A character-theoretic criterion for the existence of normal complements to subgroups of finite groups
type: Item
instance of: scholarly article
title: A character-theoretic criterion for the existence of normal complements to subgroups of finite groups
zbMATH Open document ID: 0594.20007
DOI: 10.1016/0021-8693(85)90191-7
author: Trevor Hawkes
author: John F. Humphreys
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: A subgroup H of a finite group G is said to have the CR property in G if each irreducible complex character of H is the restriction of a character of G. \textit{H. Schmidt} and \textit{F. Richen} [J. Lond. Math. Soc., II. Ser. 7, 168-170 (1973; Zbl 0285.20025)] proved that if H has the CR property in a soluble group G and is a Carter subgroup of G then H has a normal complement in G. The authors generalize this to: Theorem. Let \({\mathcal F}\) be a saturated formation, let G be a finite soluble group, and let H be an \({\mathcal F}\)-projector of G. If H has the CR property in G, then H has a normal complement in G. This result is proved by first studying the properties of a minimal counterexample and then arriving to a contradiction.
reviewed by: B.G.Basmaji
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 3955193
zbMATH Keywords: CR property
zbMATH Keywords: irreducible complex character
zbMATH Keywords: saturated formation
zbMATH Keywords: finite soluble group
zbMATH Keywords: normal complement
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90191-7
OpenAlex ID: W2029442616
cites work: The \(\mathfrak F\)-normalizers of a finite soluble group
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cites work: Endliche Gruppen I
cites work: Q4103728
cites work: A Character-Theoretic Complementation Theorem for Carter Subgroups
cites work: Existence of normal complements and extension of characters in finite groups
description: scientific article; zbMATH DE number 3955193