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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
700,154 | KOU-2.84 | Kourovka Notebook Problem 2.84 | Suppose that a locally finite group $G$ is a
product of two locally nilpotent subgroups. Is $G$ necessarily
locally soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 2nd issue (1967), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 2.84.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,967 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,155 | KOU-2.88 | Kourovka Notebook Problem 2.88 | Is every Hall $\pi$-subgroup of an arbitrary group a maximal
$\pi$-subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 2nd issue (1967). Original problem number: 2.88.
Proposed by: M. I. \`Eidinov
Published resolution:
Not always (R. Baer, *J. Reine Angew. Math.*, **239/240** (1969),
109--144). | 2 | solved | M. I. \`Eidinov | 1,967 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 always (R. Baer, *J. Reine Angew. Math.*, **239/240** (1969),\n109--144)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,156 | KOU-3.2 | Kourovka Notebook Problem 3.2 | Classify the faithful irreducible
(infinite-dimensional) representations of the nilpotent group
defined by generators $a,b,c$ and relations $[a,b]=c$, $ac=ca$,
$bc=cb$. (A condition for a representation to be monomial is given
in (A. E. Zalesskiĭ, *Math. Notes*, **9** (1971),
117--123).) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.2.
Proposed by: S. D. Berman, A. E. Zalesski\u{\i
Published resolution:
Classification of these representations up to equivalence does not seem to be feasible. D. Segal (*Math. P... | 2 | solved | S. D. Berman, A. E. Zalesski\u{\i | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: Classification of these representations up to equivalence does not seem to be feasible. D. Segal (*Math. Proc. Cambridge Phil. Soc.*, **81** (1977), 201--208) pr... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,157 | KOU-3.3 | Kourovka Notebook Problem 3.3 | (Well-known problem). Describe the automorphism
group of the free associative algebra with $n$ generators, $n \geq
2$.
It is known
that all automorphisms are tame for $n=2$ (A. J. Czerniakiewicz, *Trans. Amer. Math. Soc.*, **160** (1971), 393--401; **171** (1972), 309--315; \
L. G. Makar-Limanov, *Funct. Anal. Appl.*,... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.3.
Proposed by: L. A. Bokut' | 2 | open | L. A. Bokut' | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,158 | KOU-3.4 | Kourovka Notebook Problem 3.4 | (Well-known problems). a) Is there an algorithm that decides,
for any set of group words $f_1,\ldots ,f_m$ (in a fixed set of
variables $ x_1,x_2,\ldots $) and a separate word $f$, whether
$f=1$ is a consequence of $ f_1=1,\ldots , f_m=1$?
b) Given words $f_1,\ldots ,f_m$, is there an algorithm that
decides, for any w... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.4.
Proposed by: L. A. Bokut'
Published resolution:
No, in both cases (Yu. G. Kleiman, *Soviet Math. Dokl.*,
**20** (1979), 115--119). | 2 | solved | L. A. Bokut' | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, in both cases (Yu. G. Kleiman, *Soviet Math. Dokl.*,\n**20** (1979), 115--119)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,159 | KOU-3.5 | Kourovka Notebook Problem 3.5 | Can a subgroup of a relatively free group be radicable? In particular, can a verbal subgroup of a relatively free group be radicable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.5.
Proposed by: N. R. Brumberg | 2 | open | N. R. Brumberg | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,160 | KOU-3.6 | Kourovka Notebook Problem 3.6 | Describe the insoluble finite groups in which every soluble
subgroup is either $2$-closed, $2'$-closed, or isomorphic to $\mathbb{S}
_4$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.6.
Proposed by: V. A. Vedernikov
Published resolution:
This follows from (D. Gorenstein, J. H. Walter, *Illinois
J. Math.*, **6** (1962) 553--593; \ V. D. Mazurov, *Soviet Math. ... | 2 | solved | V. A. Vedernikov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 follows from (D. Gorenstein, J. H. Walter, *Illinois\nJ. Math.*, **6** (1962) 553--593; \\ V. D. Mazurov, *Soviet Math. Dokl.*, **7** (1966), 681--682; \\\n... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,161 | KOU-3.7 | Kourovka Notebook Problem 3.7 | An automorphism $\sigma$ of a group $G$ is called *algebraic* if, for
any $g\in G$, the minimal $\sigma$-invariant subgroup of $G$
containing $g$ is finitely generated. An automorphism $\sigma$ of
$G$ is called an *$e$-automorphism* if, for any
$\sigma$-invariant subgroups $A$ and $B$, where $A$ is a proper
subgroup of... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.7.
Proposed by: V. G. Vilyatser
Published resolution:
Yes, it is (A. V. Yagzhev, *Algebra and Logic*, **28**
(1989), 83--85). | 2 | solved | V. G. Vilyatser | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (A. V. Yagzhev, *Algebra and Logic*, **28**\n(1989), 83--85)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,162 | KOU-3.8 | Kourovka Notebook Problem 3.8 | Let $G$ be the free product of free groups $A$ and $B$, and $V$
the verbal subgroup of $G$ corresponding to the equation $ x^4 =
1$. Is it true that, if $ a\in A\setminus V$ and $ b\in B\setminus
V$, then $ (ab)^2\notin V$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.8.
Proposed by: V. G. Vilyatser
Published resolution:
No, it is not. If $A$ and $B$ are free groups with bases $a_1,a_2$
and $b_1,b_2$, respectively, then, for example, the commu... | 2 | solved | V. G. Vilyatser | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it is not. If $A$ and $B$ are free groups with bases $a_1,a_2$\nand $b_1,b_2$, respectively, then, for example, the commutators\n$a=[[a_1,a_2,a_2],[a_1,a_2]]... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,163 | KOU-3.9 | Kourovka Notebook Problem 3.9 | Does there exist an infinite periodic group
with a finite maximal subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.9.
Proposed by: Yu. M. Gorchakov
Published resolution:
Yes, there does (S. I. Adian, *Proc. Steklov Inst. Math.*,
**112** (1971), 61--69). | 2 | solved | Yu. M. Gorchakov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (S. I. Adian, *Proc. Steklov Inst. Math.*,\n**112** (1971), 61--69)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,164 | KOU-3.10 | Kourovka Notebook Problem 3.10 | Need the number of finite non-abelian simple groups contained
in a proper variety of groups be finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.10.
Proposed by: Yu. M. Gorchakov
Published resolution:
Yes, mod CFSG; see, for example, (G. A. Jones, *J. Austral.
Math. Soc.*, **17** (1974), 162--173). | 2 | solved | Yu. M. Gorchakov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, mod CFSG; see, for example, (G. A. Jones, *J. Austral.\nMath. Soc.*, **17** (1974), 162--173)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,165 | KOU-3.11 | Kourovka Notebook Problem 3.11 | An element $g$ of a group $G$ is said to be *generalized
periodic* if there exist $ x_1,\ldots ,x_n\in G$ such that $
x_1^{-1}gx_1\cdots x_n^{-1}gx_n= 1$. Does there exist a finitely
generated torsion-free group all of whose elements are
generalized periodic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.11.
Proposed by: Yu. M. Gorchakov
Published resolution:
Yes, there does (A. P. Goryushkin, *Siberian Math. J.*, **14** (1973), 146--148). Another example is $ G=\left< a,b\mid
(b... | 2 | solved | Yu. M. Gorchakov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (A. P. Goryushkin, *Siberian Math. J.*, **14** (1973), 146--148). Another example is $ G=\\left< a,b\\mid\n(b^2)^{a}=b^{-2},\\;\\; (a^2)^{b}=a^{-... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,166 | KOU-3.12 | Kourovka Notebook Problem 3.12 | (Well-known problem). Is a locally finite group
with a full Sylow basis locally soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.12.
Proposed by: Yu. M. Gorchakov
Discussion and literature:
[Solved] No, not always (AI solution confirmed by A. A. Buturlakin: \ Achyuth Jayadevan, *Preprint... | 2 | solved | Yu. M. Gorchakov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 A. A. Buturlakin: \\ 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 |
700,167 | KOU-3.13 | Kourovka Notebook Problem 3.13 | Is the elementary theory of subgroup lattices of finite abelian
groups decidable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.13.
Proposed by: Yu. L. Ershov, A. I. Kokorin
Published resolution:
No (G. T. Kozlov, *Algebra and Logic*, **9** (1970),
104--107). | 2 | solved | Yu. L. Ershov, A. I. Kokorin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (G. T. Kozlov, *Algebra and Logic*, **9** (1970),\n104--107)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,168 | KOU-3.14 | Kourovka Notebook Problem 3.14 | Let $G$ be a finite group of $ n\times n$ matrices over a
skew field $T$ of characteristic zero. Prove that $G$ has a
soluble normal subgroup $H$ whose index in $G$ is not greater
than some number $f(n)$ not depending on $G$ or $T$. This
problem is related to the representation theory of finite groups
(the Schur index)... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.14.
Proposed by: A. E. Zalesski\u{\i
Published resolution:
This is proved mod CFSG (B. Hartley, M. A. Shahabi Shojaei, *Math. Proc. Cambridge Phil. Soc.*, **92** (1982), 55--64). | 2 | solved | A. E. Zalesski\u{\i | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This is proved mod CFSG (B. Hartley, M. A. Shahabi Shojaei, *Math. Proc. Cambridge Phil. Soc.*, **92** (1982), 55--64)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,169 | KOU-3.15 | Kourovka Notebook Problem 3.15 | A group $G$ is said to be *$U$-embeddable* in a class $
{\mathfrak{K}}$ of groups if, for any finite submodel $M\subset G$,
there is a group $ A\in {\mathfrak{K}}$ such that $M$ is isomorphic
to some submodel of $A$. Are the following groups $U$-embeddable
in the class of finite groups:
a) every group with one definin... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.15.
Proposed by: M. I. Kargapolov
Published resolution:
No, in both cases (A. I. Budkin, V. A. Gorbunov, *Algebra
and Logic*, **14** (1975), 73--84).
Another example. The group $... | 2 | solved | M. I. Kargapolov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, in both cases (A. I. Budkin, V. A. Gorbunov, *Algebra\nand Logic*, **14** (1975), 73--84).\nAnother example. The group $ G=\\left< a,b\\mid\n(b^2)^a=b^3\\rig... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,170 | KOU-3.16 | Kourovka Notebook Problem 3.16 | The word problem for a group admitting a single
defining relation in the variety of soluble groups of derived
length $n$, $n \geq 3$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.16.
Proposed by: M. I. Kargapolov | 2 | open | M. I. Kargapolov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,171 | KOU-3.17 | Kourovka Notebook Problem 3.17 | a) Is a non-abelian group with a unique linear ordering
necessarily simple?
b) Is a non-commutative orderable group simple if it
has no non-trivial normal relatively convex subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.17.
Proposed by: A. I. Kokorin
Published resolution:
Not always, in both cases (V. V. Bludov, *Algebra and
Logic*, **13** (1974), 343--360). | 2 | solved | A. I. Kokorin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 always, in both cases (V. V. Bludov, *Algebra and\nLogic*, **13** (1974), 343--360)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,172 | KOU-3.18 | Kourovka Notebook Problem 3.18 | (B. H. Neumann). A group $U$ is called *universal* for a
class ${\mathfrak{K}}$ of groups if $U$ contains an isomorphic image
of every member of ${\mathfrak{K}}$. Does there exist a countable
group that is universal for the class of countable orderable
groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.18.
Proposed by: A. I. Kokorin
Published resolution:
No (M. I. Kargapolov, *Algebra and Logic*, **9** (1970),
257--261; \ D. B. Smith, *Pacific J. Math.*, **35**
(1970), 499--502... | 2 | solved | A. I. Kokorin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (M. I. Kargapolov, *Algebra and Logic*, **9** (1970),\n257--261; \\ D. B. Smith, *Pacific J. Math.*, **35**\n(1970), 499--502)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,173 | KOU-3.19 | Kourovka Notebook Problem 3.19 | (A. I. Mal'cev). For an arbitrary linearly ordered group $G$,
does there exist a linearly ordered abelian group with the same
order-type as $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.19.
Proposed by: A. I. Kokorin
Published resolution:
No (W. C. Holland, A. H. Mekler, S. Shelah, *Order*, **1**
(1985), 383--397). | 2 | solved | A. I. Kokorin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (W. C. Holland, A. H. Mekler, S. Shelah, *Order*, **1**\n(1985), 383--397)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,174 | KOU-3.20 | Kourovka Notebook Problem 3.20 | Does the class of orderable groups coincide with
the smallest axiomatizable class containing pro-orderable groups?
(See 1.35.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.20.
Proposed by: A. I. Kokorin
Discussion and literature:
[Unverified AI claim] No, it does not (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026, h... | 2 | open | A. I. Kokorin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, it does not (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026, htt... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,175 | KOU-3.21 | Kourovka Notebook Problem 3.21 | Let ${\mathfrak{K}}$ be the class of one-based models of signature
$\sigma$, \ $\vartheta$ a property that makes sense for models in
${\mathfrak{K}}$, and ${\mathfrak{K}}_0$ the class of two-based models whose
first base is a set $M$ taken from ${\mathfrak{K}}$ and whose second
base consists of all submodels of $M$ wit... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.21.
Proposed by: A. I. Kokorin
Published resolution:
No, in all cases (for a), c): G. T. Kozlov, *Algebra and
Logic*, **9** (1970), 104--107, \ *Algebra*, no. 1, Irkutsk
Univ., 1... | 2 | solved | A. I. Kokorin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, in all cases (for a), c): G. T. Kozlov, *Algebra and\nLogic*, **9** (1970), 104--107, \\ *Algebra*, no. 1, Irkutsk\nUniv., 1972, 21--23 (Russian); \\ for b):... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,176 | KOU-3.22 | Kourovka Notebook Problem 3.22 | Let $\xi = \{ G_{\alpha}, \;\pi_{\beta}^{\alpha}\mid \alpha
,\beta \in I\}$ be a projective system (over a directed set $I$)
of finitely generated free abelian groups. If all the
projections $
\pi_{\beta}^{\alpha}$
are epimorphisms and all the $G_{\alpha}$ are non-zero, does it
follow that $ \lim\limits _{\longleftarro... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.22.
Proposed by: V. I. Kuz'minov
Published resolution:
[Solved] Under the assumption of the continuum hypothesis, not always
(E. A. Palyutin, *Siberian Math. J.*, **19** (1978),
... | 2 | solved | V. I. Kuz'minov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: [Solved] Under the assumption of the continuum hypothesis, not always\n(E. A. Palyutin, *Siberian Math. J.*, **19** (1978),\n1415--1417). Another example,\nnot u... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,177 | KOU-3.26 | Kourovka Notebook Problem 3.26 | (F. Gross). Is it true that finite groups of exponent
$p^{\alpha}q^{\beta}$ have nilpotent length $\leq\alpha
+\beta$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.26.
Proposed by: V. D. Mazurov
Published resolution:
No, not always (E. I. Khukhro, *Algebra and Logic*, **17** (1978), 473--482). | 2 | solved | V. D. Mazurov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always (E. I. Khukhro, *Algebra and Logic*, **17** (1978), 473--482)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,178 | KOU-3.27 | Kourovka Notebook Problem 3.27 | (J. G. Thompson). Is every finite simple group with a nilpotent
maximal subgroup isomorphic to some $PSL_2(q)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.27.
Proposed by: V. D. Mazurov
Published resolution:
Yes, it is (B. Baumann, *J. Algebra*, **38** (1976),
119--135). | 2 | solved | V. D. Mazurov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (B. Baumann, *J. Algebra*, **38** (1976),\n119--135)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,179 | KOU-3.28 | Kourovka Notebook Problem 3.28 | If $G$ is a finite 2-group with cyclic centre and every abelian
normal subgroup 2-generated, then is every abelian subgroup
of $G$ \ $3$-generated? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.28.
Proposed by: V. D. Mazurov
Published resolution:
Not always (Ya. G. Berkovich, *Algebra and Logic*, **9**
(1970), 75). | 2 | solved | V. D. Mazurov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 always (Ya. G. Berkovich, *Algebra and Logic*, **9**\n(1970), 75)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,180 | KOU-3.29 | Kourovka Notebook Problem 3.29 | Under what conditions can a wreath product of matrix groups
over a field be represented by matrices over a field? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.29.
Proposed by: Yu. I. Merzlyakov
Published resolution:
Conditions have been found (Yu. E. Vapne, *Soviet Math.
Dokl.*, **11** (1970), 1396--1399). | 2 | solved | Yu. I. Merzlyakov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: Conditions have been found (Yu. E. Vapne, *Soviet Math.\nDokl.*, **11** (1970), 1396--1399)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,181 | KOU-3.30 | Kourovka Notebook Problem 3.30 | A torsion-free abelian group is called *factor-decomposable* if, in all its factor groups, the
periodic part is a direct summand. Characterize these groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.30.
Proposed by: A. P. Mishina
Published resolution:
This is done (L. Bican, *Commentat. Math. Univ. Carolinae*, **19** (1978), 653--672). | 2 | solved | A. P. Mishina | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 done (L. Bican, *Commentat. Math. Univ. Carolinae*, **19** (1978), 653--672)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,182 | KOU-3.31 | Kourovka Notebook Problem 3.31 | Find necessary and sufficient conditions under which every pure
subgroup of a completely decomposable torsion-free abelian group
is itself completely decomposable. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.31.
Proposed by: A. P. Mishina
Published resolution:
These have been found (L. Bican, *Czech. Math. J.*, **24**
(1974), 176--191; \ A. A. Kravchenko, *Vestnik Moskov.
Univ. Ser. ... | 2 | solved | A. P. Mishina | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: These have been found (L. Bican, *Czech. Math. J.*, **24**\n(1974), 176--191; \\ A. A. Kravchenko, *Vestnik Moskov.\nUniv. Ser. 1 Mat. Mekh.*, **1980**, no. 3, p... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,183 | KOU-3.33 | Kourovka Notebook Problem 3.33 | Are two groups
necessarily isomorphic if each of them can be defined by a single
relation and is a homomorphic image of the other one? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.33.
Proposed by: D. I. Moldavanski\u{\i
Published resolution:
No, not necessarily
(A. V. Borshchev, D. I. Moldavanskiĭ, *Math. Notes*,
**79**, no. 1 (2006), 31--40). | 2 | solved | D. I. Moldavanski\u{\i | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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\n(A. V. Borshchev, D. I. Moldavanskiĭ, *Math. Notes*,\n**79**, no. 1 (2006), 31--40)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,184 | KOU-3.34 | Kourovka Notebook Problem 3.34 | (Well-known problem). The conjugacy problem for
groups with a single defining relation. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.34.
Proposed by: D. I. Moldavanski\u{\i | 2 | open | D. I. Moldavanski\u{\i | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,185 | KOU-3.35 | Kourovka Notebook Problem 3.35 | (K. Ross). Suppose that a group $G$ admits two topologies $\sigma$
and $\tau$ yielding locally compact topological groups $G_{\sigma}$ and $
G_{\tau}$. If the sets of closed subgroups in $G_{\sigma}$ and $G_{\tau}$
are the same, does it follow that $G_{\sigma}$ and $G_{\tau}$ are
topologically isomorphic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.35.
Proposed by: Yu. N. Mukhin
Published resolution:
Not always (A. I. Moskalenko, *Ukrain. Math. J.*, **30** (1978), 199--201). | 2 | solved | Yu. N. Mukhin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 always (A. I. Moskalenko, *Ukrain. Math. J.*, **30** (1978), 199--201)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,186 | KOU-3.37 | Kourovka Notebook Problem 3.37 | Suppose that every finitely generated subgroup of a locally
compact group $G$ is pronilpotent. Then is it true that every
maximal closed subgroup of $G$ contains the derived subgroup
$G'$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.37.
Proposed by: Yu. N. Mukhin
Published resolution:
Yes, it is (I. V. Protasov, *Soviet Math. Dokl.*, **19**
(1978), 1208--1210). | 2 | solved | Yu. N. Mukhin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (I. V. Protasov, *Soviet Math. Dokl.*, **19**\n(1978), 1208--1210)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,187 | KOU-3.38 | Kourovka Notebook Problem 3.38 | Describe the topological groups which have no
proper closed subgroups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.38.
Proposed by: Yu. N. Mukhin | 2 | open | Yu. N. Mukhin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,188 | KOU-3.39 | Kourovka Notebook Problem 3.39 | Describe the finite groups with a
self-centralizing subgroup of prime order. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.39.
Proposed by: V. T. Nagrebetski\u{\i
Published resolution:
They are described. Self-centralizing subgroups of prime order are
CC-subgroups. Finite groups with a CC-subgroup we... | 2 | solved | V. T. Nagrebetski\u{\i | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. Self-centralizing subgroups of prime order are\nCC-subgroups. Finite groups with a CC-subgroup were fully classified\nin (Z. Arad, W. Herfort... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,189 | KOU-3.40 | Kourovka Notebook Problem 3.40 | (I. R. Shafarevich). Let $SL_2(\mathbb{Z} ) \hat{}$ and $SL_2(\mathbb{Z} )^{-}$
denote the completions of $SL_2(\mathbb{Z} )$ determined by all subgroups
of finite index and all congruence subgroups, respectively, and
let $\psi : SL_2(\mathbb{Z} ) \hat{}\rightarrow SL_2(\mathbb{Z} )^{-}$ be the
natural homomorphism. Is... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.40.
Proposed by: V. P. Platonov
Published resolution:
Yes, it is (O. V. Mel'nikov, *Soviet Math. Dokl.*, **17** (1976), 867--870). | 2 | solved | V. P. Platonov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (O. V. Mel'nikov, *Soviet Math. Dokl.*, **17** (1976), 867--870)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,190 | KOU-3.41 | Kourovka Notebook Problem 3.41 | Is every compact periodic group locally finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.41.
Proposed by: V. P. Platonov
Published resolution:
Yes, it is (E. I. Zel'manov, *Israel J. Math.*, **77**
(1992), 83--95). | 2 | solved | V. P. Platonov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (E. I. Zel'manov, *Israel J. Math.*, **77**\n(1992), 83--95)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,191 | KOU-3.42 | Kourovka Notebook Problem 3.42 | (Kneser--Tits conjecture). Let $G$ be a simply connected $k$-defined simple algebraic group, and $E_k(G)$ the subgroup generated by unipotent $k$-elements. If $E_k(G)\ne 1$, then $G_k=E_k(G)$. The proof is known for $k$-decomposable groups (C. Chevalley) and for local fields (V. P. Platonov). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.42.
Proposed by: V. P. Platonov
Published resolution:
The conjecture was refuted (V. P. Platonov, *Math. USSR--Izv.*, **10**, no. 2 (1976), 211--243). | 2 | solved | V. P. Platonov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: The conjecture was refuted (V. P. Platonov, *Math. USSR--Izv.*, **10**, no. 2 (1976), 211--243)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,192 | KOU-3.43 | Kourovka Notebook Problem 3.43 | Let $\mu$ be an infinite cardinal number. A group
$G$ is said to be $\mu $-*overnilpotent* if every
cyclic subgroup of $G$ is a member of some ascending normal series
of length less than $\mu$ reaching $G$. It is not difficult to
show that the class of $\mu $-overnilpotent groups is a
radical class. Is it true that if ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.43.
Proposed by: B. I. Plotkin
Discussion and literature:
[Unverified AI claim] Yes, it is true (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026,
... | 2 | open | B. I. Plotkin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] Yes, it is true (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2026,\n\n... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,193 | KOU-3.44 | Kourovka Notebook Problem 3.44 | Suppose that a group is generated by its
ascendant soluble subgroups. Is it necessarily locally soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.44.
Proposed by: B. I. Plotkin | 2 | open | B. I. Plotkin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,194 | KOU-3.45 | Kourovka Notebook Problem 3.45 | Let ${\mathfrak{X}}$ be a hereditary radical. Is it
true that, in a locally nilpotent torsion-free group $G$, the
subgroup ${\mathfrak{X}}(G)$ is isolated? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.45.
Proposed by: B. I. Plotkin | 2 | open | B. I. Plotkin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,195 | KOU-3.46 | Kourovka Notebook Problem 3.46 | Does there exist a group having more than one,
but finitely many maximal locally soluble normal subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.46.
Proposed by: B. I. Plotkin
Discussion and literature:
[Solved] Yes, such groups exist (AI solution confirmed by M. Trombetti: \ P. Monticone, *Preprint*, 2... | 2 | solved | B. I. Plotkin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, such groups exist (AI solution confirmed by M. Trombetti: \\ P. Monticone, *Preprint*, 2026,\nhttps... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,196 | KOU-3.47 | Kourovka Notebook Problem 3.47 | (Well-known problem of A. I. Maltsev).
Is it true that every locally
nilpotent group is a homomorphic image of some
torsion-free locally nilpotent group?
*Editors' comment:* An affirmative answer is
known for periodic groups (E. M. Levich, A. I. Tokarenko,
*Siberian Math. J.*, **11** (1970), 1033--1034) and for
counta... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.47.
Proposed by: B. I. Plotkin | 2 | open | B. I. Plotkin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,197 | KOU-3.48 | Kourovka Notebook Problem 3.48 | It can be shown that hereditary radicals form a
semigroup with respect to taking the products of classes. It is an
interesting problem to find all indecomposable elements of this
semigroup. In particular, we point out the problem of finding all
indecomposable radicals contained in the class of locally finite
$p
$-group... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.48.
Proposed by: B. I. Plotkin | 2 | open | B. I. Plotkin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,198 | KOU-3.49 | Kourovka Notebook Problem 3.49 | Does the semigroup generated by all
indecomposable radicals satisfy any identity? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.49.
Proposed by: B. I. Plotkin | 2 | open | B. I. Plotkin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,199 | KOU-3.50 | Kourovka Notebook Problem 3.50 | Let $G$ be a group of order $p^{\alpha} \cdot
m$, where $p$ is a prime, $p$ and $m$ are coprime, and let $k$ be
an algebraically closed field of characteristic $p$. Is it true
that if the indecomposable projective module corresponding to the
1-representation of $G$ has $k $-dimension $p^{\alpha}$, then
$G$ has a Hall $... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.50.
Proposed by: A. I. Saksonov
Published resolution:
No, not always: see Example 4.5 in (W. Willems, *Math. Z.*, **171** (1980), 163--174). | 2 | solved | A. I. Saksonov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, not always: see Example 4.5 in (W. Willems, *Math. Z.*, **171** (1980), 163--174)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,200 | KOU-3.51 | Kourovka Notebook Problem 3.51 | Is it true that every finite group with a group of
automorphisms $\Phi$ which acts regularly on the set of conjugacy
classes of $G$ (that is, leaves only the identity class fixed) is
soluble? The answer is known to be affirmative in the case where
$\Phi$ is a cyclic group generated by a regular automorphism. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.51.
Proposed by: A. I. Saksonov
Published resolution:
No, not always (Y. Fine,
*J. Group Theory*, **22**, no. 6 (2019),
1077--1087). | 2 | solved | A. I. Saksonov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (Y. Fine,\n*J. Group Theory*, **22**, no. 6 (2019),\n1077--1087)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,201 | KOU-3.52 | Kourovka Notebook Problem 3.52 | Can the quasivariety generated by the free group of rank 2 be
defined by a system of quasi-identities in finitely many
variables? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.52.
Proposed by: D. M. Smirnov
Published resolution:
No (A. I. Budkin, *Algebra and Logic*, **15** (1976),
25--33). | 2 | solved | D. M. Smirnov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (A. I. Budkin, *Algebra and Logic*, **15** (1976),\n25--33)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,202 | KOU-3.53 | Kourovka Notebook Problem 3.53 | Let $L({\mathfrak{N}}_4)$ denote the lattice of subvarieties of the
variety ${\mathfrak{N}}_4$ of nilpotent groups of class at most $4$. Is
$L({\mathfrak{N}}_4)$ distributive? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.53.
Proposed by: D. M. Smirnov
Published resolution:
No (Yu. A. Belov, *Algebra and Logic*, **9** (1970), 371--374). | 2 | solved | D. M. Smirnov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (Yu. A. Belov, *Algebra and Logic*, **9** (1970), 371--374)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,203 | KOU-3.55 | Kourovka Notebook Problem 3.55 | Is every binary soluble group all of whose abelian
subgroups have finite rank, locally soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.55.
Proposed by: S. P. Strunkov | 2 | open | S. P. Strunkov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,204 | KOU-3.56 | Kourovka Notebook Problem 3.56 | Is a 2-group with the minimum condition for abelian subgroups
locally finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.56.
Proposed by: S. P. Strunkov
Published resolution:
Yes, it is (V. P. Shunkov, *Algebra and Logic*, **9**
(1970), 291--297). | 2 | solved | S. P. Strunkov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. P. Shunkov, *Algebra and Logic*, **9**\n(1970), 291--297)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,205 | KOU-3.57 | Kourovka Notebook Problem 3.57 | Determine the laws of distribution of non-soluble
and simple group-theoretic numbers in the sequence of natural
numbers. See 2.78. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.57.
Proposed by: P. I. Trofimov | 2 | open | P. I. Trofimov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,206 | KOU-3.58 | Kourovka Notebook Problem 3.58 | Let $G$ be a compact $0$-dimensional topological group all of
whose Sylow $p $-subgroups are direct products of cyclic groups
of order $p$. Then is every normal subgroup of $G$ complementable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.58.
Proposed by: V. S. Charin
Published resolution:
Yes, it is (M. I. Kabenyuk, *Siberian Math. J.*, **13**
(1972), 654--657). | 2 | solved | V. S. Charin | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (M. I. Kabenyuk, *Siberian Math. J.*, **13**\n(1972), 654--657)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,207 | KOU-3.59 | Kourovka Notebook Problem 3.59 | Let $H$ be an insoluble minimal normal subgroup of a finite
group $G$, and suppose that $H$ has cyclic Sylow $p $-subgroups for
every prime $p$ dividing $ |G : H|$. Prove that $H$ has at least
one complement in $G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.59.
Proposed by: L. A. Shemetkov
Published resolution:
This was proved (S. A. Syskin, *Siberian Math. J.*,
**12** (1971), 342--344; \ L. A. Shemetkov, *Soviet Math.
Dokl.*, **11*... | 2 | solved | L. A. Shemetkov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 was proved (S. A. Syskin, *Siberian Math. J.*,\n**12** (1971), 342--344; \\ L. A. Shemetkov, *Soviet Math.\nDokl.*, **11** (1970), 1436--1438)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,208 | KOU-3.60 | Kourovka Notebook Problem 3.60 | The notion of the *$p
$-length*
of an arbitrary finite group was introduced in (L. A. Shemetkov,
*Math. USSR Sbornik*, **1** (1968), 83--92). Investigate the
relations between the $p
$-length of a finite group and
the invariants $c_p$, $d_p$, $e_p$ of its Sylow $p
$-subgroup. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 3.60.
Proposed by: L. A. Shemetkov | 2 | open | L. A. Shemetkov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,209 | KOU-3.61 | Kourovka Notebook Problem 3.61 | Let $\sigma$ be an automorphism of prime order
$p$ of a finite group $G$, which has a Hall $\pi $-subgroup with
cyclic Sylow subgroups. Suppose that $p \in \pi$. Does the
centralizer $C_G(\sigma )$ have at least one Hall $\pi
$-subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.61.
Proposed by: L. A. Shemetkov
Published resolution:
Yes, it does, mod CFSG (V. D. Mazurov, *Algebra and Logic*,
**31**, no. 6 (1992), 360--366). | 2 | solved | L. A. Shemetkov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, mod CFSG (V. D. Mazurov, *Algebra and Logic*,\n**31**, no. 6 (1992), 360--366)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,210 | KOU-3.62 | Kourovka Notebook Problem 3.62 | (Well-known problem). A finite group is said to
be a {\it $D_{\pi} $-group} if any two of its maximal $\pi
$-subgroups
are conjugate. Is an extension of a $D_{\pi} $-group by a $D_{\pi
} $-group always a $D_{\pi} $-group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.62.
Proposed by: L. A. Shemetkov
Published resolution:
Yes, it is mod CFSG
(E. P. Vdovin, D. O. Revin, *Contemp.
Math.*, **402** (2006), 229--263). | 2 | solved | L. A. Shemetkov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 mod CFSG\n(E. P. Vdovin, D. O. Revin, *Contemp.\nMath.*, **402** (2006), 229--263)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,211 | KOU-3.64 | Kourovka Notebook Problem 3.64 | Describe the finite simple groups with a Sylow
2-subgroup of the following type: $\langle a,t\mid a^{2^n}=t^2=1,\;\;
tat=a^{2^{n-1}-1}\rangle$, \ $n>2$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 3rd issue (1969). Original problem number: 3.64.
Proposed by: V. P. Shunkov
Published resolution:
This was done (J. L. Alperin, R. Brauer, D. Gorenstein,
*Trans. Amer. Math. Soc.*, **151** (1970), 1--261). | 2 | solved | V. P. Shunkov | 1,969 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 was done (J. L. Alperin, R. Brauer, D. Gorenstein,\n*Trans. Amer. Math. Soc.*, **151** (1970), 1--261)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,212 | KOU-4.1 | Kourovka Notebook Problem 4.1 | Find an infinite finitely generated group with
an identical relation of the form $x^{2^n}=1$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.1.
Proposed by: S. I. Adian
Published resolution:
It has been found (S. V. Ivanov, *Int. J. Algebra Comput.*,
**4**, no. 1--2 (1994), 1--308; I. G. Lys\"enok, *Izv. Math.*, **60*... | 2 | solved | S. I. Adian | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 has been found (S. V. Ivanov, *Int. J. Algebra Comput.*,\n**4**, no. 1--2 (1994), 1--308; I. G. Lys\\\"enok, *Izv. Math.*, **60**, no. 3 (1996), 453--654)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,213 | KOU-4.2 | Kourovka Notebook Problem 4.2 | a) Find an infinite finitely generated group of
exponent $< 100$.
b) Do there exist such groups of exponent 5? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.2.
Proposed by: S. I. Adian | 2 | open | S. I. Adian | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,214 | KOU-4.3 | Kourovka Notebook Problem 4.3 | Construct a finitely presented group with
insoluble word problem and satisfying a non-trivial law. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.3.
Proposed by: S. I. Adian
Published resolution:
This has been done (O. G. Kharlampovich, *Math. USSR--Izv.*,
**19** (1982), 151--169). | 2 | solved | S. I. Adian | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 done (O. G. Kharlampovich, *Math. USSR--Izv.*,\n**19** (1982), 151--169)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,215 | KOU-4.4 | Kourovka Notebook Problem 4.4 | Construct a finitely presented group with
undecidable word problem all of whose non-trivial defining
relations have the form $A^2=1$. This problem is interesting for
topologists. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.4.
Proposed by: S. I. Adian
Published resolution:
It is constructed (O. A. Sarkisyan, *The word problem for
some classes of groups and semigroups*, Candidate disser., Moscow
Univ... | 2 | solved | S. I. Adian | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 constructed (O. A. Sarkisyan, *The word problem for\nsome classes of groups and semigroups*, Candidate disser., Moscow\nUniv., 1983 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,216 | KOU-4.5 | Kourovka Notebook Problem 4.5 | b) Is it true that an arbitrary finitely
presented group
has either polynomial or exponential growth?
[Archived solved part(s) of 4.5:]
a) (J. Milnor). Is it true that an arbitrary finitely generated
group has either polynomial or exponential growth?
c) Is it true that every finitely generated group with
undecidable
... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 4.5.
Proposed by: S. I. Adian
Archive of Solved Problems entry:
a) No, c) No (R. I. Grigorchuk, *Soviet Math. Dokl.*, **28** (1983), 23--26). | 2 | partially_solved | S. I. Adian | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (a) No, c) No (R. I. Grigorchuk, *Soviet Math. Dokl.*, **28** (1983), 23--26).); 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 |
700,217 | KOU-4.6 | Kourovka Notebook Problem 4.6 | (P. Hall). Are the projective groups in the
variety of metabelian groups free? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.6.
Proposed by: V. A. Artamonov | 2 | open | V. A. Artamonov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,218 | KOU-4.7 | Kourovka Notebook Problem 4.7 | (Well-known problem). For which ring epimorphisms
$R \rightarrow Q$ is the corresponding group homomorphism $SL_n(R)
\rightarrow SL_n(Q)$ an epimorphism (for a fixed $n \geq 2$)? In
particular, for what rings $R$ does the equality $SL_n(R)=E_n(R)$
hold? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.7.
Proposed by: V. A. Artamonov | 2 | open | V. A. Artamonov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,219 | KOU-4.8 | Kourovka Notebook Problem 4.8 | Suppose $G$ is a finitely generated free-by-cyclic
group. Is $G$ finitely presented? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.8.
Proposed by: G. Baumslag
Published resolution:
Yes, it is;
moreover, every finitely generated subgroup is finitely presented. For infinite cyclic extensions this is proved in ... | 2 | solved | G. Baumslag | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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;\nmoreover, every finitely generated subgroup is finitely presented. For infinite cyclic extensions this is proved in (M. Feighn, M. Handel, *Ann. Mat... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,220 | KOU-4.9 | Kourovka Notebook Problem 4.9 | Let $G$ be a finitely generated torsion-free
nilpotent group. Are there only a finite number of non-isomorphic
groups in the sequence $\alpha G$, $\alpha ^2G, \ldots $ ? Here
$\alpha G$ denotes the automorphism group of $G$ and $\alpha
^{n+1}G=\alpha (\alpha ^nG)$ for $n=1, 2,\ldots $ | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.9.
Proposed by: G. Baumslag | 2 | open | G. Baumslag | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,221 | KOU-4.10 | Kourovka Notebook Problem 4.10 | A group $G$ is called *locally indicable* if every
non-trivial finitely generated subgroup of $G$ has an infinite
cyclic factor group. Is every torsion-free one-relator group
locally indicable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.10.
Proposed by: G. Baumslag
Published resolution:
Yes, it is (S. D. Brodskiĭ, Dep. no. 2214-80, VINITI,
Moscow, 1980 (Russian)). | 2 | solved | G. Baumslag | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (S. D. Brodskiĭ, Dep. no. 2214-80, VINITI,\nMoscow, 1980 (Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,222 | KOU-4.11 | Kourovka Notebook Problem 4.11 | Let $F$ be the free group of rank 2 in some
variety of groups. If $F$ is not finitely presented, is the
multiplicator of $F$ necessarily infinitely generated? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.11.
Proposed by: G. Baumslag | 2 | open | G. Baumslag | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,223 | KOU-4.12 | Kourovka Notebook Problem 4.12 | Let $G$ be a finite group and $A$ a group of
automorphisms of $ G$ stabilizing a series of subgroups
beginning with $G$ and ending with its Frattini subgroup. Then
is $A$ nilpotent? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.12.
Proposed by: Ya. G. Berkovich
Published resolution:
Yes, it is (P. Schmid, *Math. Ann.*, **202** (1973), 57--69). | 2 | solved | Ya. G. Berkovich | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (P. Schmid, *Math. Ann.*, **202** (1973), 57--69)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,224 | KOU-4.13 | Kourovka Notebook Problem 4.13 | Prove that every finite non-abelian $p
$-group admits an automorphism of order $p$ which is not an inner
one. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.13.
Proposed by: Ya. G. Berkovich | 2 | open | Ya. G. Berkovich | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,225 | KOU-4.14 | Kourovka Notebook Problem 4.14 | Let $p$ be a prime number. What are necessary and
sufficient conditions for a finite group $G$ in order that the
group algebra of $G$ over a field of characteristic $p$ be
indecomposable as a two-sided ideal? There exist some nontrivial
examples, for instance, the group algebra of the Mathieu group
$M_{24}$ is indecomp... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.14.
Proposed by: R. Brauer
Published resolution:
A necessary and sufficient condition can be extracted from
(G. R. Robinson, *J. Algebra*, **84** (1983),
493--502); another solut... | 2 | solved | R. Brauer | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 necessary and sufficient condition can be extracted from\n(G. R. Robinson, *J. Algebra*, **84** (1983),\n493--502); another solution, which requires considerin... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,226 | KOU-4.16 | Kourovka Notebook Problem 4.16 | Suppose that ${\mathfrak{K}}$ is a class of groups meeting the
following requirements: 1) subgroups and epimorphic images of
${\mathfrak{K}}$-groups are ${\mathfrak{K}}$-groups; \ 2) if the group $ G =
UV$ is the product of $ {\mathfrak{K}}$-subgroups $U$ and $V$ (neither
of which need be normal), then $ G\in {\mathfra... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.16.
Proposed by: R. Baer
Published resolution:
No (S. A. Syskin, *Siberian Math. J.*, **20** (1979),
475--476). | 2 | solved | R. Baer | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (S. A. Syskin, *Siberian Math. J.*, **20** (1979),\n475--476)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,227 | KOU-4.17 | Kourovka Notebook Problem 4.17 | If $ {\mathfrak{V}}$ is a variety of groups, denote by
$\widetilde{{\mathfrak{V}}}$ the class of all finite ${\mathfrak{V}}
$-groups. How to characterize the classes of finite groups of the
form $\widetilde{{\mathfrak{V}}}$ for ${\mathfrak{V}}$ a variety? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.17.
Proposed by: R. Baer
Discussion and literature:
[Unverified AI claim] A characterization is given (unverified AI solution: Achyuth Jayadevan, *Preprint*, 2... | 2 | open | R. Baer | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 characterization is given (unverified AI solution: Achyuth Jayadevan, *Preprint... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,228 | KOU-4.18 | Kourovka Notebook Problem 4.18 | Characterize the classes ${\mathfrak{K}}$ of groups,
meeting the following requirements: subgroups, epimorphic images
and groups of automorphisms of ${\mathfrak{K}} $-groups are $ {\mathfrak{K}}
$-groups, but not every countable group is a ${\mathfrak{K}} $-group.
Note that the class of all finite groups and the class ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.18.
Proposed by: R. Baer | 2 | open | R. Baer | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,229 | KOU-4.19 | Kourovka Notebook Problem 4.19 | Denote by ${\mathfrak{C}}$ the class of all groups $G$
with the following property: if $U$ and $V$ are maximal locally
soluble subgroups of $G$, then $U$ and $V$ are conjugate in $G$
(or at least isomorphic). It is almost obvious that a finite group
$G$ belongs to ${\mathfrak{C}}$ if and only if $G$ is soluble. What ca... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.19.
Proposed by: R. Baer | 2 | open | R. Baer | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,230 | KOU-4.20 | Kourovka Notebook Problem 4.20 | a) Let $F$ be a free group, and $N$ a normal subgroup of it. Is it true that the Cartesian square of $N$ is $m$-reducible to $N$ (that is, there is an algorithm that from a pair of words $w_1,w_2\in F$ constructs a word $w\in F$ such that
$w_1\in N$ and $w_2\in N$ if and only if $w\in N$)?
b) (Well-known problem). Do ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.20.
Proposed by: M. K. Valiev
Published resolution:
a) No, it is not true (O. V. Belegradek, *Siberian Math. J.*, **19** (1978), 867--870).
b) No, there exist $m$-degrees which ... | 2 | solved | M. K. Valiev | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) No, it is not true (O. V. Belegradek, *Siberian Math. J.*, **19** (1978), 867--870).\n\nb) No, there exist $m$-degrees which do not contain the word problem o... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,231 | KOU-4.21 | Kourovka Notebook Problem 4.21 | Let $G$ be a finite group, $p$ an odd prime number, and $P$ a
Sylow $p $-subgroup of $G$. Let the order of every non-identity
normal subgroup of $G$ be divisible by $p$. Suppose $P$ has an
element $x$ that is conjugate to no other from $P$. Does $x$
belong to the centre of $G$? For $ p = 2$, the answer is positive
(G. ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.21.
Proposed by: G. Glauberman
Published resolution:
Yes, it does, mod CFSG (O. D. Artemovich, *Ukrain. Math.
J.*, **40** (1988), 343--345). | 2 | solved | G. Glauberman | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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, mod CFSG (O. D. Artemovich, *Ukrain. Math.\nJ.*, **40** (1988), 343--345)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,232 | KOU-4.22 | Kourovka Notebook Problem 4.22 | (J. G. Thompson). Let $G$ be a
finite group, $A$ a group of automorphisms of $G$ such that
$|A|$ and $|G|$ are coprime. Does there exist an $A
$-invariant soluble subgroup $H$ of $G$ such that
$C_A(H)=1$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.22.
Proposed by: G. Glauberman
Published resolution:
Yes, it does (S. A. Syskin, *Siberian Math. J.*, **32**,
no. 6 (1991), 1034--1037). | 2 | solved | G. Glauberman | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (S. A. Syskin, *Siberian Math. J.*, **32**,\nno. 6 (1991), 1034--1037)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,233 | KOU-4.23 | Kourovka Notebook Problem 4.23 | Let $G$ be a finite simple group, $\tau$ some element of prime
order, and $\alpha$ an automorphism of $G$ whose order is
coprime to $ |G|$. Suppose $\alpha$ centralizes $C_G(\tau )$.
Is $\alpha = 1$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.23.
Proposed by: G. Glauberman
Published resolution:
Not always; for example, $G=Sz(8)$, \ $|\tau |=5$, \ $|\alpha
|=3$ \
(N. D. Podufalov, *Letter of September, 3, 1975*). | 2 | solved | G. Glauberman | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 always; for example, $G=Sz(8)$, \\ $|\\tau |=5$, \\ $|\\alpha\n|=3$ \\\n(N. D. Podufalov, *Letter of September, 3, 1975*)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,234 | KOU-4.24 | Kourovka Notebook Problem 4.24 | Suppose $T$ is a non-abelian Sylow 2-subgroup
of a finite simple group $G$.
a) Suppose $T$ has nilpotency class $n$. The
best possible bound for the exponent of the center of $T$ is
$2^{n-1}$. This easily implies the bound $2^{n(n-1)}$ for the
exponent of $T$, however, this is almost certainly too crude. What
is the b... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 4.24.
Proposed by: D. M. Goldschmidt
Archive of Solved Problems entry:
b) Yes, it is. For example, the Sylow 2-subgroups of the
alternating groups $ {\mathbb{... | 2 | partially_solved | D. M. Goldschmidt | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (b) Yes, it is. For example, the Sylow 2-subgroups of the\nalternating groups $ {\\mathbb{A}_{14}$ and $ {\\mathbb{A}_{15}$ are\nisomorphic to the Sylow 2-subgrou... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,235 | KOU-4.27 | Kourovka Notebook Problem 4.27 | Describe all finite
simple groups $G$ which
can be
represented in the form $G=ABA$, where $A$ and $B$ are abelian
subgroups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.27.
Proposed by: I. P. Doktorov
Published resolution:
They are described mod CFSG (D. L. Zagorin, L. S. Kazarin,
*Dokl. Math.*, **53**, no. 2 (1996), 237--239; \
D. L. Zagorin, *... | 2 | solved | I. P. Doktorov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 mod CFSG (D. L. Zagorin, L. S. Kazarin,\n*Dokl. Math.*, **53**, no. 2 (1996), 237--239; \\\nD. L. Zagorin, *Some problems of $ABA$-factorizati... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,236 | KOU-4.28 | Kourovka Notebook Problem 4.28 | For a given field $k$ of characteristic $p > 0$,
characterize the locally finite groups with semisimple group
algebras over $k$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.28.
Proposed by: A. E. Zalesskii
Published resolution:
They are characterized: simple groups in (D. S. Passman,
A. E. Zalesskii, *Proc. London Math. Soc. (3)*, **67**
(1993), 243... | 2 | solved | A. E. Zalesskii | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 characterized: simple groups in (D. S. Passman,\nA. E. Zalesskii, *Proc. London Math. Soc. (3)*, **67**\n(1993), 243--276); the general case in (D. S. P... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,237 | KOU-4.29 | Kourovka Notebook Problem 4.29 | Classify the irreducible matrix groups over a finite field that
are generated by reflections, that is, by matrices with Jordan
form $\mathrm{diag}(-1, 1, \ldots , 1)$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.29.
Proposed by: A. E. Zalesski\u{\i
Published resolution:
These are classified (A. E. Zalesskiĭ, V. N. Ser\"ezhkin, *Math.
USSR--Izv.*, **17** (1981), 477--503; \ A. Wagner, *Ge... | 2 | solved | A. E. Zalesski\u{\i | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: These are classified (A. E. Zalesskiĭ, V. N. Ser\\\"ezhkin, *Math.\nUSSR--Izv.*, **17** (1981), 477--503; \\ A. Wagner, *Geom.\nDedicata*, **9** (1980), 239--253... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,238 | KOU-4.30 | Kourovka Notebook Problem 4.30 | Describe the groups (finite groups, abelian
groups) that are the full automorphism groups of topological
groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.30.
Proposed by: M. I. Kargapolov | 2 | open | M. I. Kargapolov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,239 | KOU-4.31 | Kourovka Notebook Problem 4.31 | Describe the lattice of quasivarieties of
nilpotent groups of nilpotency class 2. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.31.
Proposed by: M. I. Kargapolov | 2 | open | M. I. Kargapolov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,240 | KOU-4.32 | Kourovka Notebook Problem 4.32 | The conjugacy problem for metabelian groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.32.
Proposed by: M. I. Kargapolov
Published resolution:
This is algorithmically soluble (G. A. Noskov, *Math.
Notes*, **31** (1982), 252--258). | 2 | solved | M. I. Kargapolov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 algorithmically soluble (G. A. Noskov, *Math.\nNotes*, **31** (1982), 252--258)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,241 | KOU-4.33 | Kourovka Notebook Problem 4.33 | Let ${\mathfrak{K}}_n$ be the class of all groups with
a single defining relation in the variety of soluble groups of
derived length $n$.
a) Under what conditions does a ${\mathfrak{K}}_n $-group have non-trivial center? Can a $ {\mathfrak{K}}_n $-group,
$n \geq 2$, that cannot be generated by two elements have
non-tr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.33.
Proposed by: M. I. Kargapolov | 2 | open | M. I. Kargapolov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,242 | KOU-4.34 | Kourovka Notebook Problem 4.34 | Let $v$ be a group word, and let ${\mathfrak{K}}_v$ be
the class of groups $G $ such that there exists a positive integer
$n=n(G)$ such that each element of the verbal subgroup $vG$ can be
represented as a product of $n$ values of the word $v$ on the
group $G$.
a) For which $v$ do all finitely generated
soluble groups... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.34.
Proposed by: M. I. Kargapolov | 2 | open | M. I. Kargapolov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,243 | KOU-4.35 | Kourovka Notebook Problem 4.35 | (Well-known problem). Is there an infinite locally finite
simple group satisfying the minimum condition for $p $-subgroups
for every prime $p$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.35.
Proposed by: O. H. Kegel
Published resolution:
No (V. V. Belyaev, *Algebra and Logic*, **20** (1981),
393--401). | 2 | solved | O. H. Kegel | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. V. Belyaev, *Algebra and Logic*, **20** (1981),\n393--401)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,244 | KOU-4.36 | Kourovka Notebook Problem 4.36 | Is there an infinite locally finite simple group $G$ with an
involution $i$ such that the centralizer $C_G(i)$ is a Chernikov
group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.36.
Proposed by: O. H. Kegel
Published resolution:
No (A. O. Asar, *Proc. London Math. Soc.*, **45** (1982), 337--364). | 2 | solved | O. H. Kegel | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (A. O. Asar, *Proc. London Math. Soc.*, **45** (1982), 337--364)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,245 | KOU-4.37 | Kourovka Notebook Problem 4.37 | Is there an infinite locally finite simple group $G$ that
cannot be represented by matrices over a field and is such that
for some prime $p$ the $p $-subgroups of $G$ are either of
bounded derived length or of finite exponent? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.37.
Proposed by: O. H. Kegel
Published resolution:
No (mod CFSG) by Theorem 4.8 of (O. H. Kegel,
B. A. F. Wehrfritz, *Locally
finite groups*, North Holland, Amsterdam, 1973). | 2 | solved | O. H. Kegel | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (mod CFSG) by Theorem 4.8 of (O. H. Kegel,\nB. A. F. Wehrfritz, *Locally\nfinite groups*, North Holland, Amsterdam, 1973)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,246 | KOU-4.38 | Kourovka Notebook Problem 4.38 | Classify the composition factors of automorphism groups of
finite (nilpotent of class 2) groups of prime exponent. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.38.
Proposed by: V. D. Mazurov
Published resolution:
Any finite simple group can be such a composition factor
(P. M. Beletskiĭ, *Russian Math. Surveys*, **33**,
no. 6 (1980), 85-... | 2 | solved | V. D. Mazurov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: Any finite simple group can be such a composition factor\n(P. M. Beletskiĭ, *Russian Math. Surveys*, **33**,\nno. 6 (1980), 85--86)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,247 | KOU-4.39 | Kourovka Notebook Problem 4.39 | A countable group $U$ is said to be {\it ${\sc
SQ}$-universal} if
every countable group is isomorphic to a subgroup of a quotient
group of $ U$. Let $G$ be a group that has a presentation with $ r
\geq 2$ generators and at most $r-2$ defining relations. Is $G$ \
${\sc SQ}$-universal? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.39.
Proposed by: A. M. Macbeath, P. M. Neumann
Published resolution:
Yes, it is (B. Baumslag, S. J. Pride, *J. London Math. Soc.*, **17** (1978), 425--426). | 2 | solved | A. M. Macbeath, P. M. Neumann | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (B. Baumslag, S. J. Pride, *J. London Math. Soc.*, **17** (1978), 425--426)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,248 | KOU-4.40 | Kourovka Notebook Problem 4.40 | Let $C$ be a fixed non-trivial group (for
instance, $C={\mathbb{Z}/2{\mathbb{Z}$). As it is shown in
(Yu. I. Merzlyakov, *Algebra and Logic*, **9**, no. 5
(1970), 326--337), for any two groups $A$ and $B$, all split
extensions of $B$ by $A$ can be imbedded in a certain unified way
into the direct product $A \times \mat... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.40.
Proposed by: Yu. I. Merzlyakov
Discussion and literature:
[Unverified AI claim] A description is given (unverified AI solution: Achyuth Jayadevan, *Preprin... | 2 | open | Yu. I. Merzlyakov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 description is given (unverified AI solution: Achyuth Jayadevan, *Preprint*, 20... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,249 | KOU-4.41 | Kourovka Notebook Problem 4.41 | A point $z$ in the complex plane is called *free* if the
matrices {\small $\left(
\begin{array}{cc}1&2\\0&1\end{array}\right)$} and
{\small $\left( \begin{array}{cc}1&0\\z&1\end{array}\right)$} generate
a
free group. Are all points outside the rhombus with vertices
$\pm 2$, $\pm i$ free? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.41.
Proposed by: Yu. I. Merzlyakov
Published resolution:
No (Yu. A. Ignatov, *Math. USSR--Sb.*, **35** (1979),
49--56; \ A. I. Shkuratskiĭ, *Math. Notes*, **24**
(1978), 720--721... | 2 | solved | Yu. I. Merzlyakov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (Yu. A. Ignatov, *Math. USSR--Sb.*, **35** (1979),\n49--56; \\ A. I. Shkuratskiĭ, *Math. Notes*, **24**\n(1978), 720--721)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,250 | KOU-4.42 | Kourovka Notebook Problem 4.42 | For what natural numbers $n$ does the
following equality hold: $$GL_n({\mathbb{R})=D_n({\mathbb{R})\cdot O_n({\mathbb{R})\cdot UT_n({\mathbb{R})\cdot
GL_n({\mathbb{Z})? $$ For notation, see,
for instance, (M. I. Kargapolov, Ju. I. Merzljakov, *Fundamentals of the Theory of Groups*, Springer, New York, 1979).
For given ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.42.
Proposed by: Yu. I. Merzlyakov | 2 | open | Yu. I. Merzlyakov | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,251 | KOU-4.44 | Kourovka Notebook Problem 4.44 | (Well-known problem). Describe the groups whose
automorphism groups are abelian. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 4.44.
Proposed by: V. T. Nagrebetski\u{\i | 2 | open | V. T. Nagrebetski\u{\i | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,252 | KOU-4.45 | Kourovka Notebook Problem 4.45 | Let $G$ be a free product amalgamating proper
subgroups $H$ and $K$ of $A$ and $B$, respectively.
a) Suppose that $H,K$ are finite and $|A:H| > 2$, \ $|B:K| \geq
2$. Is $G$ ${\sc SQ}
$-universal?
b) Suppose that $A,B,H,K$ are free groups of finite ranks. Can $G$
be simple? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 4th issue (1973). Original problem number: 4.45.
Proposed by: P. M. Neumann
Published resolution:
a) Yes, it is (K. I. Lossov, *Siberian Math. J.*, **27**,
no. 6 (1986), 890--899).
b) Yes, it can (M. Burger, S. Mozes, *Ins... | 2 | solved | P. M. Neumann | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) Yes, it is (K. I. Lossov, *Siberian Math. J.*, **27**,\nno. 6 (1986), 890--899).\n\nb) Yes, it can (M. Burger, S. Mozes, *Inst. Hautes \\'Etudes Sci. Publ. Ma... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,253 | KOU-4.46 | Kourovka Notebook Problem 4.46 | a) We call a variety of groups a *limit
variety* if it cannot be defined by finitely many laws, while
each of its proper subvarieties has a finite basis of identities.
It follows from Zorn's lemma that every variety that has no finite
basis of identities contains a limit subvariety. Give explicitly
(by means of identit... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 4th issue (1973), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 4.46.
Proposed by: A. Yu. Olshanskii
Archive of Solved Problems entry:
No, there are continuum of such varieties (P. A. Kozhevnikov,
*On varieties of groups o... | 2 | partially_solved | A. Yu. Olshanskii | 1,973 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (No, there are continuum of such varieties (P. A. Kozhevnikov,\n*On varieties of groups of large odd exponent*,\nDep. 1612-V00, VINITI, Moscow, 2000 (Russian); \\... | 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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