Inductive sources and subnormal subgroups.
type: Item
instance of: scholarly article
title: Inductive sources and subnormal subgroups.
zbMATH Open document ID: 1055.20006
author: I. M. Isaacs
author: Mark L. Lewis
published in: Archiv der Mathematik
publication date: 2004-11-05T00:00:00Z
review text: By a character pair in a finite group \(G\) is meant a pair \((H,\theta)\), where \(H\leq G\) and \(\theta\in\text{Irr}(H)\). The group \(G\) acts on the set of character pairs by \((H,\theta)^g=(H^g,\theta^g)\), where \(g\in G\). The character \(\theta^g\) of \(H^g\) is defined by the formula \(\theta^g(h^g)=\theta(h)\) for \(h\in H\). The stabilizer of \((H,\theta)\) in this action coincides with the inertia group of \(\theta\) in \(N_G(H)\). The character pair \((H,\theta)\) is an `inductive source' for \(G\), if induction defines an injection from \(\text{Irr}(T|\theta)\) into \(\text{Irr}(G)\), where \(T\) is the stabilizer of \((H,\theta)\) in \(G\). If, for example, \(H\) is normal in \(G\), then \((H,\theta)\) is an inductive source for \(G\). On the other hand, if \(H\) is not normal in \(G\), then the pair \((H,1_H)\) is never an inductive source. In this note a necessary condition for a character pair \((H,\theta)\) to be an inductive source is given. It appears that this condition is also sufficient in the case where \(H\) is subnormal in \(G\). We write \((H,\theta)\leq(K,\psi)\) if \(H\leq K\) and \(\langle\theta,\psi_H\rangle>0\). We say that a character pair \((H,\theta)\) is `conjugate stable' in \(G\), if whenever there is a pair \((K,\psi)\) in \(G\) and \(g\in G\) such that both \((H,\theta)\leq(K,\psi)\) and \((H,\theta)^g\leq(K,\psi)\), then there is \(k\in K\) such that \((H,\theta)^g=(H,\theta)^k\). Theorem A. Let \((H,\theta)\) be an inductive pair in a group \(G\). Then \((H,\theta)\) is conjugate stable in \(G\).
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D35
zbMATH DE Number: 2113839
zbMATH Keywords: character pairs
zbMATH Keywords: finite groups
zbMATH Keywords: inertia groups
zbMATH Keywords: inductive sources
zbMATH Keywords: stabilizers
zbMATH Keywords: subnormal subgroups
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: s00013-004-4938-7
OpenAlex ID: W2044476892
DOI: 10.1007/S00013-004-4938-7
description: scientific article; zbMATH DE number 2113839