Characters, coprime actions, and operator groups
type: Item
instance of: scholarly article
title: Characters, coprime actions, and operator groups
zbMATH Open document ID: 0917.20008
author: Mark L. Lewis
published in: Archiv der Mathematik
publication date: 1999-08-03T00:00:00Z
review text: The following result is proved. Main Theorem. Let \(M\) be a normal subgroup of a finite group \(G\). Assume that \(M\) has a normal Hall subgroup \(N\) of odd order and that \(S\) is a complement for \(N\) in \(M\). If \(\chi\in\text{Irr}(N)\) is a \(G\)-invariant character, then \(\chi\) extends to \(G\) if and only if \(\chi^*\) extends to \(N_G(S)\), where \(\chi^*\in\text{Irr}(C_M(S))\) is the Glauberman-Isaacs correspondent of \(\chi\). Corollary. Let \(S\) act on \(G\) so that \((| S|,| G|)=1\) and that \(| G|\) is odd. Assume that \(A\) acts on \(SG\) so that \(S\) and \(G\) are both \(A\)-invariant under its action. If the \(S\)-invariant character \(\chi\) is \(A\)-invariant, then \(\chi\) extends to \(AG\) if and only if \(\chi^*\) extends to \(AC_G(S)\), where \(\chi^*\in\text{Irr}(C_G(S))\) is the Glauberman-Isaacs correspondent of \(\chi\).
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 1145411
zbMATH Keywords: finite groups
zbMATH Keywords: normal Hall subgroups
zbMATH Keywords: Glauberman-Isaacs correspondence
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: s000130050145
OpenAlex ID: W2026398447
DOI: 10.1007/S000130050145
description: scientific article; zbMATH DE number 1145411