Subgroups with the character restriction property
type: Item
instance of: scholarly article
title: Subgroups with the character restriction property
zbMATH Open document ID: 0588.20007
DOI: 10.1016/0021-8693(86)90084-0
author: I. M. Isaacs
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: A subgroup H of a finite group G satisfies the CR (character restriction) property in G if each ordinary irreducible character of H is the restriction of one of G. It is sufficient for this that H have a normal complement in G, but not necessary. In this paper, some necessary conditions for the CR property are given. First, if \(\pi\) is a set of primes, H is maximal among solvable \(\pi\)- subgroups of G and the CR property holds, then H is a Hall \(\pi\)-subgroup of G, and G has a normal \(\pi\)-complement. Second, if H is the normalizer of a Sylow subgroup and satisfies the CR property in G, then H has a normal complement in G. Third, if G is solvable, \({\mathcal F}\) is a saturated formation, and an \({\mathcal F}\)-normalizer of H in G satisfies the CR property, then H has a normal complement in G. Along the way, a variation on the difficult Orbit Theorem of \textit{T. O. Hawkes} and \textit{G. R. Jones} [J. Algebra 94, 364-381 (1985)] is obtained.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 3944123
zbMATH Keywords: character restriction
zbMATH Keywords: irreducible character
zbMATH Keywords: CR property
zbMATH Keywords: solvable \(\pi \) -subgroups
zbMATH Keywords: Hall \(\pi \) -subgroup
zbMATH Keywords: normal \(\pi \) -complement
zbMATH Keywords: saturated formation
zbMATH Keywords: normal complement
reviewed by: William H. Gustafson
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90084-0
OpenAlex ID: W2132405791
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cites work: On the structure of a group whose orbits on a finite module are the orbits of a proper subgroup
cites work: Q4103728
cites work: Characters of \(\pi\)-separable groups
cites work: Existence of normal complements and extension of characters in finite groups
description: scientific article; zbMATH DE number 3944123