Second Maximal Invariant Subgroups and Solubility of Finite Groups

Abstract

Let G be a finite group and assume that a group of automorphisms A is acting on G such that A and G have coprime orders. We prove that the fact of imposing specific properties on the second maximal A-invariant subgroups of G determines that G is either soluble or isomorphic to a few non-soluble groups such as PSL(2, 5) or SL(2, 5).

Keywords

Invariant subgroups / Coprime action / Second maximal subgroup / Trivial intersection subgroup

Funding

National Nature Science Fund of China(12071181)

Nature Science Fund of Shandong Province(ZR2019MA044)

Ministerio de Ciencia Tecnología y Telecomunicaciones (CR)(PGC2018-096872-B-100)

Universitat Jaume I(UJI-B2019-03)

Generalitat Valenciana(Proyecto AICO/2020/298)