A coprime action version of a solubility criterion of Deskins
type: Item
instance of: scholarly article
title: A coprime action version of a solubility criterion of Deskins
zbMATH Open document ID: 1466.20018
author: Changguo Shao
author: Antonio Beltrán
published in: Monatshefte für Mathematik
publication date: 2019-02-28T00:00:00Z
full work available at URL: http://hdl.handle.net/10234/182282
review text: The main goal of this short well-written article is to provide a proof of the following result: Theorem: Let G and A be finite groups of co-prime orders, and assume that A acts on G by automorphisms. If G has a maximal A-invariant subgroup that is nilpotent with a Sylow 2-subgroup of class less than 3, then G is soluble. This result is in the spirit of a solubility criterion of W.E.Deskins who showed that a finite group possessing a maximal subgroup that is nilpotent and of class less than 3 is soluble [\textit{W. E. Deskins}, Ill. J. Math. 5, 306--313 (1961; Zbl 0097.25701)]. The proof uses the classification of finite simple groups.
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 7031253
zbMATH Keywords: finite groups
zbMATH Keywords: maximal subgroups
zbMATH Keywords: co-prime action
zbMATH Keywords: soluble groups
reviewed by: Inna Capdeboscq
MaRDI profile type: Publication
OpenAlex ID: W2907990908
cites work: Restrictions on maximal invariant subgroups implying solvability of finite groups
cites work: A condition for the solvability of a finite group
cites work: Finite Groups with Sylow 2-Subgroups of Class Two. I
cites work: The classification of finite simple groups I. Simple groups and local analysis
cites work: Endliche Gruppen I
cites work: Normalteiler und maximale Untergruppen endlicher Gruppen
cites work: Normalizers of the Sylow 2-subgroups in finite simple groups.
cites work: The theory of finite groups. An introduction.
DOI: 10.1007/S00605-018-1256-X
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description: scientific article; zbMATH DE number 7031253