MATHEMATICS

Finite groups with permuted strongly generalized maximal subgroups

Yu. V. Gorbatova

Russian Presidential Academy of National Economy and Public Administration (Bryansk Branch), Bryansk, Russian Federation

Abstract: The structure of finite groups in which any strictly 2-maximal subgroup permutes with an arbitrary strictly 3-maximal subgroup is described. It is shown that the class of groups with this property coincides with the class of groups in which any 2-maximal subgroup permutes with an arbitrary 3-maximal subgroup, and, as a consequence, such groups are solvable. As auxiliary results, we describe the structure of groups in which any strictly 2-maximal subgroup permutes with an arbitrary maximal subgroup. In particular, it is shown that the class of such groups coincides with the class of groups in which any 2-maximal subgroup commutes with all maximal subgroups, and, as a consequence, such groups are supersoluble.

Keywords: solvable group, $i$-maximal subgroup, strongly $i$-maximal subgroup, normal subgroup, nilpotent group, supersolvable group, Schmidt group.

UDC: 512.542

MSC: 20E28

Received: 16.09.2021
Accepted: December 1, 2022

DOI: 10.17223/19988621/80/3

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