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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
700,454 | KOU-8.2 | Kourovka Notebook Problem 8.2 | Let $G$ be the amalgamated free product of two
polycyclic groups, amalgamating a normal subgroup of each. Is $G$
isomorphic to a linear group? $G$ is residually finite
(G. Baumslag, *Trans. Amer. Math. Soc.*, **106**, no. 2
(1963), 193--209) and the answer is ``yes'' if the amalgamated part
is torsion-free, abelian (B.... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.2.
Proposed by: B. A. F. Wehrfritz | 2 | open | B. A. F. Wehrfritz | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,455 | KOU-8.3 | Kourovka Notebook Problem 8.3 | Let $m$ and $n$ be positive integers and $p$ a
prime. Let $P_n({\mathbb{Z}_{p^m})$ be the group of all $n \times n$
matrices $(a_{ij})$ over the integers modulo $p^m$ such that
$a_{ii} \equiv 1$ for all $i$ and $a_{ij} \equiv 0 (\mathrm{mod}
p)$ for all $i > j$. The group $P_n({\mathbb{Z}_{p^m})$ is a finite
$p
$-group... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.3.
Proposed by: B. A. F. Wehrfritz | 2 | open | B. A. F. Wehrfritz | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,456 | KOU-8.4 | Kourovka Notebook Problem 8.4 | Construct a finite nilpotent loop with no finite
basis for its laws. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.4.
Proposed by: M. R. Vaughan-Lee | 2 | open | M. R. Vaughan-Lee | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,457 | KOU-8.5 | Kourovka Notebook Problem 8.5 | Prove that if $X$ is a finite group, $F$ is any
field, and $M$ is a non-trivial irreducible $FX $-module
then $\frac{1}{|X|}\sum_{x\in X} \dim \mathrm{fix} (x) \leq
\frac{1}{2}\dim M$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.5.
Proposed by: M. R. Vaughan-Lee
Published resolution:
Proved, even with ${...}\leq
\frac{1}{p} \dim M$, where $p$ is the least prime divisor of $|G|$ \ (R. M. Guralnick,
A. Mar... | 2 | solved | M. R. Vaughan-Lee | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: Proved, even with ${...}\\leq\n\\frac{1}{p} \\dim M$, where $p$ is the least prime divisor of $|G|$ \\ (R. M. Guralnick,\nA. Mar\\'oti, *Adv. Math.*, **226**, no... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,458 | KOU-8.6 | Kourovka Notebook Problem 8.6 | (M. M. Day). Do the classes of amenable groups and elementary
groups coincide? The latter consists of groups that can be
obtained from commutative and finite groups by forming
subgroups, factor-groups, extensions, and direct limits. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.6.
Proposed by: R. I. Grigorchuk
Published resolution:
No (R. I. Grigorchuk, *Soviet Math. Dokl.*, **28**
(1983), 23--26). | 2 | solved | R. I. Grigorchuk | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (R. I. Grigorchuk, *Soviet Math. Dokl.*, **28**\n(1983), 23--26)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,459 | KOU-8.7 | Kourovka Notebook Problem 8.7 | Does there exist a non-amenable finitely presented group which has
no free subgroups of rank 2? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.7.
Proposed by: R. I. Grigorchuk
Published resolution:
Yes, there does (A. Yu. Olshanskii, M. V. Sapir, *Publ.
Math. Inst. Hautes \'Etud. Sci.*, **96** (2002), 43--169). | 2 | solved | R. I. Grigorchuk | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. Yu. Olshanskii, M. V. Sapir, *Publ.\nMath. Inst. Hautes \\'Etud. Sci.*, **96** (2002), 43--169)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,460 | KOU-8.8 | Kourovka Notebook Problem 8.8 | b) (D. V. Anosov). Does there exist a non-cyclic
finitely presented group $G$ which contains an element $a$ such
that each element of $G$ is conjugate to some power of $a$?
[Archived solved part(s) of 8.8:]
(D. V. Anosov). a) Does there exist a non-cyclic
finitely-generated group $G$ containing an element $a$ such tha... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 8.8.
Proposed by: R. I. Grigorchuk
Archive of Solved Problems entry:
Yes, there does (V. S. Guba, *Math. USSR--Izv.*, **29**
(1986), 233--277). | 2 | partially_solved | R. I. Grigorchuk | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (Yes, there does (V. S. Guba, *Math. USSR--Izv.*, **29**\n(1986), 233--277).); 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 |
700,461 | KOU-8.9 | Kourovka Notebook Problem 8.9 | (C. Chou). We say that a group $G$ has *property
$P$* if for every finite subset $F$ of $G$, there is a finite
subset $S \supset F$ and a subset $X \subset G$ such that $x_1S
\cap x_2S$ is empty for any $x_1, x_2 \in X$, $x_1 \ne x_2$, and
$G=\bigcup\limits_{x\in X}Sx$. Does every group have property $P$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.9.
Proposed by: R. I. Grigorchuk | 2 | open | R. I. Grigorchuk | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,462 | KOU-8.10 | Kourovka Notebook Problem 8.10 | Is the group $G=\left< a,b\mid a^n=1,\;
ab=b^3a^3\right>$ finite or infinite for $n=7$, $n=9$, and $n=15$? All
other cases known, see Archive, 7.7. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.10.
Proposed by: D. L. Johnson
Published resolution:
Infinite in all three cases. For $n=15$ by (D. J. Seal, *Proc. Roy. Soc. Edinburgh
(A)*, **92** (1982), 181--192). For $n=9$ ... | 2 | solved | D. L. Johnson | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: Infinite in all three cases. For $n=15$ by (D. J. Seal, *Proc. Roy. Soc. Edinburgh\n(A)*, **92** (1982), 181--192). For $n=9$ (and $15$) by (M. I. Prishchepov, *... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,463 | KOU-8.11 | Kourovka Notebook Problem 8.11 | Consider the group $$M=\left<
x,y,z,t\mid [x,y]=[y,z]=[z,x]=(x,t)=(y,t)=(z,t)=1\right> $$
where $[x, y]=x^{-1}y^{-1}xy$ and
$(x,t)=x^{-1}t^{-1}x^{-1}txt$. The subgroup $H=\left<
x,t,y\right>$ is isomorphic to the braid group ${\mathfrak{B}}_4$,
and is normally complemented by $N=\left< (zx^{-1})^M\right>
$. Is $N$ a f... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.11.
Proposed by: D. L. Johnson
Discussion and literature:
[Solved] As pointed by Y. Antol\'{i}n, the group $M$ is the Artin group of type $D_4$; it is proved i... | 2 | solved | D. L. Johnson | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] As pointed by Y. Antol\\'{i}n, the group $M$ is the Artin group of type $D_4$; it is proved in (B. Perro... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,464 | KOU-8.12 | Kourovka Notebook Problem 8.12 | Let $D0$ denote the class of finite groups of
deficiency zero, i. e. having a presentation $\left< X\mid
R\right>$ with $|X|=|R|$.
a) Does $D0$ contain any 3-generator
$p
$-group for $p \geq 5$?
c) Do soluble $D0
$-groups have bounded
derived length?
d) Which non-abelian simple groups can occur
as composition
factor... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 8.12.
Proposed by: D. L. Johnson,
E. F. Robertson
Archive of Solved Problems entry:
No, not always (G. Havas, E. F. Robertson, *Commun.
Algebra*, **24** (1996... | 2 | partially_solved | D. L. Johnson,
E. F. Robertson | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (No, not always (G. Havas, E. F. Robertson, *Commun.\nAlgebra*, **24** (1996), 3483--3487).); 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 |
700,465 | KOU-8.13 | Kourovka Notebook Problem 8.13 | Let $G$ be a simple algebraic group over an
algebraically closed field of
characteristic $p$ and ${\mathfrak{G}}$ be the Lie algebra of $G$.
Is the number of orbits of nilpotent elements of ${\mathfrak{G}}$ under the
adjoint action of $G$ finite? This is known to be true if $p$
is not too small (i. e. if $p$ is not a `... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.13.
Proposed by: R. W. Carter
Published resolution:
Yes, it is (D. F. Holt, N. Spaltenstein,
*J. Austral. Math. Soc. (A)*, **38** (1985), 330--350). | 2 | solved | R. W. Carter | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (D. F. Holt, N. Spaltenstein,\n*J. Austral. Math. Soc. (A)*, **38** (1985), 330--350)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,466 | KOU-8.14 | Kourovka Notebook Problem 8.14 | b) Assume a group $G$ is existentially closed in one
of the classes $L{\mathfrak{N}}^{+}$, \ $L{\mathfrak{S}}_{\pi}$, \ $L{\mathfrak{S}}^{+}$, \ $L{\mathfrak{S}}$ of, respectively, all locally nilpotent
torsion-free groups, all locally soluble $\pi $-groups, all locally
soluble torsion-free groups, or all locally solub... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 8.14.
Proposed by: O. H. Kegel
Archive of Solved Problems entry:
Not always (S. R. Thomas, *Arch. Math.*, **44** (1985), 98--109). | 2 | partially_solved | O. H. Kegel | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (Not always (S. R. Thomas, *Arch. Math.*, **44** (1985), 98--109).); 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 |
700,467 | KOU-8.15 | Kourovka Notebook Problem 8.15 | (Well-known problem). Is every $\widetilde{N}
$-group a $ \overline{\! Z} $-group? For definitions, see
(M. I. Kargapolov, Yu. I. Merzlyakov, *Fundamentals of the
Theory of Groups*, 3rd Edition, Moscow, Nauka, 1982, p. 205
(Russian)). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.15.
Proposed by: Sh. S. Kemkhadze | 2 | open | Sh. S. Kemkhadze | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,468 | KOU-8.16 | Kourovka Notebook Problem 8.16 | Is the class of all $ \overline{\! Z} $-groups
closed under taking normal subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.16.
Proposed by: Sh. S. Kemkhadze
Discussion and literature:
[Solved] No, it is not (AI solution confirmed by J. S. Wilson: \ Achyuth Jayadevan, *Preprint*, 20... | 2 | solved | Sh. S. Kemkhadze | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it is not (AI solution confirmed by J. S. Wilson: \\ Achyuth Jayadevan, *Preprint*, 2026, https://ze... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,469 | KOU-8.17 | Kourovka Notebook Problem 8.17 | An {\it $\widetilde{RN}$-group} is one
whose every homomorphic image is an $RN$-group. Is the class of
$\widetilde{RN}$-groups closed under taking normal subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.17.
Proposed by: Sh. S. Kemkhadze
Published resolution:
No (J. S. Wilson, *Arch. Math.*, **25** (1974), 574--577). | 2 | solved | Sh. S. Kemkhadze | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (J. S. Wilson, *Arch. Math.*, **25** (1974), 574--577)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,470 | KOU-8.18 | Kourovka Notebook Problem 8.18 | Is every countably infinite
abelian group a verbal subgroup of some finitely generated (soluble)
relatively free group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.18.
Proposed by: Yu. G. Kleiman
Published resolution:
Yes, it is (A. Storozhev, *Commun. Algebra*, **22**, no. 7
(1994), 2677--2701). | 2 | solved | Yu. G. Kleiman | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. Storozhev, *Commun. Algebra*, **22**, no. 7\n(1994), 2677--2701)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,471 | KOU-8.19 | Kourovka Notebook Problem 8.19 | Which of the following properties of (soluble)
varieties of groups are equivalent to one another:
1) to satisfy the minimum condition for
subvarieties;
2) to have at most countably
many subvarieties;
3) to have no infinite independent system of
identities? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.19.
Proposed by: Yu. G. Kleiman | 2 | open | Yu. G. Kleiman | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,472 | KOU-8.20 | Kourovka Notebook Problem 8.20 | What is the cardinality of the set of all varieties covering an
abelian (nilpotent? Cross? hereditarily finitely based?) variety
of groups? The question is related to 4.46 and 4.73. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.20.
Proposed by: Yu. G. Kleiman
Published resolution:
There are continually many varieties covering the variety $A$ of
abelian groups, as well as the variety $A_n$ of abelian gro... | 2 | solved | Yu. G. Kleiman | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: There are continually many varieties covering the variety $A$ of\nabelian groups, as well as the variety $A_n$ of abelian groups of\nsufficiently large odd expon... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,473 | KOU-8.21 | Kourovka Notebook Problem 8.21 | If the commutator subgroup $G'$ of a relatively
free group $G$ is periodic, must $G'$ have finite exponent? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.21.
Proposed by: L. G. Kov\'acs | 2 | open | L. G. Kov\'acs | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,474 | KOU-8.22 | Kourovka Notebook Problem 8.22 | If $G$ is a (non-abelian) finite
group contained in a join ${\mathfrak{A}} \vee {\mathfrak{B}}$ of two varieties
${\mathfrak{A}}, {\mathfrak{B}}$ of groups, must there exist finite groups $A
\in {\mathfrak{A}}$, $B \in {\mathfrak{B}}$ such that $G$ is a section of the
direct product $A \times B$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.22.
Proposed by: L. G. Kov\'acs
Published resolution:
Not always (A. Storozhev, *Bull. Austral. Math. Soc.*, **51**, no. 2 (1995), 287--290). | 2 | solved | L. G. Kov\'acs | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. Storozhev, *Bull. Austral. Math. Soc.*, **51**, no. 2 (1995), 287--290)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,475 | KOU-8.23 | Kourovka Notebook Problem 8.23 | If the dihedral group $D$ of order 18 is a
section of a direct product \mbox{$A \times B$}, must at least one
of $A$ and $B$ have a section isomorphic to $D$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.23.
Proposed by: L. G. Kov\'acs | 2 | open | L. G. Kov\'acs | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,476 | KOU-8.24 | Kourovka Notebook Problem 8.24 | Prove that every linearly ordered group of finite
rank is soluble. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.24.
Proposed by: V. M. Kopytov | 2 | open | V. M. Kopytov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,477 | KOU-8.25 | Kourovka Notebook Problem 8.25 | Does there exist an algorithm which recognizes by
an identity whether it defines a
variety of groups that has at most countably many subvarieties? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.25.
Proposed by: A. V. Kuznetsov | 2 | open | A. V. Kuznetsov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,478 | KOU-8.26 | Kourovka Notebook Problem 8.26 | We call a variety *passable* if there exists an unrefinable chain of its subvarieties
which is well-ordered by inclusion. For example, every variety
generated by its finite groups is passable --- this is an easy
consequence of (H. Neumann, *Varieties of groups*, Berlin et
al., Springer, 1967, Chapter 5). Do there exist... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.26.
Proposed by: A. V. Kuznetsov
Published resolution:
Yes, there do (M. I. Anokhin, *Moscow Univ. Math. Bull.*,
**51**, no. 1 (1996), 48--49). | 2 | solved | A. V. Kuznetsov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there do (M. I. Anokhin, *Moscow Univ. Math. Bull.*,\n**51**, no. 1 (1996), 48--49)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,479 | KOU-8.27 | Kourovka Notebook Problem 8.27 | Does the lattice of all varieties of groups
possess non-trivial automorphisms? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.27.
Proposed by: A. V. Kuznetsov | 2 | open | A. V. Kuznetsov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,480 | KOU-8.28 | Kourovka Notebook Problem 8.28 | Is the variety of groups finitely
based if it is generated by a finitely based quasivariety of groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.28.
Proposed by: A. V. Kuznetsov
Published resolution:
No, not always (M. I. Anokhin, *Sb. Math.*, **189**,
no. 7--8 (1998), 1115--1124). | 2 | solved | A. V. Kuznetsov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (M. I. Anokhin, *Sb. Math.*, **189**,\nno. 7--8 (1998), 1115--1124)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,481 | KOU-8.29 | Kourovka Notebook Problem 8.29 | Do there exist locally nilpotent groups with
trivial centre satisfying the weak maximal
condition for normal
subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.29.
Proposed by: L. A. Kurdachenko | 2 | open | L. A. Kurdachenko | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,482 | KOU-8.30 | Kourovka Notebook Problem 8.30 | Let ${\mathfrak{X}}$, ${\mathfrak{Y}}$ be Fitting classes
of soluble groups which satisfy the Lockett condition, i. e.
${\mathfrak{X}} \cap {\mathfrak{S}}_{\displaystyle{*}}={\mathfrak{X}}_{\displaystyle{*}}$, \ ${\mathfrak{Y}} \cap {\mathfrak{S}}_
{\displaystyle{*}} = {\mathfrak{Y}}_{\displaystyle{*}}$ where ${\mathfr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.30.
Proposed by: H. Lausch | 2 | open | H. Lausch | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,483 | KOU-8.31 | Kourovka Notebook Problem 8.31 | Describe the finite groups in which every proper
subgroup has a complement in some larger subgroup. Among these
groups are, for example, $PSL_2(7)$ and all Sylow subgroups of
symmetric groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.31.
Proposed by: V. M. Levchuk
Published resolution:
These groups are described independently in (V. M. Levchuk, A. G. Likharev, *Siberian Math. J.*, **47**, no. 4 (2006), 659--6... | 2 | solved | V. M. Levchuk | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 groups are described independently in (V. M. Levchuk, A. G. Likharev, *Siberian Math. J.*, **47**, no. 4 (2006), 659--668) and in (V. N. Tyutyanov, *Proc. ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,484 | KOU-8.32 | Kourovka Notebook Problem 8.32 | Suppose $G$ is a finitely
generated group such that, for any set $\pi$ of primes and any
subgroup $H$ of $G$, if $G/\left< H^G\right>$ is a finite $\pi
$-group then $|G:H|$ is a finite $\pi $-number. Is $G$ nilpotent?
This is true for finitely generated soluble groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.32.
Proposed by: J. C. Lennox
Published resolution:
Not always (V. N. Obraztsov, *J. Austral. Math. Soc. (A)*,
**61**, no. 2 (1996), 267--288). | 2 | solved | J. C. Lennox | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (V. N. Obraztsov, *J. Austral. Math. Soc. (A)*,\n**61**, no. 2 (1996), 267--288)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,485 | KOU-8.33 | Kourovka Notebook Problem 8.33 | Let $a, b$ be two elements of a group, $a$ having
infinite order. Find a necessary and sufficient condition for
$\bigcap_{n=1}^{\infty} \left< a^n, b\right> =\left< b\right> .$ | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.33.
Proposed by: F. N. Liman | 2 | open | F. N. Liman | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,486 | KOU-8.34 | Kourovka Notebook Problem 8.34 | Let $G$ be a finite group. Is it true that
indecomposable projective ${\mathbb{Z}G $-modules are finitely
generated (and hence locally free)? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.34.
Proposed by: P. A. Linnell
Published resolution:
Yes, it is; see Remark 2.13 and Corollary 4.2 in
(P. P{\v{r}}{\'{\i}}hoda, *Rend. Semin. Mat. Univ. Padova*, **123** (2010), ... | 2 | solved | P. A. Linnell | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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; see Remark 2.13 and Corollary 4.2 in\n(P. P{\\v{r}}{\\'{\\i}}hoda, *Rend. Semin. Mat. Univ. Padova*, **123** (2010), 141--167)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,487 | KOU-8.35 | Kourovka Notebook Problem 8.35 | c) Determine the conjugacy classes of maximal
subgroups in the sporadic simple group $F_1$. See the current
status in *Atlas of Finite Group Representations*
(http://brauer.maths.qmul.ac.uk/Atlas/spor/M/).
[Archived solved part(s) of 8.35:]
Determine the conjugacy classes
of maximal subgroups in the sporadic simple gr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 8.35.
Proposed by: V. D. Mazurov
Archive of Solved Problems entry:
a) They are determined mod CFSG (S. A. Linton, R. A. Wilson,
*Proc. London Math. Soc.*, **6... | 2 | partially_solved | V. D. Mazurov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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) They are determined mod CFSG (S. A. Linton, R. A. Wilson,\n*Proc. London Math. Soc.*, **63**, no. 1 (1991), 113--164).\n\nb) They are determined mod CFSG (R. ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,488 | KOU-8.37 | Kourovka Notebook Problem 8.37 | (R. Griess). \ a) Is $M_{11}$ a section in $O'N$?
b) The same question for
$M_{24}$ in $F_2$; \ $J_1$ in $F_1$
and in $F_2$; \ $J_2$ in $F_{23}$, $F_{24}'$, and $F_2$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.37.
Proposed by: V. D. Mazurov
Published resolution:
a) Yes (A. A. Ivanov, S. V. Shpektorov, *Abs. 18 All-Union
Algebra Conf.*, Part 1, Kishinev, 1985, 209 (Russian); \
S. Yoshia... | 2 | solved | V. D. Mazurov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (A. A. Ivanov, S. V. Shpektorov, *Abs. 18 All-Union\nAlgebra Conf.*, Part 1, Kishinev, 1985, 209 (Russian); \\\nS. Yoshiara, *J. Fac. Sci. Univ. Tokyo (IA... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,489 | KOU-8.39 | Kourovka Notebook Problem 8.39 | b) Describe the irreducible subgroups of $SL_6(q)$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.39.
Proposed by: V. D. Mazurov
Published resolution:
They are described mod CFSG (A. S. Kondratiev, *Algebra and
Logic*, **28** (1989), 122--138; \ P. Kleidman, *Low-dimensional ... | 2 | solved | V. D. Mazurov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (A. S. Kondratiev, *Algebra and\nLogic*, **28** (1989), 122--138; \\ P. Kleidman, *Low-dimensional finite classical groups and their ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,490 | KOU-8.40 | Kourovka Notebook Problem 8.40 | Describe the finite groups generated by a conjugacy
class $D$ of involutions which satisfies the following property: if
$a,b \in D$ and $|ab|=4$, then $[a,b] \in D$. This condition is
satisfied, for example, in the case where $D$ is a conjugacy class of
involutions of a known finite simple group such that $\left<
C_D(a... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.40.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,491 | KOU-8.41 | Kourovka Notebook Problem 8.41 | What finite groups $G$ contain a normal set of
involutions $D$ which contains a non-empty proper subset $T$
satisfying the following properties:
1) $C_D(a) \subseteq T$ for any $a \in T$;
2) if $a,b \in T$ and $ab=ba \ne 1$, then
$C_D(ab) \subseteq
T$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.41.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,492 | KOU-8.42 | Kourovka Notebook Problem 8.42 | Describe the finite groups all of whose soluble
subgroups of odd indices have 2-length 1. For example, the groups
$L_n(2^m)$ are known to satisfy this condition. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.42.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,493 | KOU-8.43 | Kourovka Notebook Problem 8.43 | (F. Timmesfeld). Let $T$ be a Sylow 2-subgroup
of a finite group $G$. Suppose that $\left< N(B)\mid B \right.$ is
a non-trivial characteristic subgroup of $\left. T\right> $ is a
proper subgroup of $G$. Describe the group $G$ if $F^
{\displaystyle{*}}(M)=O_2(M)$ for any 2-local subgroup $M$
containing $T$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.43.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,494 | KOU-8.44 | Kourovka Notebook Problem 8.44 | Prove or disprove that for all, but finitely
many, primes $p$ the group $$G_p=\left< a, b\mid
a^2=b^p=(ab)^3=(b^rab^{-2r}a)^2=1\right> ,$$
where $r^2 + 1 \equiv 0 (\mathrm{mod} p)$, is infinite. A
solution of this problem would have interesting topological
applications. It was proved with the aid of computer that
$G_p$... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.44.
Proposed by: J. Mennicke | 2 | open | J. Mennicke | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,495 | KOU-8.45 | Kourovka Notebook Problem 8.45 | When it follows that a group is residually a
$({\mathfrak{X}} \cap {\mathfrak{Y}} )$-group, if it is both residually a
${\mathfrak{X}} $-group and residually a ${\mathfrak{Y}} $-group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.45.
Proposed by: Yu. I. Merzlyakov | 2 | open | Yu. I. Merzlyakov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,496 | KOU-8.46 | Kourovka Notebook Problem 8.46 | Describe the automorphisms of the symplectic group $Sp_{2n}$
over an arbitrary commutative ring. *Conjecture:* they are all
standard. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.46.
Proposed by: Yu. I. Merzlyakov
Published resolution:
The conjecture was proved (V. M. Petechuk, *Algebra and
Logic*, **22** (1983), 397--405). | 2 | solved | Yu. I. Merzlyakov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 proved (V. M. Petechuk, *Algebra and\nLogic*, **22** (1983), 397--405)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,497 | KOU-8.47 | Kourovka Notebook Problem 8.47 | Do there exist finitely presented soluble groups in which the
maximum condition for normal subgroups fails but all central sections
are finitely generated? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.47.
Proposed by: Yu. I. Merzlyakov
Published resolution:
Yes (Yu. V. Sosnovskiĭ, *Math. Notes*, **36** (1984),
577--580). | 2 | solved | Yu. I. Merzlyakov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (Yu. V. Sosnovskiĭ, *Math. Notes*, **36** (1984),\n577--580)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,498 | KOU-8.48 | Kourovka Notebook Problem 8.48 | If a finite group $G$ can be written as the product of two
soluble subgroups of odd index, then is $G$ soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.48.
Proposed by: V. S. Monakhov
Published resolution:
Yes, it is (mod CFSG) (L. S. Kazarin, *Ukrain. Math. J.*, **43** (1991), 883--886). | 2 | solved | V. S. Monakhov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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) (L. S. Kazarin, *Ukrain. Math. J.*, **43** (1991), 883--886)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,499 | KOU-8.49 | Kourovka Notebook Problem 8.49 | Let $G$ be a $p $-group acting transitively as a permutation
group on a set $\Omega$, let $F$ be a field of characteristic $p$, and
regard $F\Omega$ as an $FG$-module. Then do the descending and
ascending Loewy series of $F\Omega$ coincide? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.49.
Proposed by: P. M. Neumann
Published resolution:
Yes, they do (J. L. Alperin, *Quart. J. Math. Oxford (2)*,
**39**, no. 154 (1988), 129--133). | 2 | solved | P. M. Neumann | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, they do (J. L. Alperin, *Quart. J. Math. Oxford (2)*,\n**39**, no. 154 (1988), 129--133)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,500 | KOU-8.50 | Kourovka Notebook Problem 8.50 | At a conference in Oberwolfach in 1979 I
exhibited a finitely generated soluble group $G$ (of derived
length 3) and a non-zero cyclic ${\mathbb{Z}G $-module $V$ such that
$V \cong V \oplus V$. Can this be done with $G$ metabelian? I
conjecture that it cannot. For background of this problem and for
details of the constr... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.50.
Proposed by: P. M. Neumann | 2 | open | P. M. Neumann | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,501 | KOU-8.51 | Kourovka Notebook Problem 8.51 | (J. McKay). If $G$ is a finite group and $p$ a
prime let $ m_p(G)$ denote the number of ordinary irreducible
characters of $G$ whose degree is not divisible by $p$. Let $P$ be
a Sylow $p
$-subgroup of $G$. Is it true that
$m_p(G)=m_p(N_G(P))$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.51.
Proposed by: J. B. Olsson
Discussion and literature:
[Solved] Yes, it is true (M. Cabanes, B. Sp\"ath, *Preprint*, 2024, https://arxiv.org/abs/2410.20392) | 2 | solved | J. B. Olsson | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (M. Cabanes, B. Sp\\\"ath, *Preprint*, 2024, https://arxiv.org/abs/2410.20392)"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,502 | KOU-8.52 | Kourovka Notebook Problem 8.52 | (Well-known problem). Do there exist infinite
finitely presented periodic groups? Compare with 6.3. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.52.
Proposed by: A. Yu. Olshanskii | 2 | open | A. Yu. Olshanskii | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,503 | KOU-8.53 | Kourovka Notebook Problem 8.53 | a) Let $n$ be a sufficiently large odd number.
Describe the automorphisms of the free Burnside group $B(m,n)$
of exponent $n$ on $m$ generators.
[Archived solved part(s) of 8.53:]
b) Let $n$ be a sufficiently large odd number. Is it true
that every non-cyclic subgroup of $B(m, n)$ has a subgroup
isomorphic to $B(2, n)... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 8.53.
Proposed by: A. Yu. Olshanskii
Archive of Solved Problems entry:
Yes, it is true for odd $n\geq 1003$
(V. S. Atabekyan, *Izv. Math.*, **73**, no. 5 (200... | 2 | partially_solved | A. Yu. Olshanskii | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (Yes, it is true for odd $n\\geq 1003$\n(V. S. Atabekyan, *Izv. Math.*, **73**, no. 5 (2009), 861--892).); 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 |
700,504 | KOU-8.54 | Kourovka Notebook Problem 8.54 | a) (Well-known problem). Classify metabelian
varieties of groups (or show that this is, in a certain sense, a
``wild'' problem).
b) Describe the identities of 2-generated
metabelian groups,
that is, classify varieties generated by such groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.54.
Proposed by: A. Yu. Olshanskii
Discussion and literature:
[Unverified AI claim] b) These varieties are classified (unverified AI solution: Achyuth Jayadeva... | 2 | open | A. Yu. Olshanskii | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] b) These varieties are classified (unverified AI solution: Achyuth Jayadevan, *Pr... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,505 | KOU-8.55 | Kourovka Notebook Problem 8.55 | It is easy to see that the set of all
quasivarieties of groups, in each of which quasivarieties a
non-trivial identity holds, is a semigroup under multiplication of
quasivarieties. Is this semigroup free? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.55.
Proposed by: A. Yu. Olshanskii | 2 | open | A. Yu. Olshanskii | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,506 | KOU-8.56 | Kourovka Notebook Problem 8.56 | Let $X$ be a finite set and $f$ a mapping from the set of
subsets of $X$ to
the positive integers. Under the requirement that in a group
generated by $X$, every subgroup $\left< Y\right>$, \
$Y\subseteq X$, be nilpotent of class $\leq f(Y)$, is it true
that the free group $G_f$ relative to this condition is
torsion-fre... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.56.
Proposed by: A. Yu. Olshanskii
Published resolution:
Not always (V. V. Bludov, V. F. Kleimenov, E. V. Khlamov, *Algebra and Logic*, **29** (1990), 95--96). | 2 | solved | A. Yu. Olshanskii | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (V. V. Bludov, V. F. Kleimenov, E. V. Khlamov, *Algebra and Logic*, **29** (1990), 95--96)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,507 | KOU-8.58 | Kourovka Notebook Problem 8.58 | Does a locally compact locally nilpotent nonabelian group without elements of finite order contain a proper closed isolated normal subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.58.
Proposed by: V. M. Poletskikh | 2 | open | V. M. Poletskikh | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,508 | KOU-8.59 | Kourovka Notebook Problem 8.59 | Suppose that, in a locally compact locally
nilpotent group $G$, all proper closed normal subgroups
are compact. Does
$G$ contain an open compact normal subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.59.
Proposed by: V. M. Poletskikh,
I. V. Protasov | 2 | open | V. M. Poletskikh,
I. V. Protasov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,509 | KOU-8.60 | Kourovka Notebook Problem 8.60 | Describe the locally compact primary locally
soluble groups which are covered by compact subgroups and all of
whose closed abelian subgroups have finite rank. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.60.
Proposed by: V. M. Poletskikh,
V. S. Charin | 2 | open | V. M. Poletskikh,
V. S. Charin | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,510 | KOU-8.61 | Kourovka Notebook Problem 8.61 | Suppose that a locally compact group $G$ contains a subgroup that is topologically isomorphic to the additve group of the field of real numbers with natural topology. Is the space of all closed subgroups of $G$ connected in the Chabauty topology? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.61.
Proposed by: I. V. Protasov
Published resolution:
No, not always: let $H$ be the group of matrices of the form
$\begin{pmatrix}
1&z_1&r\\
0&1&z_2\\
0&0&1\end{pmatrix}$, where... | 2 | solved | I. V. Protasov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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: let $H$ be the group of matrices of the form\n$\\begin{pmatrix}\n1&z_1&r\\\\\n0&1&z_2\\\\\n0&0&1\\end{pmatrix}$, where $z_1,z_2\\in \\mathbb{Z}$ ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,511 | KOU-8.62 | Kourovka Notebook Problem 8.62 | Describe the locally compact locally pronilpotent
groups for which the space of closed normal subgroups is compact
in the $ E $-topology. The corresponding problem has been solved
for discrete groups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.62.
Proposed by: I. V. Protasov | 2 | open | I. V. Protasov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,512 | KOU-8.63 | Kourovka Notebook Problem 8.63 | Suppose that the space of all closed subgroups of a locally
compact group $G$ is $\sigma$-compact in the $E$-topology. Is
it true that the set of closed non-compact subgroups of $G$ is
at most countable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.63.
Proposed by: I. V. Protasov
Published resolution:
Yes, it is (A. G. Piskunov, *Ukrain. J. Math.*, **40**
(1988), 679--683). | 2 | solved | I. V. Protasov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. G. Piskunov, *Ukrain. J. Math.*, **40**\n(1988), 679--683)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,513 | KOU-8.64 | Kourovka Notebook Problem 8.64 | Does the class of all finite groups possess an
independent basis of quasiidentities? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.64.
Proposed by: A. K. Rumyantsev, D. M. Smirnov | 2 | open | A. K. Rumyantsev, D. M. Smirnov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,514 | KOU-8.66 | Kourovka Notebook Problem 8.66 | Construct examples of residually finite groups which
would separate Shunkov's classes of groups with $(a,b)
$-finiteness condition, (weakly) conjugacy biprimitively finite
groups and (weakly) biprimitively finite groups (see, in particular,
6.57). Can one derive such examples from Golod's construction? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.66.
Proposed by: A. I. Sozutov
Published resolution:
Such examples are constructed (L. Hammoudi, *Nil-alg\`ebres
non-nilpotentes et groupes p\'eriodiques infinis $($Doctor
Thesis... | 2 | solved | A. I. Sozutov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Such examples are constructed (L. Hammoudi, *Nil-alg\\`ebres\nnon-nilpotentes et groupes p\\'eriodiques infinis $($Doctor\nThesis*), Strasbourg, 1996; \\ A. V. R... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,515 | KOU-8.67 | Kourovka Notebook Problem 8.67 | Do there exist Golod groups all of whose abelian
subgroups have finite ranks? Here a *Golod group* means a
finitely generated non-nilpotent subgroup of the adjoint group
of a nil-ring. The answer is negative for the whole adjoint
group of a nil-ring (Ya. P. Sysak, *Abstracts of the 17th
All--Union Algebraic Conf., part... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.67.
Proposed by: A. I. Sozutov | 2 | open | A. I. Sozutov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,516 | KOU-8.68 | Kourovka Notebook Problem 8.68 | Let $G=\left< a,b\mid r=1\right> $ where $r$ is
a cyclically reduced word that is not a proper power of any word in
$a,b$. If $G$ is residually finite, is $G_t=\left<a,b\mid
r^t=1\right> $, $t > 1$, residually finite?
*Editors' comment:* An analogous fact was
proved for relative one-relator groups (S. J. Pride, *Proc.... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.68.
Proposed by: C. Y. Tang
Published resolution:
Yes, it is (D. T. Wise, *The structure of groups with a quasiconvex hierarchy*,
Annals of Mathematics Studies, **209**, Princeto... | 2 | solved | C. Y. Tang | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (D. T. Wise, *The structure of groups with a quasiconvex hierarchy*,\nAnnals of Mathematics Studies, **209**, Princeton University Press, Princeton, N... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,517 | KOU-8.69 | Kourovka Notebook Problem 8.69 | Is every 1-relator group with non-trivial torsion
conjugacy separable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.69.
Proposed by: C. Y. Tang
Published resolution:
Yes, it is (A. Minasyan, P. Zalesskii, *J. Algebra*, **382** (2013), 39--45). | 2 | solved | C. Y. Tang | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. Minasyan, P. Zalesskii, *J. Algebra*, **382** (2013), 39--45)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,518 | KOU-8.70 | Kourovka Notebook Problem 8.70 | Let $A, B$ be polycyclic-by-finite groups. Let
$G=A*_H B$ where $H$ is cyclic. Is $G$ conjugacy separable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.70.
Proposed by: C. Y. Tang
Published resolution:
Yes, it is (L. Ribes, D. Segal, P. A. Zalesskii, *J. London Math. Soc. (2)*, **57**, no. 3 (1998), 609--628). | 2 | solved | C. Y. Tang | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (L. Ribes, D. Segal, P. A. Zalesskii, *J. London Math. Soc. (2)*, **57**, no. 3 (1998), 609--628)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,519 | KOU-8.71 | Kourovka Notebook Problem 8.71 | Is every countable conjugacy-separable group embeddable in a
2-generator conjugacy-separable group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.71.
Proposed by: C. Y. Tang
Published resolution:
Yes, it is (V. A. Roman'kov, *Embedding theorems for residually
finite groups*, Preprint 84-515, Comp. Centre, Novosibirsk, 1984... | 2 | solved | C. Y. Tang | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. A. Roman'kov, *Embedding theorems for residually\nfinite groups*, Preprint 84-515, Comp. Centre, Novosibirsk, 1984\n(Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,520 | KOU-8.72 | Kourovka Notebook Problem 8.72 | Does there exist a finitely presented group $G$
which is not free or cyclic of prime order, having the property
that every proper subgroup of $G$ is free? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.72.
Proposed by: C. Y. Tang | 2 | open | C. Y. Tang | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,521 | KOU-8.73 | Kourovka Notebook Problem 8.73 | We say that a finite group $G$ *separates cyclic
subgroups* if, for any cyclic subgroups $A$ and $B$ of $G$, there
is a $g\in G$ such that $ A\cap B^g=1$. Is it true that $G$ separates
cyclic subgroups if and only if $G$ has no non-trivial cyclic
normal subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.73.
Proposed by: J. G. Thompson
Published resolution:
Not always. For $p_1=2$, $p_2=5$, $p_3=11$, $p_4=17$ let
$R_i$ be an elementary abelian group of order $p_i^2$ and $\varphi
... | 2 | solved | J. G. Thompson | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 $p_1=2$, $p_2=5$, $p_3=11$, $p_4=17$ let\n$R_i$ be an elementary abelian group of order $p_i^2$ and $\\varphi\n_i$ a regular automorphism of orde... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,522 | KOU-8.74 | Kourovka Notebook Problem 8.74 | A subnormal subgroup $H\mbox{$\lhd\lhd$} G$
of a group $G$ is said
to be *good* if and only if $\left< H,J\right>
\mbox{$\lhd\lhd$} G$ whenever $J \mbox{$\lhd\lhd$} G$. Is it
true that when $H$ and $K$ are good subnormal subgroups of $G$,
then $H \cap K$ is good? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.74.
Proposed by: J. G. Thompson
Discussion and literature:
[Solved] No, not always (K. Muliarchyk, *Preprint of 14 September 2026*, https://kourovkanotebookorg... | 2 | solved | J. G. Thompson | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (K. Muliarchyk, *Preprint of 14 September 2026*, https://kourovkanotebookorg.wordpress.co... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,523 | KOU-8.75 | Kourovka Notebook Problem 8.75 | (A known problem). Suppose $G$ is a finite
primitive permutation group on $\Omega$, and $\alpha , \beta $ are
distinct points of $\Omega$. Does there exist an element $g \in G
$ such that $\alpha g=\beta $ and $g$ fixes no point of $\Omega$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.75.
Proposed by: J. G. Thompson
Discussion and literature:
[Solved] Not always (P. M\"uller, *Preprint*, 2023, https://arxiv.org/pdf/2304.08459.pdf). | 2 | solved | J. G. Thompson | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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] Not always (P. M\\\"uller, *Preprint*, 2023, https://arxiv.org/pdf/2304.08459.pdf)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,524 | KOU-8.76 | Kourovka Notebook Problem 8.76 | Give a realistic upper bound for the torsion-free rank of a
finitely generated nilpotent group in terms of the ranks of its
abelian subgroups. More precisely, for each integer $n$ let $f(n)$
be the largest integer $h$ such that there is a finitely generated
nilpotent group of torsion-free rank $h$ with the property tha... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.76.
Proposed by: John S. Wilson
Published resolution:
Yes, it is (M. V. Milenteva, *J. Group Theory*, **7**,
no. 3 (2004), 403--408). | 2 | solved | John S. Wilson | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. V. Milenteva, *J. Group Theory*, **7**,\nno. 3 (2004), 403--408)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,525 | KOU-8.77 | Kourovka Notebook Problem 8.77 | Do there exist strongly regular graphs with
parameters $\lambda =0$, \ $\mu =2$ of degree $k > 10$? Such graphs
are known for $k=5$ and $ k=10$, their automorphism groups are
primitive permutation groups of rank 3. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.77.
Proposed by: D. G. Fon-Der-Flaass | 2 | open | D. G. Fon-Der-Flaass | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,526 | KOU-8.78 | Kourovka Notebook Problem 8.78 | It is known that there exists a countable locally
finite group that contains a copy of every other countable locally
finite group. For which other classes of countable groups does a
similar ``largest'' group exist? In particular, what about periodic
locally soluble groups? periodic locally nilpotent groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.78.
Proposed by: B. Hartley | 2 | open | B. Hartley | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,527 | KOU-8.79 | Kourovka Notebook Problem 8.79 | Does there exist a countable infinite locally
finite group $G$ that is complete, in the sense that $G$ has
trivial centre and no outer automorphisms? An uncountable one
exists (K. K. Hickin, *Trans. Amer. Math. Soc.*, **239**
(1978), 213--227). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.79.
Proposed by: B. Hartley | 2 | open | B. Hartley | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,528 | KOU-8.80 | Kourovka Notebook Problem 8.80 | Let $G$ be a locally finite group containing a
maximal subgroup which is Chernikov. Is $G$ almost soluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.80.
Proposed by: B. Hartley
Published resolution:
Yes, it is (B. Hartley, *Algebra and Logic*, **37**, no. 1
(1998), 101--106). | 2 | solved | B. Hartley | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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. Hartley, *Algebra and Logic*, **37**, no. 1\n(1998), 101--106)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,529 | KOU-8.81 | Kourovka Notebook Problem 8.81 | Let $G$ be a finite $p $-group admitting an automorphism $\alpha
$ of prime order $q$ with $|C_G(a)|\leq n$.
a) If $ p = q$, then does $ G$ have a nilpotent subgroup of
class at most 2 and index bounded by a function of $n$?
b) If $p\ne q$, then does $G$ have a nilpotent subgroup of
class bounded by a function of $q$... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.81.
Proposed by: B. Hartley
Published resolution:
a) Not necessarily (E. I. Khukhro, *Math. Notes*, **38**
(1985), 867--870).
b) Yes, it does (E. I. Khukhro, *Math. USSR--Sb.*, ... | 2 | solved | B. Hartley | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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) Not necessarily (E. I. Khukhro, *Math. Notes*, **38**\n(1985), 867--870).\n\nb) Yes, it does (E. I. Khukhro, *Math. USSR--Sb.*, **71**,\nno. 1 (1992), 51--63)... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,530 | KOU-8.82 | Kourovka Notebook Problem 8.82 | Let ${\mathfrak{H}}={\mathbb{C} \times {\mathbb{R} =\{
(z,r)\mid z \in {\mathbb{C},\; r > 0\}$ be the three-dimensional
Poincar\'e space which admits the following action of the group
$SL_2({\mathbb{C})$:
$$(z,r)
\left(\!\!\begin{array}{cc}a&b\cr
c&d\end{array}\!\!\right)=\left(
\frac{(az+b)(\overline{cz+d})+a \overlin... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.82.
Proposed by: H. Helling | 2 | open | H. Helling | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,531 | KOU-8.83 | Kourovka Notebook Problem 8.83 | In the notation of 8.82, for $r(\pi ) > 0$, one
can define via Hecke algebras a formal Dirichlet series with Euler
multiplication (see G. Shimura, *Introduction to the
arithmetic theory of automorphic functions*, Princeton Univ.
Press, 1971). Does there exist an algebraic Hasse--Weil variety
whose $ \zeta $-function is... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.83.
Proposed by: H. Helling | 2 | open | H. Helling | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,532 | KOU-8.84 | Kourovka Notebook Problem 8.84 | We say that an automorphism $\varphi$ of the
group $G$ is a *pseudo-identity* if, for all $x \in G$,
there exists a finitely generated subgroup $K_x$ of $G$ such that
$x \in K_x$ and $\varphi |_{K_{\scriptstyle{x}}}$ is an
automorphism of $K_x$. Let $G$ be generated by subgroups $H, K$
and let $G$ be locally nilpotent.... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.84.
Proposed by: P. Hilton
Published resolution:
No, not necessarily (A. V. Yagzhev, *Math. Notes*, **56**,
no. 5 (1994), 1205--1207). | 2 | solved | P. Hilton | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 (A. V. Yagzhev, *Math. Notes*, **56**,\nno. 5 (1994), 1205--1207)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,533 | KOU-8.85 | Kourovka Notebook Problem 8.85 | Construct a finite $p
$-group $G$
whose Hughes subgroup $H_p(G)=\left< x \in G\mid \right.$ $\left.
|x|
\ne
p\right> $ is non-trivial and has index $p^3$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.85.
Proposed by: E. I. Khukhro | 2 | open | E. I. Khukhro | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,534 | KOU-8.86 | Kourovka Notebook Problem 8.86 | (Well-known problem). Suppose that all proper
closed subgroups of a locally compact locally nilpotent group $G$
are compact. Is $G$ abelian if it is non-compact? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 8.86.
Proposed by: V. S. Charin | 2 | open | V. S. Charin | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,535 | KOU-8.87 | Kourovka Notebook Problem 8.87 | Find all hereditary local formations ${\mathfrak{F}}$ of
finite groups satisfying the following condition: every finite
minimal non-${\mathfrak{F}} $-group is biprimary. A finite group is said
to be *biprimary* if its order is divisible by precisely two
distinct primes. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 8th issue (1982). Original problem number: 8.87.
Proposed by: L. A. Shemetkov
Published resolution:
They are found (V. N. Semenchuk, *Problems of Algebra: Proc.
of the Gomel' State Univ.*, no. 1 (15) (1999), 92--102
(Russia... | 2 | solved | L. A. Shemetkov | 1,982 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: They are found (V. N. Semenchuk, *Problems of Algebra: Proc.\nof the Gomel' State Univ.*, no. 1 (15) (1999), 92--102\n(Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,536 | KOU-9.1 | Kourovka Notebook Problem 9.1 | A group $G$ is said to be *potent* if, to each
$x\in G$ and each positive integer $n$ dividing the order of
$x$ (we suppose $ \infty$ is divisible by every positive integer),
there exists a finite homomorphic image of $G$ in which the image of
$x$ has order precisely $ n$. Is the free product of two potent
groups again... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.1.
Proposed by: R. B. J. T. Allenby | 2 | open | R. B. J. T. Allenby | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,537 | KOU-9.2 | Kourovka Notebook Problem 9.2 | Is $G=\left<a,b\mid a^l=b^m=(ab)^n=1\right> $
conjugacy separable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.2.
Proposed by: R. B. J. T. Allenby
Published resolution:
Yes, it is (B. Fine, G. Rosenberger, *Contemp. Math.*,
**109** (1990), 11--18). | 2 | solved | R. B. J. T. Allenby | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (B. Fine, G. Rosenberger, *Contemp. Math.*,\n**109** (1990), 11--18)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,538 | KOU-9.3 | Kourovka Notebook Problem 9.3 | Suppose that a countable locally finite group $G$
contains no proper subgroups isomorphic to $G$ itself and suppose
that all Sylow subgroups of $G$ are finite. Does $G$ possess a
non-trivial finite normal subgroup? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.3.
Proposed by: V. V. Belyaev
Published resolution:
Yes, it does
(S. D. Bell, {\it Locally finite groups with \v{C}ernikov Sylow
subgroups}, (*Ph. D. Thesis*), University of Manc... | 2 | solved | V. V. Belyaev | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does\n(S. D. Bell, {\\it Locally finite groups with \\v{C}ernikov Sylow\nsubgroups}, (*Ph. D. Thesis*), University of Manchester,\n1994)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,539 | KOU-9.4 | Kourovka Notebook Problem 9.4 | It is known that if ${\mathfrak{M}}$ is a variety
(quasivariety, pseudovariety) of groups, then the class $I{\mathfrak{M}}$
of all quasigroups that are isotopic to groups in ${\mathfrak{M}}$ is also
a variety (quasivariety, pseudovariety), and $I{\mathfrak{M}}$ is finitely
based if and only if ${\mathfrak{M}}$ is finit... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.4.
Proposed by: A. A. Gvaramiya
Discussion and literature:
[Solved] No, it is not true (AI solutions verified by K. Muliarchyk: \ M. Kinyon, J. Urban, *Preprin... | 2 | solved | A. A. Gvaramiya | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it is not true (AI solutions verified by K. Muliarchyk: \\ M. Kinyon, J. Urban, *Preprint of 17 Sept... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,540 | KOU-9.5 | Kourovka Notebook Problem 9.5 | A variety of groups is called *primitive* if
each of its subquasivarieties is a variety. Describe all primitive
varieties of groups. Is every primitive variety of groups locally
finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.5.
Proposed by: V. A. Gorbunov | 2 | open | V. A. Gorbunov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,541 | KOU-9.6 | Kourovka Notebook Problem 9.6 | Is it true that an independent basis of
quasiidentities of any finite group is finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.6.
Proposed by: V. A. Gorbunov | 2 | open | V. A. Gorbunov | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,542 | KOU-9.7 | Kourovka Notebook Problem 9.7 | (A. M. St\"epin). Does there exist an infinite
finitely generated amenable group of bounded exponent? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.7.
Proposed by: R. I. Grigorchuk | 2 | open | R. I. Grigorchuk | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,543 | KOU-9.8 | Kourovka Notebook Problem 9.8 | Does there exist a finitely generated simple group
of intermediate growth? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.8.
Proposed by: R. I. Grigorchuk
Published resolution:
Yes, it does (V. Nekrashevych, *Ann. Math.*, **87**, no. 3 (2018), 667--719). | 2 | solved | R. I. Grigorchuk | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it does (V. Nekrashevych, *Ann. Math.*, **87**, no. 3 (2018), 667--719)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,544 | KOU-9.9 | Kourovka Notebook Problem 9.9 | Does there exist a finitely generated group which
is not nilpotent-by-finite and whose growth function has as a
majorant a function of the form $c^{\sqrt{n}}$ where $ c$ is a
constant greater than 1? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.9.
Proposed by: R. I. Grigorchuk | 2 | open | R. I. Grigorchuk | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,545 | KOU-9.10 | Kourovka Notebook Problem 9.10 | Do there exist
finitely generated groups different from ${\mathbb{Z}/2{\mathbb{Z}$ which
have precisely two conjugacy classes? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.10.
Proposed by: V. S. Guba
Published resolution:
Yes, there do
(D. Osin, *Ann. Math.*, **172**, no. 1 (2010), 1--39). | 2 | solved | V. S. Guba | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there do\n(D. Osin, *Ann. Math.*, **172**, no. 1 (2010), 1--39)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,546 | KOU-9.11 | Kourovka Notebook Problem 9.11 | Is an abelian minimal normal subgroup $A$ of a
group $G$ an elementary $p
$-group if the factor-group $
G/A$ is a soluble group of finite rank? The answer is affirmative
under the additional hypothesis that $ G/A$ is locally polycyclic
(D. I. Zaitsev, *Algebra and Logic*, **19**, no. 2
(1980), 94--105). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.11.
Proposed by: D. I. Zaitsev | 2 | open | D. I. Zaitsev | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,547 | KOU-9.12 | Kourovka Notebook Problem 9.12 | Is there a soluble group with the following properties:
torsion-free of finite rank, not finitely generated, and having
a faithful irreducible representation over a finite field? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.12.
Proposed by: D. I. Zaitsev
Published resolution:
Yes, there is (A. V. Tushev, *Ukrain. Math. J.*, **42**
(1990), 1233--1238). | 2 | solved | D. I. Zaitsev | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there is (A. V. Tushev, *Ukrain. Math. J.*, **42**\n(1990), 1233--1238)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,548 | KOU-9.13 | Kourovka Notebook Problem 9.13 | Is a soluble torsion-free group minimax if it
satisfies the weak minimum condition for normal subgroups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.13.
Proposed by: D. I. Zaitsev | 2 | open | D. I. Zaitsev | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,549 | KOU-9.14 | Kourovka Notebook Problem 9.14 | Is a group locally finite if it decomposes into a
product of periodic abelian subgroups which commute pairwise? The
answer is affirmative in the case of two subgroups. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.14.
Proposed by: D. I. Zaitsev | 2 | open | D. I. Zaitsev | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,550 | KOU-9.15 | Kourovka Notebook Problem 9.15 | Describe, without using CFSG, all subgroups $ L$
of a finite Chevalley group $G$ such that $G=PL$ for some
parabolic subgroup $P$ of $G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 9.15.
Proposed by: A. S. Kondratiev | 2 | open | A. S. Kondratiev | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,551 | KOU-9.16 | Kourovka Notebook Problem 9.16 | (Well-known problem). The *prime graph* of a finite group
$G$ is the graph with vertex set $\pi (G)$ and an edge joining
$p$ and $q$ if and only if $G$ has an element of order $pq$.
Describe all finite Chevalley groups over a field of
characteristic $2$ whose prime graph is not connected and
describe the connected comp... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.16.
Proposed by: A. S. Kondratiev
Published resolution:
This was done in (A. S. Kondratiev, *Math. USSR--Sb.*, **67**
(1990), 235--247). | 2 | solved | A. S. Kondratiev | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: This was done in (A. S. Kondratiev, *Math. USSR--Sb.*, **67**\n(1990), 235--247)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,552 | KOU-9.17 | Kourovka Notebook Problem 9.17 | b) Let $G$ be a locally normal residually finite
group. Does there exist a normal subgroup $H$ of $G$ which is
embeddable in a direct product of finite groups and is such that
$G/H$ is a divisible abelian group?
[Archived solved part(s) of 9.17:]
a) Let $G$ be a locally normal residually finite group. Can $G$
be embed... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 9.17.
Proposed by: L. A. Kurdachenko
Archive of Solved Problems entry:
Not always (L. A. Kurdachenko, *Math. Notes*, **39**
(1986), 273--279). | 2 | partially_solved | L. A. Kurdachenko | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (Not always (L. A. Kurdachenko, *Math. Notes*, **39**\n(1986), 273--279).); 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 |
700,553 | KOU-9.18 | Kourovka Notebook Problem 9.18 | Let ${\mathfrak{S}}_{\displaystyle{*}}$ be the smallest
normal
Fitting
class. Are there
Fitting classes which are maximal in ${\mathfrak{S}}_{\displaystyle{*}}$ (with
respect to
inclusion)? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 9th issue (1984). Original problem number: 9.18.
Proposed by: H. Lausch
Published resolution:
No, there are no such classes (N. T. Vorob'y\"ev, *Dokl.
Akad. Nauk Belorus. SSR*, **35**, no. 6 (1991), 485--487
(Russian)). | 2 | solved | H. Lausch | 1,984 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, there are no such classes (N. T. Vorob'y\\\"ev, *Dokl.\nAkad. Nauk Belorus. SSR*, **35**, no. 6 (1991), 485--487\n(Russian))."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
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