Algebraic and logical methods in computer science and artificial intelligence

$G$-permutable subgroups in $\operatorname{PSL}_2(q)$ and hereditarily $G$-permutable subgroups in $\operatorname{PSU}_3(q)$

Alexey A. Galt^a, Valentin N. Tyutyanov<sup>b

a</sup> Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russian Federation
^b MITSO International University, Gomel, Belarus

Abstract: The concept of $X$-permutable subgroup, introduced by A. N. Skiba, generalizes the classical concept of a permutable subgroup. Many classes of finite groups have been characterized in terms of $X$-permutable subgroups. In particular, W. Guo, A. N. Skiba and K. P. Shum obtained a characterization of the classes of solvable, supersolvable and nilpotent groups. Nevertheless, the further application of this concept in solving various problems in group theory is restrained by the lack of information about $G$-permutable and hereditarily $G$-permutable subgroups lying in the composition factors of groups. In this regard, the following problems were posed in the Kourovka Notebook: which finite nonabelian simple groups $G$ have a proper $G$-permutable subgroup and a proper hereditarily $G$-permutable subgroup? In this paper, an answer is obtained to the first question for simple linear groups of dimension two and to the second question for simple unitary groups of dimension three.

Keywords: simple linear group, simple unitary group, $G$-permutable subgroup, hereditarily $G$-permutable subgroup.

UDC: 512.542

MSC: 20D06

Received: 31.01.2025
Revised: 19.03.2025
Accepted: 21.03.2025

Language: English

DOI: 10.26516/1997-7670.2025.53.156