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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
700,954 | KOU-13.36 | Kourovka Notebook Problem 13.36 | For a finitely generated pro-$p
$-group $G$ set $a_n(G)=\mathrm{dim}_{ {\mathbb{F}_p}I^n/I^{n+1}$,
where $I$ is the augmentation ideal of the group ring
${\mathbb{F}_p[[G]]$.
We define the *growth* of $G$ to be the growth of the
sequence
$\{ a_n(G)\} _{n\in {\mathbb{N}}$.
a) If the growth of $G$ is exponential, does i... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 13.36.
Proposed by: O. V. Mel'nikov
Archive of Solved Problems entry:
b) Not necessarily. Let $G$ be the Nottingham
group over ${\mathbb{F}_p$. The growth of... | 2 | partially_solved | O. V. Mel'nikov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (b) Not necessarily. Let $G$ be the Nottingham\ngroup over ${\\mathbb{F}_p$. The growth of $c_n(G):=\\log_p |G:\\omega_n(G)|$,\nwhere $\\{\\omega_n(G)\\}$ is the ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,955 | KOU-13.37 | Kourovka Notebook Problem 13.37 | Let $G$ be a torsion-free pro-$p
$-group, $U$ an open subgroup of $G$. Suppose that $U$ is a
pro-$p
$-group with a single defining relation. Is it
true that then $G$ is also a pro-$p
$-group with a
single defining relation? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.37.
Proposed by: O. V. Mel'nikov | 2 | open | O. V. Mel'nikov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,956 | KOU-13.38 | Kourovka Notebook Problem 13.38 | Let $G=F/R$ be a pro-$p
$-group with one
defining relation, where
$R$ is the normal subgroup of a free pro-$p
$-group
$F$ generated by a single element $r\in F^p[F, F]$.
a) Suppose that $r=t^p$ for some $t\in F$; can $G$
contain a Demushkin group as a subgroup?
b) Do there exist two pro-$p
$-groups
$G_1\supset G_2$ w... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.38.
Proposed by: O. V. Mel'nikov
Published resolution:
a) Yes, it can. b) Yes,
they exist. The pro-$p $-group with presentation
$\left< a,b\mid [a,b]^p=1\right>$ contains the D... | 2 | solved | O. V. Mel'nikov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: a) Yes, it can. b) Yes,\nthey exist. The pro-$p $-group with presentation\n$\\left< a,b\\mid [a,b]^p=1\\right>$ contains the Demushkin group\n$\\left< x_1,\\ldot... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,957 | KOU-13.39 | Kourovka Notebook Problem 13.39 | Let $A$ be an associative ring with unity and with
torsion-free additive group, and let $F^A$ be the tensor product
of a free group $F$ by $A$ (A. G. Myasnikov,
V. N. Remeslennikov, *Siberian Math. J.*, **35**,
no. 5 (1994), 986--996);
then $F^A$ is a *free exponential group over $A$*; in
(A. G. Myasnikov, V. N. Remesl... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 13.39.
Proposed by: A. G. Myasnikov,
V. N. Remeslennikov
Archive of Solved Problems entry:
a) Yes, it is (A. Jaikin-Zapirain, *Ann. Sci. \'Ec. Norm. Sup\'er.... | 2 | partially_solved | A. G. Myasnikov,
V. N. Remeslennikov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Part(s) solved in Archive of Solved Problems (a) Yes, it is (A. Jaikin-Zapirain, *Ann. Sci. \\'Ec. Norm. Sup\\'er. (4)*, **57**, no. 4 (2024), 1101--1133).\n\nc) Yes, it is faithful (A. Jaikin-Zapirain, *Ann... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,958 | KOU-13.40 | Kourovka Notebook Problem 13.40 | A group $G$ is said to be $\omega$-*residually free* if, for every finite set of non-trivial
elements of $G$, there is a homomorphism of $G$ into a free group
such that the images of all these elements are non-trivial. Is
every finitely-generated $\omega$-residually free group
embeddable in a free ${\mathbb{Z} [x]$-gro... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.40.
Proposed by: A. G. Myasnikov, V. N. Remeslennikov
Published resolution:
Yes, it is (O. Kharlampovich,
A. Myasnikov, *J. Algebra*, **200** (1998), 472--570). | 2 | solved | A. G. Myasnikov, V. N. Remeslennikov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is (O. Kharlampovich,\nA. Myasnikov, *J. Algebra*, **200** (1998), 472--570)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,959 | KOU-13.41 | Kourovka Notebook Problem 13.41 | Is the elementary theory of the class of all groups
acting freely on $\Lambda $-trees decidable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.41.
Proposed by: A. G. Myasnikov, V. N. Remeslennikov | 2 | open | A. G. Myasnikov, V. N. Remeslennikov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,960 | KOU-13.42 | Kourovka Notebook Problem 13.42 | Prove that the tensor $A $-completion (see
A. G. Myasnikov, V. N. Remeslennikov, *Siberian Math. J.*, **35**, no. 5 (1994), 986--996)
of a free nilpotent group can be
non-nilpotent. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.42.
Proposed by: A. G. Myasnikov, V. N. Remeslennikov
Discussion and literature:
[Unverified AI claim] An example is produced (unverified AI solution: M. Kin... | 2 | open | A. G. Myasnikov, V. N. Remeslennikov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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] An example is produced (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,961 | KOU-13.43 | Kourovka Notebook Problem 13.43 | (G. R. Robinson). Let $G$ be a finite group and $B$
be a $p
$-block of characters of $G$. *Conjecture:* If the defect group $D=D(B)$ of the block $B$ is
non-abelian, and if $|D:Z(D)| = p^a$, then each character in $B$
has height strictly less than $a$.
*G. R. Robinson's comment of 2020*:
the conjecture is proved mod C... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.43.
Proposed by: J. Olsson
Discussion and literature:
[Solved] The conjecture is proved (R. Kessar, G. Malle, *J. London Math. Soc. (2)*, **111**, no. 2 (202... | 2 | solved | J. Olsson | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] The conjecture is proved (R. Kessar, G. Malle, *J. London Math. Soc. (2)*, **111**, no. 2 (2025), Articl... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,962 | KOU-13.44 | Kourovka Notebook Problem 13.44 | For any partition of an arbitrary group
$G$ into finitely many subsets
$G=A_1\cup \cdots \cup A_n$, there exists a subset of the
partition $A_i$ and a finite subset $F\subseteq G$, such that
$G=A_i^{-1}A_iF$ (I. V. Protasov, *Siberian Math. J.*, **34**, no. 5 (1993), 938--952). Can the subset $F$ always be chosen
so th... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.44.
Proposed by: I. V. Protasov | 2 | open | I. V. Protasov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,963 | KOU-13.45 | Kourovka Notebook Problem 13.45 | Every infinite group $G$ of regular cardinality
${\mathfrak{m}}$ can be partitioned into two subsets
$G=A_1\cup A_2$ so that
$A_1F\ne G$ and $A_2F\ne G$ for every subset
$F\subset
G$ of cardinality less than
${\mathfrak{m}}$. Is this statement true for groups of singular
cardinality? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.45.
Proposed by: I. V. Protasov
Published resolution:
No, not always. The answer depends on the algebraic structure of $G$. In particular, this is true for a free group, but
th... | 2 | solved | I. V. Protasov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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. The answer depends on the algebraic structure of $G$. In particular, this is true for a free group, but\nthe statement does not hold for every Ab... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,964 | KOU-13.46 | Kourovka Notebook Problem 13.46 | Can every uncountable abelian group of finite odd
exponent be partitioned into two subsets so that neither of them
contains cosets of infinite subgroups? Among countable abelian
groups, such partitions exist for groups with finitely many
involutions. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.46.
Proposed by: I. V. Protasov
Published resolution:
Yes, it can (E. G. Zelenyuk, *Math. Notes*, **67** (2000),
599--602). | 2 | solved | I. V. Protasov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it can (E. G. Zelenyuk, *Math. Notes*, **67** (2000),\n599--602)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,965 | KOU-13.47 | Kourovka Notebook Problem 13.47 | Can every countable abelian group with finitely
many involutions be partitioned into two subsets that are dense in
every group topology? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.47.
Proposed by: I. V. Protasov
Published resolution:
Yes, it can
(E. G. Zelenyuk, *Ukrain. Math. J.*, **51**, no. 1
(1999), 44--50). | 2 | solved | I. V. Protasov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it can\n(E. G. Zelenyuk, *Ukrain. Math. J.*, **51**, no. 1\n(1999), 44--50)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,966 | KOU-13.48 | Kourovka Notebook Problem 13.48 | (W. W. Comfort, J. van Mill). A
topological group is said to be *irresolvable* if every two of its dense subsets intersect
non-trivially. Does every non-discrete irresolvable group contain
an infinite subgroup of exponent 2? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.48.
Proposed by: I. V. Protasov | 2 | open | I. V. Protasov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,967 | KOU-13.49 | Kourovka Notebook Problem 13.49 | (V. I. Malykhin). Can a topological group be
partitioned into two dense subsets, if there are infinitely many free
ultrafilters on the group converging to the identity? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.49.
Proposed by: I. V. Protasov | 2 | open | I. V. Protasov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,968 | KOU-13.50 | Kourovka Notebook Problem 13.50 | Let ${\mathfrak{F}}$ be a local Fitting class. Is it true
that there are no maximal elements in the partially ordered by
inclusion set of the Fitting classes contained in
${\mathfrak{F}}$ and distinct from
${\mathfrak{F}}$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.50.
Proposed by: A. N. Skiba
Published resolution:
No, there may exist
maximal elements (N. V. Savel'eva, N. T. Vorob'ev, *Siberian Math. J.*, **49**, no. 6
(2008), 1124--1130)... | 2 | solved | A. N. Skiba | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 may exist\nmaximal elements (N. V. Savel'eva, N. T. Vorob'ev, *Siberian Math. J.*, **49**, no. 6\n(2008), 1124--1130)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,969 | KOU-13.51 | Kourovka Notebook Problem 13.51 | Is every finite modular lattice embeddable in the
lattice of formations of finite groups? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.51.
Proposed by: A. N. Skiba
Discussion and literature:
[Solved] No, not every (AI solution confirmed by A. N. Skiba: \ Achyuth Jayadevan, *Preprint*, 2026, ... | 2 | solved | A. N. Skiba | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not every (AI solution confirmed by A. N. Skiba: \\ Achyuth Jayadevan, *Preprint*, 2026, https://zen... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,970 | KOU-13.52 | Kourovka Notebook Problem 13.52 | The *dimension* of a finitely based variety of
algebras
${\mathcal{V}}$ is defined to be the maximal length of
a *basis* (that is, an independent generating set) of the
$SC $-theory $SC({\mathcal{V}})$, which consists of the strong Mal'cev
conditions satisfied on ${\mathcal{V}}$. The dimension is defined to
be infinite... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.52.
Proposed by: D. M. Smirnov | 2 | open | D. M. Smirnov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,971 | KOU-13.53 | Kourovka Notebook Problem 13.53 | Let $a, b$ be elements of finite order of the infinite
group $G=\left< a, b\right>$. Is it true that there are infinitely
many elements $g\in G$ such that the subgroup $\left< a,
b^g\right>$ is infinite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.53.
Proposed by: A. I. Sozutov | 2 | open | A. I. Sozutov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,972 | KOU-13.54 | Kourovka Notebook Problem 13.54 | a) Is it true that, for
$
p$ sufficiently large, every (finite) $p
$-group can be
a complement in some Frobenius group
(see 6.55)?
[Archived solved part(s) of 13.54:]
b) Is it true that every group is embeddable in
the kernel of some Frobenius group (see Archive, 6.53)? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 13.54.
Proposed by: A. I. Sozutov
Archive of Solved Problems entry:
Yes, it is true (V. V. Bludov, *Siberian Math. J.*, **38**,
no. 6 (1997), 1054--1056). | 2 | partially_solved | A. I. Sozutov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 (V. V. Bludov, *Siberian Math. J.*, **38**,\nno. 6 (1997), 1054--1056).); 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 |
700,973 | KOU-13.55 | Kourovka Notebook Problem 13.55 | Does there exist a
Golod group (see 9.76), which is isomorphic to an $AT $-group? For
a definition of an *$AT $-group* see (A. V. Rozhkov, *Math. Notes*, **40**, no. 5 (1986), 827--836). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.55.
Proposed by: A. V. Timofeenko | 2 | open | A. V. Timofeenko | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,974 | KOU-13.56 | Kourovka Notebook Problem 13.56 | (A. Shalev). Let $G$ be a finite $p
$-group of
sectional rank $r$, and $\varphi$ an automorphism of $G$ having
exactly $m$ fixed points. Is the derived length of $G$ bounded by
a function depending on $r$ and $m$ only? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.56.
Proposed by: E. I. Khukhro
Published resolution:
Yes, it is (A. Jaikin-Zapirain, *Israel J. Math.*, **129**
(2002), 209--220). | 2 | solved | E. I. Khukhro | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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. Jaikin-Zapirain, *Israel J. Math.*, **129**\n(2002), 209--220)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,975 | KOU-13.57 | Kourovka Notebook Problem 13.57 | Let $\varphi$ be an automorphism of prime order
$p$ of a finite group $G$ such that $C_G(\varphi )\leq
Z(G)$.
a) Is $G$ soluble if $p=3$? V. D. Mazurov and
T. L. Nedorezov proved in (*Algebra and Logic*, **35**, no. 6 (1996), 392--397) that the group $G$ is soluble for
$p=2$, and there are examples of unsoluble $G$ fo... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.57.
Proposed by: E. I. Khukhro | 2 | open | E. I. Khukhro | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,976 | KOU-13.58 | Kourovka Notebook Problem 13.58 | Let $\varphi$ be
an automorphism of prime order
$p$ of a nilpotent (periodic) group $G$ such that
$C_G(\varphi )$ is a group of finite sectional rank
$r$.
Does $G$ possess a normal subgroup $N$ which is nilpotent of class
bounded by a function of $p$ only and is such that
$G/N$ is a group of finite sectional rank bound... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.58.
Proposed by: E. I. Khukhro
Published resolution:
Yes, it does
(E. I. Khukhro, *J. London Math. Soc.*, **77** (2008),
130--148). | 2 | solved | E. I. Khukhro | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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(E. I. Khukhro, *J. London Math. Soc.*, **77** (2008),\n130--148)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,977 | KOU-13.59 | Kourovka Notebook Problem 13.59 | One can show that any extension of shape ${\mathbb{Z}^d. {\mathcal{S}L}_d({\mathbb{Z})$ is residually finite, unless possibly
$d=3$ or $d=5$. Are there in fact any extensions of this shape that
fail to be residually finite when $d=5$? As shown in (P. R. Hewitt,
*Groups/St. Andrews'93 in Galway, Vol. 2 $($London Math. S... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.59.
Proposed by: P. R. Hewitt | 2 | open | P. R. Hewitt | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,978 | KOU-13.60 | Kourovka Notebook Problem 13.60 | If a locally graded group $G$ is a product of
two subgroups of finite special rank, is $G$ of finite special
rank itself? The answer is affirmative if both factors are
periodic groups. (N. S. Chernikov, *Ukrain. Math. J.*, **42**, no. 7 (1990),
855--861). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.60.
Proposed by: N. S. Chernikov | 2 | open | N. S. Chernikov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,979 | KOU-13.61 | Kourovka Notebook Problem 13.61 | We call a metric space *narrow* if it is
quasiisometric to a subset of the real line, and *wide*
otherwise. Let $G$ be a group with the finite set of generators
$A$, and let $\Gamma =\Gamma (G,A)$ be the Cayley graph of $G$
with the natural metric. Suppose that, after deleting any narrow
subset $L$ from $\Gamma$, at mo... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.61.
Proposed by: V. A. Churkin
Published resolution:
No, it is not true in general (O. V. Bogopol'skii, *Preprint*, Novosibirsk, 1998 (Russian)). See also new problem 14.98. | 2 | solved | V. A. Churkin | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it is not true in general (O. V. Bogopol'skii, *Preprint*, Novosibirsk, 1998 (Russian)). See also new problem 14.98."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,980 | KOU-13.62 | Kourovka Notebook Problem 13.62 | Let $U$ and $V$ be non-cyclic subgroups of a
free group. Does the inclusion
$[U,
U]\leq [V, V]$ imply that $U\leq V$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.62.
Proposed by: V. P. Shaptala
Published resolution:
No, it does not
(M. J. Dunwoody, *Arch. Math.*, **16** (1965), 153--157). | 2 | solved | V. P. Shaptala | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it does not\n(M. J. Dunwoody, *Arch. Math.*, **16** (1965), 153--157)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,981 | KOU-13.63 | Kourovka Notebook Problem 13.63 | Let $\pi _e(G)$ denote the set of orders of elements of a group
$G$. For $\Gamma \subseteq {\mathbb{N}$ let $h(\Gamma )$ denote the
number of non-isomorphic finite groups $G$ with $\pi
_e(G)=\Gamma$. Is there a number $k$ such that, for every
$\Gamma$, either $h(\Gamma )\leq k$, or $h(\Gamma )=\infty$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.63.
Proposed by: W. J. Shi
Published resolution:
No, there is no such number: $h(\pi _e(L_3 (7^{3^r})))=r+1$ for any $r\geq 0$
(A. V. Zavarnitsine, *J. Group Theory*, **7**, no... | 2 | solved | W. J. Shi | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, there is no such number: $h(\\pi _e(L_3 (7^{3^r})))=r+1$ for any $r\\geq 0$\n(A. V. Zavarnitsine, *J. Group Theory*, **7**, no. 1\n(2004), 81--97)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,982 | KOU-13.64 | Kourovka Notebook Problem 13.64 | Let $\pi _e(G)$ denote the set of orders of elements of a group
$G$. A group $G$ is said to be an *$OC_n
$-group* if $\pi _e(G)=\{ 1, 2, \ldots , n\}$. Is every
$OC_n $-group locally finite? Do there exist infinite $OC_n
$-groups for $n\geq 7$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.64.
Proposed by: W. J. Shi | 2 | open | W. J. Shi | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,983 | KOU-13.65 | Kourovka Notebook Problem 13.65 | A finite simple group is called a *$K_n$-group* if its order is divisible by exactly
$n$ different primes.
The number of $K_3 $-groups is known to be 8.
The $K_4$-groups are classified mod CFSG (W. J. Shi, *in:
Group Theory in China $($Math. Appl.*, **365**), Kluwer,
1996, 163--181) and some
significant further results... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.65.
Proposed by: W. J. Shi | 2 | open | W. J. Shi | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,984 | KOU-13.66 | Kourovka Notebook Problem 13.66 | Let $F$ be a \
a) free; \
b) free metabelian \
group of finite rank. Let $M$ denote the set of all
endomorphisms of
$F$ with non-cyclic images.
Can one choose two elements $g, h\in F$ such that, for every $\varphi
, \psi \in M$, equalities $\varphi (g)=\psi (g)$ and $\varphi (h)=\psi (h)$
imply that $\varphi =\psi$,... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 13th issue (1995). Original problem number: 13.66.
Proposed by: V. E. Shpil'rain
Published resolution:
a) Yes, one can (D. Lee, *J. Algebra*, **247** (2002),
no. 2, 509--540).
b) No, not always (E. I. Timoshenko, *Math. No... | 2 | solved | V. E. Shpil'rain | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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, one can (D. Lee, *J. Algebra*, **247** (2002),\nno. 2, 509--540).\n\nb) No, not always (E. I. Timoshenko, *Math. Notes*,\n**62**, no. 5--6 (1998), 767--7... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,985 | KOU-13.67 | Kourovka Notebook Problem 13.67 | Let $G$ be a $T_0 $-group (see 12.101), $i$ an
involution in $G$ and $G=\left< i^g\mid g\in G\right>$. Is the
centralizer $C_G(i)$ residually periodic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 13.67.
Proposed by: V. P. Shunkov | 2 | open | V. P. Shunkov | 1,995 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,986 | KOU-14.1 | Kourovka Notebook Problem 14.1 | Suppose that $G$ is a finite group with no
non-trivial normal subgroups of odd order, and $\varphi$ is its
$2$-automorphism centralizing a Sylow $2$-subgroup of $G$. Is it
true that $\varphi ^2$ is an inner automorphism of $G$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.1.
Proposed by: R. Zh. Aleev
Published resolution:
Yes, it is true (G. Glauberman,
*Math. Z.*, **107** (1968), 1--20). | 2 | solved | R. Zh. Aleev | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is true (G. Glauberman,\n*Math. Z.*, **107** (1968), 1--20)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,987 | KOU-14.2 | Kourovka Notebook Problem 14.2 | (S. D. Berman). Prove that every automorphism of
the centre of the integral group ring of a finite group induces a
monomial permutation on the set of the class sums. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.2.
Proposed by: R. Zh. Aleev | 2 | open | R. Zh. Aleev | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,988 | KOU-14.3 | Kourovka Notebook Problem 14.3 | Is it true that every central unit of the
integral group ring of a finite group is a product of a central
element of the group and a symmetric central unit? (A unit is *symmetric* if it is fixed by the canonical antiinvolution that
transposes the
coefficients at the mutually inverse elements.) | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.3.
Proposed by: R. Zh. Aleev | 2 | open | R. Zh. Aleev | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,989 | KOU-14.4 | Kourovka Notebook Problem 14.4 | a) Is it true that there exists a nilpotent group
$G$ for which the lattice ${\mathcal{L}}(G)$ of all group topologies is
not modular? (It is known that for abelian groups the lattice
${\mathcal{L}}(G)$ is modular and that there are groups for which
this lattice is not modular: V. I. Arnautov, A. G. Topale,
*Izv. Akad.... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.4.
Proposed by: V. I. Arnautov
Discussion and literature:
[Solved] a) Yes, it is true (V. Arnautov, A. Topal\u a, {\it Bul. Acad. \c{S}tiin\c{t}.
Repub. Mold... | 2 | solved | V. I. Arnautov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] a) Yes, it is true (V. Arnautov, A. Topal\\u a, {\\it Bul. Acad. \\c{S}tiin\\c{t}.\nRepub. Moldova, Mat.... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,990 | KOU-14.5 | Kourovka Notebook Problem 14.5 | Let $G$ be an infinite group admitting
non-discrete Hausdorff group topologies, and ${\mathcal{L}}(G)$ the
lattice of all group topologies on $G$.
a) Is it true that for any natural number
$k$ there exists a non-refinable chain $\tau_0 < \tau_1<\cdots
< \tau_k$ of length $k$ of Hausdorff topologies in ${\mathcal{L}}(G... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.5.
Proposed by: V. I. Arnautov
Discussion and literature:
[Solved] b) The inequality does hold, but it is not sharp (V. I. Arnautov,
{\it Bull. Acad. \c{S}ti... | 2 | partially_solved | V. I. Arnautov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] b) The inequality does hold, but it is not sharp (V. I. Arnautov,\n{\\it Bull. Acad. \\c{S}tii... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,991 | KOU-14.6 | Kourovka Notebook Problem 14.6 | A group $\Gamma$ is said to have {\it Property
$P_{\rm{nai}}$} if, for any finite subset $F$ of $\Gamma \setminus
\{1\},$ there exists an element $y_0 \in \Gamma$ of infinite order
such that, for each $x \in F$, the canonical epimorphism from the
free product $\langle x \rangle * \langle y_0\rangle$ onto the
subgroup $... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.6.
Proposed by: P. de la Harpe | 2 | open | P. de la Harpe | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,992 | KOU-14.7 | Kourovka Notebook Problem 14.7 | Let $\Gamma_g$ be the fundamental group of
a closed surface of genus $g \geq 2.$ For each finite system $S$
of generators of $\Gamma_g,$ let $\beta_S(n)$ denote the number of
elements of $\Gamma _g$ which can be written as products of at
most $n$ elements of $S \cup S^{-1},$ and let $\omega (\Gamma
_g,S) = \limsup_{n \... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.7.
Proposed by: P. de la Harpe | 2 | open | P. de la Harpe | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,993 | KOU-14.8 | Kourovka Notebook Problem 14.8 | Let $G$ denote the group of germs at $+\infty$ of
orientation-preserving homeomorphisms of the real line ${\mathbb{R}$.
Let $\alpha \in G$ be the germ of $x \mapsto x+1.$ What are the germs
$\beta \in G$ for which the subgroup $\langle \alpha , \beta \rangle$ of
$G$ generated by $\alpha$ and $\beta$ is free of rank $2$... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.8.
Proposed by: P. de la Harpe | 2 | open | P. de la Harpe | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,994 | KOU-14.9 | Kourovka Notebook Problem 14.9 | (Well-known problem). Let $W^*(F_k)$ denote the von Neumann
algebra of the free group of rank $k \in \{ 2, 3, \ldots ,\aleph_0 \}.$
Is $W^*(F_k)$ isomorphic to $W^*(F_l)$ for $k \ne l$?
For a group $G$, recall that $W^*(G)$ is an
appropriate completion of the group algebra ${\mathbb{C} G$ (see
e. g. S. Sakai, *$C^*$-a... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.9.
Proposed by: P. de la Harpe | 2 | open | P. de la Harpe | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,995 | KOU-14.10 | Kourovka Notebook Problem 14.10 | c) Find an explicit and *"
natural"* \ finitely presented group $\Gamma_n$ and an
embedding of $GL_n({\mathbb{Q})$ in $\Gamma _n$.
Another phrasing of the same
problems is: find a simplicial complex $X$ which covers a finite
complex such that the fundamental group of $X$ is ${\mathbb{Q}$ or,
respectively, $GL_n({\math... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 14.10.
Proposed by: P. de la Harpe
Archive of Solved Problems entry:
a) Such an embedding is found (J. Belk, J. Hyde, F. Matucci, *Bull. Amer. Math. Soc.*, *... | 2 | partially_solved | P. de la Harpe | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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) Such an embedding is found (J. Belk, J. Hyde, F. Matucci, *Bull. Amer. Math. Soc.*, **59**, no. 4 (2022), 561--567).\n\nb) Such an embedding is found\n(V. H. ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,996 | KOU-14.11 | Kourovka Notebook Problem 14.11 | (Yu. I. Merzlyakov). It is a
well-known fact that for the ring
$R={\mathbb{Q} [x,y]$ the elementary group $E_2(R)$ is distinct from
$SL_2(R)$. Find a minimal subset $A\subseteq SL_2(R)$ such that
$\left< E_2(R), A\right> =SL_2(R)$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.11.
Proposed by: V. G. Bardakov | 2 | open | V. G. Bardakov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,997 | KOU-14.12 | Kourovka Notebook Problem 14.12 | (Yu. I. Merzlyakov, J. S. Birman). Is it
true that all braid groups $B_n$, $n\geq 3$, are
conjugacy separable? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.12.
Proposed by: V. G. Bardakov | 2 | open | V. G. Bardakov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
700,998 | KOU-14.13 | Kourovka Notebook Problem 14.13 | a)
By definition the *commutator length* of
an element $z$ of the derived subgroup of a group $G$ is the least
possible number of commutators from $G$ whose product is equal to
$z$. Does
there exist a simple group on which the commutator length is not
bounded?
b)
Does there exist a finitely presented simple
group on w... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.13.
Proposed by: V. G. Bardakov
Published resolution:
a) Simple groups with this
property were constructed in (J. Barge, \'{E}. Ghys, *Math.
Ann.*, **294** (1992), 235--265), a... | 2 | solved | V. G. Bardakov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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) Simple groups with this\nproperty were constructed in (J. Barge, \\'{E}. Ghys, *Math.\nAnn.*, **294** (1992), 235--265), and finitely generated simple ones\ni... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
700,999 | KOU-14.14 | Kourovka Notebook Problem 14.14 | (C. C. Edmunds, G. Rosenberger). We call
a pair of natural numbers $(k,m)$ *admissible* if
in the derived subgroup $F_2'$ of a free group $F_2$
there is an element $w$ such that the commutator length
of the element $w^m$ is equal to $k$.
Find all admissible pairs.
It is known that every pair $(k, 2)$, $k \geq 2$,
is a... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.14.
Proposed by: V. G. Bardakov | 2 | open | V. G. Bardakov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,000 | KOU-14.15 | Kourovka Notebook Problem 14.15 | For the automorphism group $A_n=\mathrm{Aut }F_n$
of a free group $F_n$ of rank $n\geq 3$ find the supremum
$k_n$ of the commutator lengths of the
elements of $A_n'$.
It is easy to show that
$k_2=\infty$.
On the other hand,
the commutator length of any element of the derived subgroup of
$\lim\limits_{\longrightarrow}A... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.15.
Proposed by: V. G. Bardakov
Discussion and literature:
[Solved] For $n\geq 3$ we have $k_n=\infty$ because $A_n$ is acylindrically hyperbolic (A. Genevoi... | 2 | solved | V. G. Bardakov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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] For $n\\geq 3$ we have $k_n=\\infty$ because $A_n$ is acylindrically hyperbolic (A. Genevois, C. Horbez,... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,001 | KOU-14.16 | Kourovka Notebook Problem 14.16 | Following Yu. I. Merzlyakov we define the *width* of a verbal subgroup $V(G)$ of a group $G$ with respect
to the set of words $V$ as the smallest $m\in {\mathbb{N}\cup \{
\infty\}$ such that every element of
$V(G)$
can be written as a product of $\leq m$ values of words from $V\cup
V^{-1}$. It is known that the width o... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.16.
Proposed by: V. G. Bardakov | 2 | open | V. G. Bardakov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,002 | KOU-14.17 | Kourovka Notebook Problem 14.17 | Let $\mathrm{IMA}(G)$ denote the subgroup of the automorphism group
$\mathrm{Aut }G$ consisting of all automorphisms that act trivially on the
second derived quotient $G/G''$. Find generators and defining
relations for
$\mathrm{IMA}(F_n)$, where $F_n$ is a free group of rank $n\geq 3$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.17.
Proposed by: V. G. Bardakov | 2 | open | V. G. Bardakov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,003 | KOU-14.18 | Kourovka Notebook Problem 14.18 | We say that a family of groups
${\mathcal{D}}$ *discriminates* a group $G$ if for any finite subset
$\{a_1,\ldots ,a_n\}\subseteq G\setminus\{ 1\}$ there exists a
group $D\in {\mathcal{D}}$ and a homomorphism $\varphi :G\rightarrow D$ such that
$a_j\varphi \ne 1$ for all $j=1,\ldots ,n$. Is every finitely generated
gro... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.18.
Proposed by: G. Baumslag, A. G. Myasnikov,
V. N. Remeslennikov | 2 | open | G. Baumslag, A. G. Myasnikov,
V. N. Remeslennikov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,004 | KOU-14.19 | Kourovka Notebook Problem 14.19 | We say that a group $G$ has the *Noetherian
Equation Property* if every system of equations over $G$ in finitely many variables
is equivalent to some finite part of it. Does an arbitrary hyperbolic
group have the Noetherian Equation Property?
*Comment of 2013:*
the conjecture holds for torsion-free hyperbolic groups (... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.19.
Proposed by: G. Baumslag, A. G. Myasnikov,
V. N. Remeslennikov
Discussion and literature:
[Solved] Yes, it does (R. Weidmann, C. Reinfeldt, *Ann. Math. B... | 2 | solved | G. Baumslag, A. G. Myasnikov,
V. N. Remeslennikov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it does (R. Weidmann, C. Reinfeldt, *Ann. Math. Blaise Pascal*, **26**, no. 2 (2019), 125--214)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,005 | KOU-14.20 | Kourovka Notebook Problem 14.20 | Does a free product of two groups have the
Noetherian Equation Property if this property is enjoyed by the factors? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.20.
Proposed by: G. Baumslag, A. G. Myasnikov,
V. N. Remeslennikov
Discussion and literature:
[Solved] Yes, it does (Z. Sela, *Preprint*, 2010, https://arxiv... | 2 | solved | G. Baumslag, A. G. Myasnikov,
V. N. Remeslennikov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it does (Z. Sela, *Preprint*, 2010, https://arxiv.org/abs/1012.0044).\nNote also that a free produc... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,006 | KOU-14.21 | Kourovka Notebook Problem 14.21 | Does a free pro-$p $-group have the Noetherian Equation Property? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.21.
Proposed by: G. Baumslag, A. G. Myasnikov,
V. N. Remeslennikov | 2 | open | G. Baumslag, A. G. Myasnikov,
V. N. Remeslennikov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,007 | KOU-14.22 | Kourovka Notebook Problem 14.22 | Prove that any irreducible system of equations
$S(x_1, \ldots,x_n) = 1$ with coefficients in a torsion-free
linear group $G$ is equivalent over $G$ to a finite system $T(x_1,
\ldots,x_n) = 1$ satisfying an analogue of Hilbert's
Nullstellensatz, i. e. $Rad_G(T) = \sqrt T$. This is true if $G$
is a free group (O. Kharlam... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.22.
Proposed by: G. Baumslag, A. G. Myasnikov,
V. N. Remeslennikov
Discussion and literature:
[Unverified AI claim] This is not true in general (unverified A... | 2 | open | G. Baumslag, A. G. Myasnikov,
V. N. Remeslennikov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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] This is not true in general (unverified AI solutions: Achyuth Jayadevan, *Preprin... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,008 | KOU-14.23 | Kourovka Notebook Problem 14.23 | Let $F_n$ be a free group with basis $\{
x_1,\ldots ,x_n \}$, and let $|\cdot |$ be the length function
with respect to this basis. For $\alpha\in \mathrm{Aut }F_n$ we put
$|\! |\alpha |\! |=\max \{
|\alpha (x_1)|,\ldots ,|\alpha (x_n)| \}$. Is it true that there is a
recursive function $f: {\mathbb{N} \rightarrow {\ma... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.23.
Proposed by: O. V. Bogopolski
Discussion and literature:
[Solved] No, this is not true if the function is required to work simultaneously for all ranks (... | 2 | solved | O. V. Bogopolski | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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, this is not true if the function is required to work simultaneously for all ranks (O. Kharlampovich,... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,009 | KOU-14.24 | Kourovka Notebook Problem 14.24 | Let $\mathrm{Aut }F_n$ be the automorphism group
of a free group of rank $n$ with norm $|\! |\cdot |\! |$ as in
14.23. Does there exist a recursive function
$f: {\mathbb{N} \times {\mathbb{N} \rightarrow {\mathbb{N}$ with the following property:
for any two conjugate elements
$\alpha,\beta\in \mathrm{Aut }F_n$ there is... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.24.
Proposed by: O. V. Bogopolski | 2 | open | O. V. Bogopolski | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,010 | KOU-14.25 | Kourovka Notebook Problem 14.25 | Let $qG$ denote the quasivariety generated by a
group $G$. Is it true that there exists a finitely generated group
$G$ such that the set of proper maximal subquasivarieties in $qG$
is infinite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.25.
Proposed by: A. I. Budkin
Published resolution:
Yes, it is (V. V. Bludov, *Algebra and Logic*, **41**,
no. 1 (2002), 1--8). | 2 | solved | A. I. Budkin | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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. V. Bludov, *Algebra and Logic*, **41**,\nno. 1 (2002), 1--8)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,011 | KOU-14.26 | Kourovka Notebook Problem 14.26 | A quasivariety ${\mathcal{M}}$ is {\it closed under
direct ${\mathbb{Z}$-wreath products} if the direct wreath product
$G\wr {\mathbb{Z}$ belongs to ${\mathcal{M}}$ for every $G\in {\mathcal{M}}$ (here
${\mathbb{Z}$ is an infinite cyclic group). Is the quasivariety
generated by the class of all nilpotent torsion-free g... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.26.
Proposed by: A. I. Budkin
Discussion and literature:
[Unverified AI claim] Yes, it is (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,
ht... | 2 | open | A. I. Budkin | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] Yes, it is (unverified AI solution: M. Kinyon, J. Urban, *Preprint*, 2026,\n\nhtt... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,012 | KOU-14.27 | Kourovka Notebook Problem 14.27 | Let $\Gamma$ be a group generated by a finite set $S$. Assume that
there exists a nested sequence $F_1\subset F_2\subset\cdots$ of
finite subsets of $\Gamma$ such that \ (i) $F_k\ne F_{k+1}$
for all $k\geq 1$, \ \ (ii) $\displaystyle{\Gamma=\bigcup_{k
\geq 1}F_k}$, \ \ (iii)
$\displaystyle{\lim_{k\to\infty}\frac{\vert\... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.27.
Proposed by: A. G. Vaillant
Published resolution:
No, not always. These properties are enjoyed by every group of
intermediate growth (V. G. Bardakov, *Algebra and Logic*,
*... | 2 | solved | A. G. Vaillant | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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. These properties are enjoyed by every group of\nintermediate growth (V. G. Bardakov, *Algebra and Logic*,\n**40**, no. 1 (2001), 12--16)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,013 | KOU-14.28 | Kourovka Notebook Problem 14.28 | Let ${\mathfrak{F}}$ be a soluble Fitting formation of
finite groups with Kegel's property, that is, ${\mathfrak{F}}$ contains
every finite group of the form $G=AB=BC=CA$ if $A,B,C$ are in ${\mathfrak{F}}$. Is ${\mathfrak{F}}$ a saturated formation?
*Comment of 2013:*
Some progress was made in (A. Ballester-Bolinches,... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.28.
Proposed by: A. F. Vasiliev | 2 | open | A. F. Vasiliev | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,014 | KOU-14.29 | Kourovka Notebook Problem 14.29 | Is there a soluble Fitting class of finite
groups ${\mathfrak{F}}$ such that ${\mathfrak{F}}$ is not a formation and $
A_{{\mathfrak{F}}}\cap B_{{\mathfrak{F}}}\subseteq G_{{\mathfrak{F}}}$ for every
finite soluble group of the form $G=AB$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.29.
Proposed by: A. F. Vasiliev | 2 | open | A. F. Vasiliev | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,015 | KOU-14.30 | Kourovka Notebook Problem 14.30 | Let $\mathrm{lFit}{\mathfrak{X}}$ be the local Fitting
class generated by a set of groups ${\mathfrak{X}}$ and let $\Psi (G)$
be the smallest normal subgroup of a finite group $G$ such that
$\mathrm{lFit}(\Psi (G)\cap M)=\mathrm{lFit}M$ for every
$M\mbox{$\lhd\lhd$} G$ (K. Doerk, P. Hauck, *Arch. Math.*,
**35**, no. 3 ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.30.
Proposed by: N. T. Vorob'\"ev | 2 | open | N. T. Vorob'\"ev | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,016 | KOU-14.31 | Kourovka Notebook Problem 14.31 | Is the lattice of Fitting subclasses of the Fitting
class generated by a finite soluble group finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.31.
Proposed by: N. T. Vorob'\"ev | 2 | open | N. T. Vorob'\"ev | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,017 | KOU-14.32 | Kourovka Notebook Problem 14.32 | Extending the classical
definition of formations, let us define a *formation* of
(not necessarily finite) groups as a nonempty class of groups
closed under taking homomorphic images and subdirect products with
finitely many factors. Must every first order axiomatizable
formation of groups be a variety? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.32.
Proposed by: \mbox{A. M. Gaglione, D. Spellman
Published resolution:
No, every variety of
groups that contains a finite nonsolvable member contains an
axiomatic subformatio... | 2 | solved | \mbox{A. M. Gaglione, D. Spellman | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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, every variety of\ngroups that contains a finite nonsolvable member contains an\naxiomatic subformation that is not a variety (K. A. Kearnes,\n*J. Group Theor... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,018 | KOU-14.33 | Kourovka Notebook Problem 14.33 | Does there exist a finitely presented
pro-$p $-group ($p$ being a prime) which contains an isomorphic
copy of every countably based pro-$p $-group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.33.
Proposed by: R. I. Grigorchuk
Published resolution:
Yes, there does:
the Nottingham group over ${\mathbb{F} _p$ for $p>2$, which
was shown to be finitely presented in
(M. V... | 2 | solved | R. I. Grigorchuk | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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:\nthe Nottingham group over ${\\mathbb{F} _p$ for $p>2$, which\nwas shown to be finitely presented in\n(M. V. Ershov, *J. London Math. Soc.*, **7... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,019 | KOU-14.34 | Kourovka Notebook Problem 14.34 | By definition, a locally compact group has the
*Kazhdan $T$-property* if the trivial representation is an
isolated point in the natural topological space of unitary
representations of the group. Does there exist a profinite group
with two dense discrete subgroups one of which is amenable, and
the other has the Kazhdan ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.34.
Proposed by: R. I. Grigorchuk, A. Lubotzky
Published resolution:
Yes, there does
(M. Kassabov, *Invent. Math.*, **170** (2007),
297--326; \ M. Ershov, A. Jaikin-Zapirain, *... | 2 | solved | R. I. Grigorchuk, A. Lubotzky | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, there does\n(M. Kassabov, *Invent. Math.*, **170** (2007),\n297--326; \\ M. Ershov, A. Jaikin-Zapirain, *Invent. Math.*, **179**, 303--347)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,020 | KOU-14.35 | Kourovka Notebook Problem 14.35 | Is every finitely presented group of prime exponent
finite? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.35.
Proposed by: N. D. Gupta | 2 | open | N. D. Gupta | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,021 | KOU-14.36 | Kourovka Notebook Problem 14.36 | A group $G$ is called a *$T$-group* if every
subnormal subgroup of $G$ is normal, while $G$ is said to be a {\it
$ \overline{\! T}$-group} if all of its subgroups are $T$-groups.
Is it true that every non-periodic locally graded $ \overline{\!
T}$-group must be abelian? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.36.
Proposed by: F. de Giovanni | 2 | open | F. de Giovanni | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,022 | KOU-14.37 | Kourovka Notebook Problem 14.37 | Let $G(n)$ be one of the classical groups
(special, orthogonal, or symplectic) of $(n\times n)$-matrices
over an infinite field
$K$ of non-zero characteristic, and $M(n)$ the space of
all $(n\times
n)$-matrices over $K$. The group
$G(n)$ acts diagonally by conjugation on the space
$M(n)^m=\underbrace{M(n)\oplus\cdots\o... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.37.
Proposed by: A. N. Zubkov | 2 | open | A. N. Zubkov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,023 | KOU-14.38 | Kourovka Notebook Problem 14.38 | For every pro-$p $-group $G$ of $(2\times
2)$-matrices for $p\ne 2$ an analogue of the Tits Alternative
holds: either $G$ is soluble, or the variety of pro-$p $-groups
generated by $G$ contains the group {\small $\overline{\left< \left(
\!\!\begin{array}{cc}1&t\cr 0&1
\end{array}\!\!\right), \left(\!\!
\begin{array}{cc... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.38.
Proposed by: A. N. Zubkov | 2 | open | A. N. Zubkov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,024 | KOU-14.39 | Kourovka Notebook Problem 14.39 | Let $F(V)$ be a free group of some variety of
pro-$p $-groups $V$. Is there a uniform bound for the exponents
of periodic elements in
$F(V)$? This is true if the variety
$V$ is metabelian (A. N. Zubkov, *Dr. Sci.
Diss.*,
Omsk, 1997 (Russian)). | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.39.
Proposed by: A. N. Zubkov | 2 | open | A. N. Zubkov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,025 | KOU-14.40 | Kourovka Notebook Problem 14.40 | Is a free pro-$p $-group representable as an
abstract group by matrices over a commutative-associative ring with
$1$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.40.
Proposed by: A. N. Zubkov | 2 | open | A. N. Zubkov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,026 | KOU-14.41 | Kourovka Notebook Problem 14.41 | A group $G$ is said to be *para-free* if
all factors $\gamma _i(G)/\gamma _{i+1}(G)$ of its lower central
series are isomorphic to the corresponding lower central factors
of some free group, and $\bigcap_{i=1}^{\infty}\gamma _i(G) =1$.
Is an arbitrary para-free group representable by matrices over a
commutative-associa... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.41.
Proposed by: A. N. Zubkov | 2 | open | A. N. Zubkov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,027 | KOU-14.42 | Kourovka Notebook Problem 14.42 | Is a free pro-$p$-group representable by
matrices over an associative-commutative profinite ring
with 1?
A negative answer is equivalent to the fact
that every linear pro-$p$-group satisfies a non-trivial
pro-$p$-identity. This is known to be true in dimension $2$
for $p\ne 2$ (A. N. Zubkov, *Siberian Math. J.*, **28*... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.42.
Proposed by: A. N. Zubkov,
V. N. Remeslennikov | 2 | open | A. N. Zubkov,
V. N. Remeslennikov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,028 | KOU-14.43 | Kourovka Notebook Problem 14.43 | Suppose that a finite group $G$ has the form
$G=AB $, where $A$ and $B$ are nilpotent subgroups of classes
$\alpha$ and $\beta$ respectively; then $G$ is soluble
(O. H. Kegel, H. Wielandt, 1961). Although the derived length
$\mathrm{dl}(G)$ of $G$ need not be bounded by $\alpha +\beta$ (see
Archive, 5.17), can one boun... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.43.
Proposed by: L. S. Kazarin | 2 | open | L. S. Kazarin | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,029 | KOU-14.44 | Kourovka Notebook Problem 14.44 | Let $k(X)$ denote the number of conjugacy
classes of a finite group $X$. Suppose that a finite group $G=AB$
is a product of two subgroups $A,B$ of coprime orders. Is it true
that $k(AB) \leq k(A)k(B)$?
Note that one cannot drop the
coprimeness condition and the answer is positive
if one of the subgroups is normal, see... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.44.
Proposed by: L. S. Kazarin, J. Sangroniz | 2 | open | L. S. Kazarin, J. Sangroniz | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
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"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 |
701,030 | KOU-14.45 | Kourovka Notebook Problem 14.45 | Does there exist a (non-abelian simple) linearly
right-orderable group all of whose proper subgroups are cyclic? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.45.
Proposed by: U. E. Kaljulaid | 2 | open | U. E. Kaljulaid | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,031 | KOU-14.46 | Kourovka Notebook Problem 14.46 | A finite group $G$ is said to be *almost
simple* if
$T\leq G\leq \mathrm{Aut}(T)$ for some nonabelian simple group
$T$. By definition, a *finite linear space* consists of a
set $V$ of points, together with a collection of $k$-element
subsets of $V$, called *lines* ($k\geq 3$), such that every
pair of points is containe... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.46.
Proposed by: A. Camina, C. E. Praeger | 2 | open | A. Camina, C. E. Praeger | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,032 | KOU-14.47 | Kourovka Notebook Problem 14.47 | Is the lattice of all soluble Fitting classes of finite groups modular? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.47.
Proposed by: S. F. Kamornikov, A. N. Skiba | 2 | open | S. F. Kamornikov, A. N. Skiba | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,033 | KOU-14.48 | Kourovka Notebook Problem 14.48 | If an equation over a free group
$F$ has no solution in
$F$, is there a finite quotient of $F$ in which the equation has no
solution? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.48.
Proposed by: L. Comerford
Published resolution:
No, not always (T. Coulbois, A. Khelif, *Proc. Amer. Math.
Soc.*, **127**, no. 4 (1999), 963--965). | 2 | solved | L. Comerford | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 (T. Coulbois, A. Khelif, *Proc. Amer. Math.\nSoc.*, **127**, no. 4 (1999), 963--965)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,034 | KOU-14.49 | Kourovka Notebook Problem 14.49 | Is $SL_3({\mathbb{Z} )$ a factor group of the modular group $
PSL_2({\mathbb{Z} )$? Since the latter is isomorphic to the free product
of two cyclic groups of orders 2 and 3,
the
question asks if $SL_3({\mathbb{Z} )$ can be generated by two
elements of orders $2$ and $3$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.49.
Proposed by: M. Conder
Published resolution:
No, it is not (M. C. Tamburini, P. Zucca, *Atti Accad. Naz.
Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.*,
**11... | 2 | solved | M. Conder | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: No, it is not (M. C. Tamburini, P. Zucca, *Atti Accad. Naz.\nLincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.*,\n**11**, no. 1 (2000), 5--7; \\ Ya. N... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,035 | KOU-14.50 | Kourovka Notebook Problem 14.50 | (Z. I. Borevich). A subgroup $A$ of a group $G$
is said to be *paranormal* (respectively, *polynormal*)
if $A^x\leq \left< A^u \mid u\in \left< A, A^x\right> \right> $
(respectively, $A^x\leq \left< A^u \mid u\in A^{\left< x
\right>}\right> $)
for any
$x\in G$.
Is every polynormal subgroup of a finite group paranormal? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.50.
Proposed by: A. S. Kondratiev
Published resolution:
Not always (V. I. Mysovskikh, *Dokl. Math.*, **60** (1999),
71--72). | 2 | solved | A. S. Kondratiev | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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. I. Mysovskikh, *Dokl. Math.*, **60** (1999),\n71--72)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,036 | KOU-14.51 | Kourovka Notebook Problem 14.51 | (Well-known problems). Is there a finite basis
for the identities of any
a) abelian-by-nilpotent group?
b) abelian-by-finite group?
c) abelian-by-(finite nilpotent) group? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.51.
Proposed by: A. N. Krasil'nikov | 2 | open | A. N. Krasil'nikov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,037 | KOU-14.52 | Kourovka Notebook Problem 14.52 | It is known that if a finitely generated group is
residually torsion-free nilpotent, then the group is residually
finite $p $-group, for every prime $p$. Is the converse true? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.52.
Proposed by: Yu. V. Kuz'min
Published resolution:
Not always (B. Hartley, in: *Symposia Mathematica, Bologna,
Vol. XVII, Convegno sui Gruppi Infiniti, INDAM, Roma, 1973*, A... | 2 | solved | Yu. V. Kuz'min | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 (B. Hartley, in: *Symposia Mathematica, Bologna,\nVol. XVII, Convegno sui Gruppi Infiniti, INDAM, Roma, 1973*, Academic\nPress, London, 1976, 225--234... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,038 | KOU-14.53 | Kourovka Notebook Problem 14.53 | *Conjecture:* Let $G$ be a profinite group
such that the set of solutions of the equation $x^n=1$ has
positive Haar measure. Then $G$ has an open subgroup $H$ and an
element $t$ such that all elements of the coset $tH$ have order
dividing $n$.
This is true in the case $n=2$. It would be
interesting to see whether simi... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.53.
Proposed by: L. Levai, L. Pyber | 2 | open | L. Levai, L. Pyber | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,039 | KOU-14.54 | Kourovka Notebook Problem 14.54 | Let $k(G)$ denote the number of conjugacy
classes of a finite group $G$. Is it true that $k(G)\leq |N|$ for
some nilpotent subgroup
$N$ of $G$? This is true if $G$ is simple; besides, always $k(G)\leq
|S|$ for
some soluble subgroup $S\leq G$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.54.
Proposed by: M. W. Liebeck, L. Pyber
Discussion and literature:
[Solved] \! No, not always (S. Tertooy, *Preprint*, 2026,
https://arxiv.org/abs/2606.214... | 2 | solved | M. W. Liebeck, L. Pyber | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 (S. Tertooy, *Preprint*, 2026,\n\nhttps://arxiv.org/abs/2606.21404)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,040 | KOU-14.55 | Kourovka Notebook Problem 14.55 | b)
Prove that the Nottingham group $J=N({\mathbb{Z}/p{\mathbb{Z})$ (as
defined in Archive, 12.24) is finitely presented
for $p=2$.
[Archived solved part(s) of 14.55:]
a)
Prove that the Nottingham group $J=N({\mathbb{Z}/p{\mathbb{Z})$ (as
defined in Archive, 12.24) is finitely presented
for $p>2$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 14.55.
Proposed by: C. R. Leedham-Green
Archive of Solved Problems entry:
Proved for $p>2$
(M. V. Ershov, *J. London Math. Soc.*, **71** (2005),
362--378). | 2 | partially_solved | C. R. Leedham-Green | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 (Proved for $p>2$\n(M. V. Ershov, *J. London Math. Soc.*, **71** (2005),\n362--378).); 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 |
701,041 | KOU-14.56 | Kourovka Notebook Problem 14.56 | Prove that if $G$ is an infinite pro-$p $-group with
$G/\gamma_{2p+1}(G)$ isomorphic to
$J/\gamma_{2p+1}(J)$ then $G$ is isomorphic to $J$, where $J$ is the
Nottingham group. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.56.
Proposed by: C. R. Leedham-Green | 2 | open | C. R. Leedham-Green | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
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"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,042 | KOU-14.57 | Kourovka Notebook Problem 14.57 | Describe the hereditarily just infinite
pro-$p $-groups of finite width.
Definitions: A pro-$p $-group $G$ has finite
*width* $p^d$ if $|\gamma _i(G)/\gamma _{i+1}(G)|\leq p^d$
for all $i$, and $G$ is *hereditarily just infinite* if $G$
is infinite and each of its open subgroups has no closed normal
subgroups of infin... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.57.
Proposed by: C. R. Leedham-Green | 2 | open | C. R. Leedham-Green | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,043 | KOU-14.58 | Kourovka Notebook Problem 14.58 | a) Suppose that $A$ is a periodic group of regular
automorphisms of an abelian group. Is $A$ cyclic if $A$ has prime
exponent?
[Archived solved part(s) of 14.58:]
b) Suppose that $A$ is a periodic group of regular
automorphisms of an abelian group.
Is $A$ finite if $A$ is generated by elements of order $3$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), updated in 21st issue (2026, arXiv:1401.0300v48). Original problem number: 14.58.
Proposed by: V. D. Mazurov
Archive of Solved Problems entry:
Yes, it is (A. Kh. Zhurtov, *Siberian Math. J.*, **41**,
no. 2 (2000), 268--275). | 2 | partially_solved | V. D. Mazurov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 (A. Kh. Zhurtov, *Siberian Math. J.*, **41**,\nno. 2 (2000), 268--275).); 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 |
701,044 | KOU-14.59 | Kourovka Notebook Problem 14.59 | Suppose that $G$ is a triply transitive
group in which a stabilizer of two points contains no involutions,
and a stabilizer of three points is trivial.
Is it true that $G$ is similar to $PGL_2(P)$ in its natural action
on the projective line
$P\cup \{ \infty \}$, for some field
$P$ of characteristic $2$? This is true u... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.59.
Proposed by: V. D. Mazurov | 2 | open | V. D. Mazurov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,045 | KOU-14.60 | Kourovka Notebook Problem 14.60 | Suppose that $H$ is a non-trivial normal
subgroup of a finite group $G$ such that the factor-group
$G/H$ is isomorphic to one of the simple groups $L_n(q)$, $n\geq 3$.
Is it true that $G$ has an element whose order is distinct from
the order of any element in $G/H$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.60.
Proposed by: V. D. Mazurov
Published resolution:
Yes, it is true: for $n\ne 4$ proved in (A. V. Zavarnitsine, *Siberian Math. J.*, **49** (2008),
246--256), and for $n=4$ i... | 2 | solved | V. D. Mazurov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Solved in Kourovka Notebook Archive of Solved Problems: Yes, it is true: for $n\\ne 4$ proved in (A. V. Zavarnitsine, *Siberian Math. J.*, **49** (2008),\n246--256), and for $n=4$ in (M. A. Grechkoseeva, S. V. Skresan... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,046 | KOU-14.61 | Kourovka Notebook Problem 14.61 | Determine all pairs $({\mathcal{S}},G)$, where $\mathcal{S}$ is a semipartial geometry and $G$ is an almost simple
flag-transitive group of automorphisms of $\mathcal{S}$. A system of
points and lines $(P,B)$ is a *semipartial geometry* with
parameters $(\alpha,s,t,\mu)$ if every point belongs to exactly
$t+1$
lines (t... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.61.
Proposed by: A. A. Makhn\"ev | 2 | open | A. A. Makhn\"ev | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,047 | KOU-14.62 | Kourovka Notebook Problem 14.62 | Suppose that $H$ is a non-soluble normal subgroup of a finite
group $G$. Does there always exist a maximal soluble subgroup $S$
of $H$ such that $G=H\cdot N_G(S)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.62.
Proposed by: V. S. Monakhov
Published resolution:
Yes, it always exists mod CFSG (V. I. Zenkov, V. S. Monakhov,
D. O. Revin, *Algebra and Logic*, **43**, no. 2 (2004),
102-... | 2 | solved | V. S. Monakhov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 always exists mod CFSG (V. I. Zenkov, V. S. Monakhov,\nD. O. Revin, *Algebra and Logic*, **43**, no. 2 (2004),\n102--108)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,048 | KOU-14.63 | Kourovka Notebook Problem 14.63 | What are the
composition factors of non-soluble finite groups all of whose
normalizers of Sylow subgroups are $2$-nilpotent, in particular,
supersoluble? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.63.
Proposed by: V. S. Monakhov
Published resolution:
Described
(L. S. Kazarin, A. A. Volochkov, *Mathematics in
Yaroslavl' Univ.: Coll. of Surveys to 30-th Anniv. of Math. Fac... | 2 | solved | V. S. Monakhov | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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: Described\n(L. S. Kazarin, A. A. Volochkov, *Mathematics in\nYaroslavl' Univ.: Coll. of Surveys to 30-th Anniv. of Math. Fac.*,\nYaroslavl', 2006, 243--255 (Russ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,049 | KOU-14.64 | Kourovka Notebook Problem 14.64 | (M. F. Newman). Classify the finite $5$-groups of
maximal class; computer calculations suggest some conjectures
(M. F. Newman, in: *Groups--Canberra, 1989 $($Lecture Notes
Math.*, **1456**), Springer, Berlin, 1990, 49--62).
*M. F. Newman's comment of 2025*:
Recent work on related conjectures can be found in (A. Cant, ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.64.
Proposed by: A. Moret\'o | 2 | open | A. Moret\'o | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,050 | KOU-14.65 | Kourovka Notebook Problem 14.65 | (Well-known problem).
For a finite group $G$ let $\rho(G)$ denote the
set of prime numbers dividing the order of some conjugacy class, and
$\sigma(G)$ the maximum number of primes dividing the order of some
conjugacy class. Is it true that $|\rho(G)|\leq3\sigma(G)$?
A possible linear bound for $|\rho(G)|$ in
terms of ... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.65.
Proposed by: A. Moret\'o | 2 | open | A. Moret\'o | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 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 |
701,051 | KOU-14.66 | Kourovka Notebook Problem 14.66 | (Well-known problem). Let $G$ be a finite soluble
group, $\pi(G)$ the set of primes dividing the order of $G$, and
$\nu(G)$ the maximum number of primes dividing the order of some
element. Does there exist a linear bound for $|\pi(G)|$ in terms
of $\nu(G)$? | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), Archive of Solved Problems, 14th issue (1999). Original problem number: 14.66.
Proposed by: A. Moret\'o
Published resolution:
Yes, it does (T. M. Keller, *J. Algebra*, **178**, no. 2
(1995),
643--652). | 2 | solved | A. Moret\'o | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (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 (T. M. Keller, *J. Algebra*, **178**, no. 2\n(1995),\n643--652)."
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,052 | KOU-14.67 | Kourovka Notebook Problem 14.67 | Suppose that $a$ is a non-trivial element
of a finite
group $G$ such that
$|C_{G}(a)| \geq |C_{G}(x)|$ for every non-trivial element
$x\in G$, and let
$H$ be a nilpotent
subgroup of $G$ which is normalized by $C_{G}(a)$.
Is it true that $H \leq C_{G}(a)$?
This is true if $H$ is abelian, or if $H$ is a
$p'$-group for so... | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.67.
Proposed by: I. T. Mukhamet'yanov, A. N. Fomin
Discussion and literature:
[Solved] No, not always (AI solution confirmed by K. Muliarchyk: \ Achyuth Jaya... | 2 | solved | I. T. Mukhamet'yanov, A. N. Fomin | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (AI solution confirmed by K. Muliarchyk: \\ Achyuth Jayadevan, *Preprint*, 2026, https://... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
701,053 | KOU-14.68 | Kourovka Notebook Problem 14.68 | (Well-known problem). Suppose that $F$ is an
automorphism of order $2$ of the polynomial ring $R_n= {\mathbb{C}
[x_1, \ldots ,x_n]$, $n> 2$. Does there exist an automorphism $G$
of $R_n$ such that $G^{-1}FG$ is a linear automorphism?
This is true for $n=2$. | Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 21st issue edition (2026, arXiv:1401.0300v48). Original problem number: 14.68.
Proposed by: M. V. Neshchadim | 2 | open | M. V. Neshchadim | 1,999 | 17 | 10 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
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