Character correspondences and subgroups of operator groups
type: Item
instance of: scholarly article
title: Character correspondences and subgroups of operator groups
zbMATH Open document ID: 0914.20007
author: Christopher Puin
published in: Journal of Algebra
publication date: 1999-03-03T00:00:00Z
review text: Let \(A\) and \(G\) be finite groups of relatively prime orders and suppose that \(A\) acts on \(G\) as a group of automorphisms. Let \(C\) be the subgroup of \(G\) consisting of the elements fixed by all automorphisms of \(A\). The Glauberman-Isaacs correspondence \(\pi(G,A)\) is a one-to-one correspondence between the set \(\text{Irr}_A(G)\) of irreducible complex characters of \(G\) fixed by all elements of \(A\) and the set \(\text{Irr}(C)\) of all irreducible complex characters of \(C\). This correspondence has been studied in great detail, especially in the theory of solvable groups. Let \(B\) be a subgroup of \(A\) and let \(\chi\in\text{Irr}_A(G)\). It was once conjectured that \(\chi\pi(G,A)\) is an irreducible constituent of the restriction of \(\chi\pi(G,B)\) to \(C\), and T. R. Wolf proved that this is true if \(| G|\) is odd, if \(B\) is subnormal in \(A\), or if \(A\) and \(B\) have the same fixed point subgroup in \(G\) and \(G\) is solvable. Despite these positive results, the author of the present paper and \textit{T. R. Wolf} showed [Bull. Lond. Math. Soc. 29, No. 3, 307-308 (1997; Zbl 0866.20005)] that the conjecture is false in general, even if \(G\) is solvable. The main result of the paper under review is that the conjecture is true if either \(G\) is supersolvable or \(\chi\) is induced from an \(A\)-invariant irreducible character of an \(A\)-invariant supersolvable subgroup of \(G\). The final section of the paper examines related problems in the context of \(\pi\)-Brauer characters of \(\pi\)-separable groups, where \(\pi\) is a set of primes.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D45
zbMATH DE Number: 1213802
zbMATH Keywords: character correspondences
zbMATH Keywords: coprime operator groups
zbMATH Keywords: supersolvable groups
zbMATH Keywords: finite groups
zbMATH Keywords: groups of automorphisms
zbMATH Keywords: Glauberman-Isaacs correspondence
zbMATH Keywords: irreducible complex characters
zbMATH Keywords: Brauer characters
zbMATH Keywords: \(\pi\)-separable groups
reviewed by: Rod Gow
MaRDI profile type: Publication
full work available at URL: jabr.1998.7495
OpenAlex ID: W1985466089
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DOI: 10.1006/JABR.1998.7495
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description: scientific article; zbMATH DE number 1213802