Restrictions on maximal invariant subgroups implying solvability of finite groups
type: Item
instance of: scholarly article
title: Restrictions on maximal invariant subgroups implying solvability of finite groups
zbMATH Open document ID: 1446.20030
author: Changguo Shao
author: Antonio Beltrán
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2019-03-26T00:00:00Z
review text: Actions of groups on groups by automorphisms play a major role in the study of finite groups. An interesting case of such actions is the coprime action: two groups \(A\) and \(G\) of coprime indices such that \(A\) acts on \(G\) as a group of automorphisms. The aim of this paper is to obtain the solubility of an \(A\)-group \(G\) such that \(\gcd(|A|, |G|)=1\) when we impose some restriction on its maximal \(A\)-invariant proper subgroups. The first main result can be understood as a generalisation of Schmidt's theorem on minimal non-nilpotent groups. Theorem A. Let \(G\) and \(A\) be finite groups of coprime orders such that \(A\) acts on \(G\) by automorphisms. If every maximal \(A\)-invariant proper subgroup of \(G\) is nilpotent but \(G\) is not, then \(G\) is soluble and \(|G|=p^aq^b\) for two distinct primes \(p\) and \(q\), and \(G\) has a normal \(A\)-invariant Sylow subgroup. The second main result generalises a theorem of Thompson. Theorem B. Let \(G\) and \(A\) be finite groups of coprime orders such that \(A\) acts on \(G\) by automorphisms. If \(G\) has a nilpotent maximal \(A\)-invariant subgroup of odd order, then \(G\) is soluble. Some extensions of Theorem A are also proved. Theorem C. Let \(G\) and \(A\) be finite groups of coprime orders such that \(A\) acts on \(G\) by automorphisms. If every non-nilpotent maximal \(A\)-invariant proper subgroup of \(G\) has prime index, then \(G\) is soluble. Theorem D. Let \(G\) and \(A\) be finite groups of coprime orders such that \(A\) acts on \(G\) by automorphisms and let \(p\) be a prime divisor of \(|G|\). If the indices of all non-nilpotent maximal \(A\)-invariant subgroups of \(G\) are powers of \(p\), then \(G\) is soluble. Theorem E. Let \(G\) and \(A\) be finite groups of coprime orders such that \(A\) acts on \(G\) by automorphisms. If every maximal \(A\)-invariant proper subgroup of \(G\) is supersoluble, then \(G\) is soluble.
reviewed by: R. Esteban-Romero
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 7041954
zbMATH Keywords: finite group
zbMATH Keywords: maximal subgroup
zbMATH Keywords: coprime action
zbMATH Keywords: group action on groups
zbMATH Keywords: soluble group
zbMATH Keywords: nilpotent group
zbMATH Keywords: supersoluble group
MaRDI profile type: Publication
full work available at URL: s10231-018-0777-1
OpenAlex ID: W2883476139
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Wikidata QID: Q129457033
DOI: 10.1007/S10231-018-0777-1
description: scientific article; zbMATH DE number 7041954