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JOURNALS // Sibirskii Matematicheskii Zhurnal // Archive

Sibirsk. Mat. Zh., 2023 Volume 64, Number 5, Pages 935–944 (Mi smj7806)

This article is cited in 1 paper

On the existence of hereditarily $G$-permutable subgroups in exceptional groups $G$ of Lie type

A. A. Galtab, V. N. Tyutyanovc

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University
c Gomel Branch of International University "MITSO"

Abstract: A subgroup $A$ of a group $G$ is $G$-permutable in $G$ if for every subgroup $B\leq G$ there exists $x\in G$ such that $AB^x=B^xA$. A subgroup $A$ is hereditarily $G$-permutable in $G$ if $A$ is $E$-permutable in every subgroup $E$ of $G$ which includes $A$. The Kourovka Notebook has Problem 17.112(b): Which finite nonabelian simple groups $G$ possess a proper hereditarily $G$-permutable subgroup? We answer this question for the exceptional groups of Lie type. Moreover, for the Suzuki groups $G\cong{^2 \operatorname{B}_2}(q)$ we prove that a proper subgroup of $G$ is $G$-permutable if and only if the order of the subgroup is $2$. In particular, we obtain an infinite series of groups with $G$-permutable subgroups.

Keywords: exceptional group of Lie type, $G$-permutable subgroup, hereditarily $G$-permutable subgroup.

UDC: 512.542

MSC: 35R30

Received: 18.03.2023
Revised: 24.07.2023
Accepted: 02.08.2023

DOI: 10.33048/smzh.2023.64.504



© Steklov Math. Inst. of RAS, 2025