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17.112 (2010)

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A subgroup $A$ of a group $G$ is said to be $G$-permutable in $G$ if for every subgroup $B$ of $G$ there exists an element $x \in G$ such that $AB^x = B^xA$. A subgroup $A$ is said to be hereditarily $G$-permutable in $G$ if $A$ is $E$-permutable in every subgroup $E$ of $G$ containing $A$. Which finite non-abelian simple groups $G$ possess
$\qquad$ a) a non-trivial $G$-permutable subgroup?
$\qquad$ b) a non-trivial hereditarily $G$-permutable subgroup?

Contributor: A. N. Skiba, V. N. Tyutyanov

Progress

Comments of 2025: Among sporadic groups, only $J_1$ has a proper $G$-permutable subgroup (A. A. Galt, V. N. Tyutyanov, Siberian Math. J., 63, no. 4 (2022), 691–698). Sporadic, alternating, and exceptional groups of Lie type have no proper hereditarily $G$-permutable subgroups (A. A. Galt, V. N. Tyutyanov, Siberian Math. J., 63, no. 4 (2022), 691–698; A. F. Vasilyev, V. N. Tyutyanov, Izv. Gomel. Gos. Univ. F. Skoriny, 2012, no. 5(74) (2012), 148–150 (Russian); A. A. Galt, V. N. Tyutyanov, Siberian Math. J., 64, no. 5 (2023), 1110–1116).

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