Kourovka Notebook Problem 17.112

Definitions and mathematical statement

For a group $G$, a subgroup $B\leq G$, and $x\in G$, write $B^x=x^{-1}Bx$. For subsets $U,V\subseteq G$, write $UV=\{uv:u\in U,\ v\in V\}$. A subgroup $A\leq G$ is $G$-permutable in $G$ if \[ \forall B\leq G\ \exists x\in G:\quad AB^x=B^xA. \] A subgroup $A\leq G$ is hereditarily $G$-permutable in $G$ if it is $E$-permutable in every subgroup $E$ of $G$ containing $A$; explicitly, \[ \forall E\ (A\leq E\leq G)\ \forall B\leq E\ \exists x\in E:\quad AB^x=B^xA. \] The conjugating element may depend on $B$, and, in the hereditary condition, on $E$ as well.

The source asks which finite non-abelian simple groups $G$ possess:

1. a non-trivial $G$-permutable subgroup;
2. a non-trivial hereditarily $G$-permutable subgroup.

Here a finite non-abelian simple group is a finite non-abelian group whose only normal subgroups are $\{1\}$ and $G$.

Unresolved qualification concerning properness. The questions say "non-trivial" without explicitly saying "proper." With the usual meaning $A\neq\{1\}$, the permitted range is $\{1\}<A\leq G$, which allows $A=G$; this subgroup satisfies both displayed definitions. The supplied comments instead discuss proper subgroups. Thus the apparent intended classification is of finite non-abelian simple groups satisfying, respectively, \[ \exists A\ (\{1\}<A<G)\ \forall B\leq G\ \exists x\in G:\quad AB^x=B^xA \] and \[ \exists A\ (\{1\}<A<G)\ \forall E\ (A\leq E\leq G)\ \forall B\leq E\ \exists x\in E:\quad AB^x=B^xA. \] These proper-subgroup formulations are explicitly conditional interpretations, not a verified correction of the source.

Supplied comments of 2025. The record reports that, among sporadic groups, only $J_1$ has a proper $G$-permutable subgroup, citing Galt and Tyutyanov (2022). It also reports that sporadic, alternating, and exceptional groups of Lie type have no proper hereditarily $G$-permutable subgroups, citing the three references listed below. These are supplied literature claims, not independently verified results. Their use of "proper" also requires a non-triviality qualification: $\{1\}$ satisfies both definitions. No precise range or low-degree qualifications for the named group families are supplied. The registry labels the problem open; this is a supplied status label rather than an independently verified assessment.

Short English statement

Classify the finite non-abelian simple groups having a subgroup that permutes with a suitable conjugate of every subgroup, and classify those having such a subgroup with the same property inside every intermediate subgroup. The apparent intended subgroup range is $\{1\}<A<G$, but properness is not explicit in the questions. The supplied 2025 partial-result comments and open status require source and mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701413$, exact source problem code catalogue entry, title Kourovka Notebook Problem 17.112. The record gives upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82 and update timestamp 2026-10-16T14:00:00Z. These are registry provenance data, not a literature-verification date.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 17.112, 17th issue (2010); the supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. Proposed by A. N. Skiba and V. N. Tyutyanov in 2010. The collection description names E. I. Khukhro and V. D. Mazurov as editors. This authoritative-source attribution is supplied by the record and has not been independently checked.
• A. A. Galt and V. N. Tyutyanov, Siberian Math. J., 63, no. 4 (2022), 691--698. Cited in the supplied comments of 2025 for the sporadic-group assertions. No article title or link is supplied.
• A. F. Vasilyev and V. N. Tyutyanov, Izv. Gomel. Gos. Univ. F. Skoriny, 2012, no. 5(74) (2012), 148--150 (Russian). Cited in the supplied comments of 2025 concerning hereditary permutability. No article title or link is supplied.
• A. A. Galt and V. N. Tyutyanov, Siberian Math. J., 64, no. 5 (2023), 1110--1116. Cited in the supplied comments of 2025 concerning hereditary permutability. No article title or link is supplied.