$ is, by
definition, an identity of an algebraic system obtained from $G$
by supplementing its signature by some set of 0-ary operations.
One can develop a theory of mixed varieties of groups on the basis
of this notion (see, for example, V. S. Anashin, *Math. USSR
Sbornik*, **57** (1987... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.2.
Proposed by: V. S. Anashin | 2 | open | V. S. Anashin | 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 |
700,620 | KOU-10.3 | Kourovka Notebook Problem 10.3 | Characterize (in terms of bases of mixed
identities or in terms of generating groups) minimal mixed
varieties of groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.3.
Proposed by: V. S. Anashin | 2 | open | V. S. Anashin | 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 |
700,621 | KOU-10.4 | Kourovka Notebook Problem 10.4 | Let $p$ be a prime number. Is it true that a
mixed variety of groups generated by an arbitrary finite
$p
$-group of sufficiently large nilpotency class is a
variety of groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.4.
Proposed by: V. S. Anashin
Discussion and literature:
[Solved] No, not always (AI solution confirmed by K. Muliarchyk: \ Achyuth Jayadevan, *Preprint*, 20... | 2 | solved | V. S. Anashin | 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] No, not always (AI solution confirmed by K. Muliarchyk: \\ Achyuth Jayadevan, *Preprint*, 2026, https://... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,622 | KOU-10.5 | Kourovka Notebook Problem 10.5 | Construct an example of a finite group whose
mixed identities do not have a finite basis. Is the group
constructed in (R. Bryant, *Bull. London Math. Soc.*, **14**, no. 2 (1982), 119--123) such an example? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.5.
Proposed by: V. S. Anashin | 2 | open | V. S. Anashin | 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 |
700,623 | KOU-10.6 | Kourovka Notebook Problem 10.6 | Is it true that in
an abelian group every non-discrete group topology can be
strengthened up to a non-discrete group topology such that the
group becomes a complete topological group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.6.
Proposed by: V. I. Arnautov
Published resolution:
This is true for group topologies satisfying the first axiom of
countability (E. I. Marin, *in: Modules, Algebras, Topologi... | 2 | solved | V. I. Arnautov | 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 is true for group topologies satisfying the first axiom of\ncountability (E. I. Marin, *in: Modules, Algebras, Topologies\n$($Mat. Issledovaniya*, **105**),... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,624 | KOU-10.7 | Kourovka Notebook Problem 10.7 | Is it true that in a countable group $G$, any non-discrete
group topology satisfying the first axiom of countability can be
strengthened up to a non-discrete group topology such that $G$
becomes a complete topological group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.7.
Proposed by: V. I. Arnautov
Published resolution:
Yes, it is (V. I. Arnautov, E. I. Kabanova, *Siberian
Math. J.*, **31** (1990), 1--10). | 2 | solved | V. I. Arnautov | 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 (V. I. Arnautov, E. I. Kabanova, *Siberian\nMath. J.*, **31** (1990), 1--10)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,625 | KOU-10.8 | Kourovka Notebook Problem 10.8 | Does there exist a topological group which cannot
be embedded in the multiplicative semigroup of a
topological ring? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.8.
Proposed by: V. I. Arnautov,
A. V. Mikhal\"ev | 2 | open | V. I. Arnautov,
A. V. Mikhal\"ev | 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 |
700,626 | KOU-10.9 | Kourovka Notebook Problem 10.9 | Let $p$ be a prime
number and let $L_p$ denote the set of all quasivarieties each of
which is generated by a finite
$
p
$-group. Is $ L_p$ a sublattice of the lattice of all
quasivarieties of groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.9.
Proposed by: A. I. Budkin
Published resolution:
No, not always (S. A. Shakhova, *Math. Notes*, **53**,
no. 3 (1993), 345--347). | 2 | solved | A. I. Budkin | 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 (S. A. Shakhova, *Math. Notes*, **53**,\nno. 3 (1993), 345--347)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,627 | KOU-10.10 | Kourovka Notebook Problem 10.10 | Is it true that a quasivariety generated by a
finitely generated torsion-free soluble group and containing a
non-abelian free metabelian group, can be defined in the class of
torsion-free groups by an independent system of quasiidentities? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.10.
Proposed by: A. I. Budkin | 2 | open | A. I. Budkin | 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 |
700,628 | KOU-10.11 | Kourovka Notebook Problem 10.11 | Is it true that every finitely presented group
contains either a free subsemigroup on two generators or a
nilpotent subgroup of finite index? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.11.
Proposed by: R. I. Grigorchuk | 2 | open | R. I. Grigorchuk | 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 |
700,629 | KOU-10.12 | Kourovka Notebook Problem 10.12 | Does there exist a finitely generated semigroup $S$
with cancellation having non-exponential growth and such that its
group of left quotients $G=S^{-1}S$ (which exists) is a group of
exponential growth? An affirmative answer would give a positive
solution to Problem 12 in (S. Wagon, *The Banach--Tarski
Paradox*, Cambri... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.12.
Proposed by: R. I. Grigorchuk | 2 | open | R. I. Grigorchuk | 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 |
700,630 | KOU-10.13 | Kourovka Notebook Problem 10.13 | Does there exist a *massive set* of
independent elements in a free group $F_2$ on free generators $a,
b$, that is, a set $ E$ of irreducible words on the alphabet $a,
b, a^{-1}, b^{-1}$ such that
1) no element $w \in E$ belongs to the
normal closure of
$E\setminus \{ w\}$, and
2) the massiveness condition is satisfie... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.13.
Proposed by: R. I. Grigorchuk
Discussion and literature:
[Solved] No, there is no such set (L. Koch-Hyde, E. Olive,
*Preprint*, 2026, https://arxiv.org/p... | 2 | solved | R. I. Grigorchuk | 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] No, there is no such set (L. Koch-Hyde, E. Olive,\n*Preprint*, 2026, https://arxiv.org/pdf/2609.00382)."... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,631 | KOU-10.14 | Kourovka Notebook Problem 10.14 | a) Does every group satisfying the
minimum condition on subgroups satisfy the weak maximum condition
on subgroups?
b)
Does every group satisfying the maximum condition on subgroups
satisfy the weak minimum condition on subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.14.
Proposed by: D. I. Zaitsev
Published resolution:
a) No, not always (V. N. Obraztsov, *Math. USSR--Sb.*, **66**, no. 2 (1990), 541--553).
b) No, not always (V. N. Obraztsov... | 2 | solved | D. I. Zaitsev | 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) No, not always (V. N. Obraztsov, *Math. USSR--Sb.*, **66**, no. 2 (1990), 541--553).\n\nb) No, not always (V. N. Obraztsov, *Siberian Math. J.*,\n**32**, no. ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,632 | KOU-10.15 | Kourovka Notebook Problem 10.15 | For every (known) finite quasisimple group and
every prime $p$, find the faithful $ p
$-modular
absolutely irreducible linear representations of minimal degree. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.15.
Proposed by: A. S. Kondratiev | 2 | open | A. S. Kondratiev | 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 |
700,633 | KOU-10.16 | Kourovka Notebook Problem 10.16 | A class of groups is called a *direct
variety* if it is closed under taking subgroups,
factor-groups, and direct products (Yu. M. Gorchakov, *Groups with Finite Classes of Conjugate Elements*, Moscow, Nauka,
1978 (Russian)). It is obvious that the class of $ FC $-groups is
a direct variety. P. Hall (*J. London Math. So... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.16.
Proposed by: L. A. Kurdachenko | 2 | open | L. A. Kurdachenko | 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 |
700,634 | KOU-10.17 | Kourovka Notebook Problem 10.17 | (M. J. Tomkinson). Let $G$ be an $FC $-group
whose derived subgroup is embeddable in a direct product of finite
groups. Must $G/Z(G)$ be embeddable in a direct product of finite
groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.17.
Proposed by: L. A. Kurdachenko | 2 | open | L. A. Kurdachenko | 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 |
700,635 | KOU-10.18 | Kourovka Notebook Problem 10.18 | (M. J. Tomkinson). Let $G$ be an $FC $-group
which is residually in the class of groups with finite derived
subgroups. Must $G/Z(G)$ be embeddable in a direct product of
finite groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.18.
Proposed by: L. A. Kurdachenko | 2 | open | L. A. Kurdachenko | 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 |
700,636 | KOU-10.19 | Kourovka Notebook Problem 10.19 | Characterize the radical associative rings such
that the set of all normal subgroups of the adjoint group
coincides with the set of all ideals of the associated Lie ring. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.19.
Proposed by: V. M. Levchuk | 2 | open | V. M. Levchuk | 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 |
700,637 | KOU-10.20 | Kourovka Notebook Problem 10.20 | In a Chevalley group of rank $\leq 6$ over a
finite field of order $\leq 9$, describe all subgroups of the form
$H=\left< H \cap U, H \cap V\right>$ that are not contained in
any proper parabolic subgroup, where $U$ and $V$ are opposite
unipotent subgroups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.20.
Proposed by: V. M. Levchuk | 2 | open | V. M. Levchuk | 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 |
700,638 | KOU-10.23 | Kourovka Notebook Problem 10.23 | Is it true that extraction of roots in braid
groups is unique up to conjugation? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.23.
Proposed by: G. S. Makanin
Published resolution:
Yes, it is true (J. Gonz\'alez-Meneses, *Algebr. Geom. Topology*, **3** (2003), 1103--1118). | 2 | solved | G. S. Makanin | 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 true (J. Gonz\\'alez-Meneses, *Algebr. Geom. Topology*, **3** (2003), 1103--1118)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,639 | KOU-10.24 | Kourovka Notebook Problem 10.24 | A braid is said
to be *coloured* if its strings represent the identity
permutation. Is it true that links obtained by closing coloured
braids are equivalent if and only if the original braids are
conjugate in the braid group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.24.
Proposed by: G. S. Makanin
Published resolution:
No. For non-oriented links: the braid words $\sigma _1^2$ and
$\sigma _1^{-2}$ both represent the Hopf link but are not
con... | 2 | solved | G. S. Makanin | 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 non-oriented links: the braid words $\\sigma _1^2$ and\n$\\sigma _1^{-2}$ both represent the Hopf link but are not\nconjugate in the braid group (easy to... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,640 | KOU-10.25 | Kourovka Notebook Problem 10.25 | (Well-known problem). Does there exist an
algorithm which decides for a given automorphism of a free group
whether this automorphism has a non-trivial fixed point? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.25.
Proposed by: G. S. Makanin
Published resolution:
Yes, it does (O. Bogopolski, O. Maslakova,
*Int. J. Algebra Comput.*, **26**, no. 1 (2016), 29--67). | 2 | solved | G. S. Makanin | 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 does (O. Bogopolski, O. Maslakova,\n*Int. J. Algebra Comput.*, **26**, no. 1 (2016), 29--67)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,641 | KOU-10.26 | Kourovka Notebook Problem 10.26 | b) Does there exist an
algorithm which
decides whether the equation of the form
$w(x_{i_1}^{\varphi
_1},\ldots ,x_{i_
n}^{\varphi _n})=1$ is soluble in a free group
where $\varphi _1,\ldots ,\varphi _n$ are automorphisms of this
group?
[Archived solved part(s) of 10.26:]
a)
Does there exist an algorithm which decides,... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 10.26.
Proposed by: G. S. Makanin
Archive of Solved Problems entry:
a) Yes, it does
(O. Bogopolski, A. Martino, O. Maslakova, E. Ventura, *Bull. London Math.... | 2 | partially_solved | G. S. Makanin | 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": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (a) Yes, it does\n(O. Bogopolski, A. Martino, O. Maslakova, E. Ventura, *Bull. London Math. Soc.*, **38**, no. 5 (2006), 787--794); further generalizations and ap... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,642 | KOU-10.27 | Kourovka Notebook Problem 10.27 | a) Let $t$ be an involution of a finite group
$G$ and suppose that the set $D=t^G \cup \{ t^xt^y\mid x,y \in G,
\; \; |t^xt^y|=2\} $ does not intersect $O_2(G)$. Prove that if $t
\in O_2(C(d))$ for any involution $d$ from $C_D(t)$, then $ D=t^G$
(and in this case the structure of the group $\left< D\right> $ is
known).... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.27.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 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 |
700,643 | KOU-10.28 | Kourovka Notebook Problem 10.28 | Is it true that finite strongly regular graphs
with $\lambda =1$ have rank 3? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.28.
Proposed by: A. A. Makhn\"ev
Published resolution:
No, not always
(A. Cossidente, T. Penttila, *J. London Math. Soc.*, **72** (2005), 731--741). | 2 | solved | A. A. Makhn\"ev | 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\n(A. Cossidente, T. Penttila, *J. London Math. Soc.*, **72** (2005), 731--741)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,644 | KOU-10.29 | Kourovka Notebook Problem 10.29 | Describe the finite groups which contain a set
of involutions $D$ such that, for any subset $D_0$ of $ D$
generating a 2-subgroup, the normalizer $N_D(D_0)$ also
generates a 2-subgroup. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.29.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 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 |
700,645 | KOU-10.30 | Kourovka Notebook Problem 10.30 | Does there exist a non-right-orderable group
which is residually finite $p
$-group for a finite set
of prime numbers $p$ containing at least two different primes? If
a group is residually finite $p
$-group for an infinite
set of primes $p$, then it admits a linear order
(A. H. Rhemtulla, *Proc. Amer. Math. Soc.*, **41*... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.30.
Proposed by: N. Ya. Medvedev
Published resolution:
Yes, there does (P. A. Linnell, *J. Algebra*, **248** (2002)
605--607). | 2 | solved | N. Ya. Medvedev | 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, there does (P. A. Linnell, *J. Algebra*, **248** (2002)\n605--607)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,646 | KOU-10.31 | Kourovka Notebook Problem 10.31 | Suppose that $G$ is an algebraic group, $H$ is a
closed normal subgroup of $G$ and $f\! : G \rightarrow G/H$ is the
canonical homomorphism. What conditions ensure that there exists a
rational section $s\! : G/H \rightarrow G$ such that $ sf=1$? For
partial results, see (Yu. I. Merzlyakov, *Rational Groups*,
Nauka, Mosc... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.31.
Proposed by: Yu. I. Merzlyakov | 2 | open | Yu. I. Merzlyakov | 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 |
700,647 | KOU-10.32 | Kourovka Notebook Problem 10.32 | A group word is said to be *universal on a
group* $G$ if its values on $G$ run over the whole of $G$. For
an arbitrary constant $c > 8/5$, J. L. Brenner, R. J. Evans,
and D. M. Silberger (*Proc. Amer. Math. Soc.*, **96**,
no. 1 (1986), 23--28) proved that there exists a number
$N_0=N_0(c)$ such that the word $x^ry^s$, ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.32.
Proposed by: Yu. I. Merzlyakov
Discussion and literature:
[Unverified AI claim] A logarithmic bound is found (unverified AI solution: M. Kinyon, J. Urban... | 2 | open | Yu. I. Merzlyakov | 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 logarithmic bound is found (unverified AI solution: M. Kinyon, J. Urban, *Prepr... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,648 | KOU-10.33 | Kourovka Notebook Problem 10.33 | For $HNN
$-extensions of the form $ G=\left< t, A\mid t^{-1}Bt=C, \;
\varphi \right> $, where $A$ is a finitely generated abelian group
and $\varphi \! : B \rightarrow C$ is an isomorphism of two of its
subgroups, find
a) a criterion to be residually finite;
b) a criterion to be Hopfian. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.33.
Proposed by: Yu. I. Merzlyakov
Published resolution:
a) Such a criterion is found (S. Andreadakis, E. Raptis,
D. Varsos, *Arch. Math.*, **50** (1988), 495--501.
b) Such a ... | 2 | solved | Yu. I. Merzlyakov | 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) Such a criterion is found (S. Andreadakis, E. Raptis,\nD. Varsos, *Arch. Math.*, **50** (1988), 495--501.\n\nb) Such a criterion is found (S. Andreadakis, E. ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,649 | KOU-10.34 | Kourovka Notebook Problem 10.34 | Does there exist a
non-soluble finite group which coincides with the product of any two
of its non-conjugate maximal subgroups?
*Editors' comment:*
the answer is negative for almost simple groups
(T. V. Tikhonenko, V. N. Tyutyanov, *Siberian Math. J.*, **51**,
no. 1 (2010), 174--177). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.34.
Proposed by: V. S. Monakhov
Discussion and literature:
[Solved] No, there are no such groups (R. Sater, *Preprint of 6 August 2026*, https://arxiv.org/ab... | 2 | solved | V. S. Monakhov | 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] No, there are no such groups (R. Sater, *Preprint of 6 August 2026*, https://arxiv.org/abs/2608.02970;\n... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,650 | KOU-10.35 | Kourovka Notebook Problem 10.35 | Is it true that every finitely generated
torsion-free subgroup of $GL_n({\mathbb{C})$ is residually in the
class of torsion-free subgroups of $GL_n(\overline{{\mathbb{Q}})$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.35.
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 |
700,651 | KOU-10.36 | Kourovka Notebook Problem 10.36 | Is it true that $SL_n({\mathbb{Z}[x_1,\ldots
,x_r])$, for $n$ sufficiently large, is a group of type $(FP)_m$
(which is defined in the same way as $(FP)_{\infty}$ was defined
in 6.3, but with that weakening that the condition of
being finitely generated is not imposed on the terms of the
resolution with numbers $ > m$)... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.36.
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 |
700,652 | KOU-10.37 | Kourovka Notebook Problem 10.37 | Suppose that $G$ is a finitely generated metabelian group all of
whose integral homology groups are finitely generated. Is it true
that $G$ is a group of finite rank? The answer is affirmative if
$G$ splits over the derived subgroup (J. R. J. Groves, *Quart. J. Math.*, **33**, no. 132 (1982), 405--420). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 10th issue (1986). Original problem number: 10.37.
Proposed by: G. A. Noskov
Published resolution:
Yes, it is (D. H. Kochloukova, *Groups St. Andrews 2001 in
Oxford*, Vol. II, Cambridge Univ. Press, 2003, 332--343). | 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: Yes, it is (D. H. Kochloukova, *Groups St. Andrews 2001 in\nOxford*, Vol. II, Cambridge Univ. Press, 2003, 332--343)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,653 | KOU-10.38 | Kourovka Notebook Problem 10.38 | (W. van der Kallen). Does $E_{n+3}({\mathbb{Z}[x_1,\ldots ,x_n])$, $n \geq 1$, have finite breadth with
respect to the set of transvections? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 10th issue (1986), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 10.38.
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 |
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