Extending characters and lattices
type: Item
instance of: scholarly article
title: Extending characters and lattices
zbMATH Open document ID: 0957.20005
author: Geoffrey R. Robinson
published in: Journal of Algebra
publication date: 2001-03-20T00:00:00Z
review text: The main result of this paper is: Theorem 1. Let \(N\) be a normal subgroup of a finite group and let \(\pi\) be a set of primes such that \(G/N\) is a \(\pi\)-group. Let \(\mu\) be a \(G\)-stable irreducible character of \(N\) whose degree is divisible by \(|N|_\pi\). Let \(h\) be the \(\pi'\)-part of the exponent of \(G\), and let \(\omega\) be a primitive complex \(h\)-th root of unity. Let \(R=\{\tfrac\alpha\beta\mid\alpha\in\mathbb{Z}[\omega]\) and \(\beta\in\mathbb{Z}[\omega]\setminus\rho\) for each prime ideal divisor \(\rho\) of \(|G|_\pi\}\). Then: (a) \(\mu\) extends to a \(\pi\)-rational irreducible character of \(G\) and furthermore, this extension is unique up to multiplication by linear characters of order 2. (b) Each extension of \(\mu\) in part (a) may be afforded by an \(R\)-free \(RG\)-module which is determined up to isomorphism by its character.
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 1429471
zbMATH Keywords: finite groups
zbMATH Keywords: \(\pi\)-groups
zbMATH Keywords: irreducible characters
zbMATH Keywords: extensions
zbMATH Keywords: linear characters
reviewed by: Morton E. Harris
MaRDI profile type: Publication
full work available at URL: jabr.1999.8160
OpenAlex ID: W2077030777
cites work: A Frobenius theorem for blocks
cites work: Extensions ofP-Groups and construction of characters
cites work: A note on extensions of p-blocks by p-groups and their characters
cites work: Q5340151
cites work: Q3922829
cites work: Zur Erweiterungstheorie der endlichen Gruppen.
cites work: Nilpotent blocks and their source algebras
cites work: On the representation theory of \(\pi\)-separable groups
DOI: 10.1006/JABR.1999.8160
Recommended article: Q3472152
Recommended article: Q5646931
Recommended article: Extending characters of fixed point algebras
Recommended article: Irreducible extensions of characters
Recommended article: Extending characters from Hall subgroups.
Recommended article: Q4299125
Recommended article: Extensions and congruences of partial lattices
Recommended article: Q3478447
Recommended article: Characterizable classes of lattices
Recommended article: Extension of characters on *-semigroups
description: scientific article; zbMATH DE number 1429471