Counting objects which behave like irreducible Brauer characters of finite groups
type: Item
instance of: scholarly article
title: Counting objects which behave like irreducible Brauer characters of finite groups
zbMATH Open document ID: 0656.20007
DOI: 10.1016/0021-8693(88)90117-2
author: I. M. Isaacs
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let \(N\triangleleft G\), where G is a finite group. Suppose \(\theta\) is an irreducible character of N such that \(\theta (gng^{-1})=\theta (n)\), for all \(g\in G\), \(n\in N\). Define Irr(G\(| \theta)\) as the set of those \(\chi\in Irr(G)\) such that \(\theta\) is a constituent of \(\chi_ N\). Schur and Gallagher gave each a description of the cardinality of \(| Irr(G| \theta)|\). Namely, Gallagher did it as follows. Define \(\bar g\in G/N\) to be \(\theta\)-good if some extension of \(\theta\) to \(<N,g>\) is invariant in C, where \(C/N=C_{G/N}(\bar g)\). He shows that \(| Irr(G| \theta)| =\#\{conjugacy\) classes of G/N, consisting of \(\theta\)-good elements\(\}\). An analogous result holds for \(\theta\in IBr(N)\), where IBr(N) stands for the irreducible Brauer characters with respect to some prime p. Supposing \(\theta\) to be G-invariant, it holds that \(| IBr(G| \theta)|\) is equal to the number of \(\theta\)-good p-regular conjugacy classes in G/N. In this paper the above research area is also done for ``\(\theta\)-good \(\pi\)-classes'' where \(\pi\) is a set of primes. The real purpose of this paper is to prove an analog of Gallagher's theorem in case G is a \(\pi\)- separable group. Your reviewer had to be short here, so the interested reader is referred to the paper's introduction.
reviewed by: Robert W. van der Waall
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 4071171
zbMATH Keywords: finite group
zbMATH Keywords: conjugacy classes
zbMATH Keywords: \(\theta \) -good elements
zbMATH Keywords: irreducible Brauer characters
zbMATH Keywords: \(\theta \) -good p-regular conjugacy classes
zbMATH Keywords: \(\pi \) - separable group
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90117-2
OpenAlex ID: W2103827434
cites work: The number of conjugacy classes in a finite group
cites work: Characters of \(\pi\)-separable groups
cites work: On blocks of twisted group algebras
description: scientific article; zbMATH DE number 4071171