Induction of characters, kernels and local subgroups
type: Item
instance of: scholarly article
title: Induction of characters, kernels and local subgroups
zbMATH Open document ID: 0997.20013
author: Gabriel Navarro
published in: Rocky Mountain Journal of Mathematics
publication date: 2002-11-14T00:00:00Z
full work available at URL: CONT31-3.html
review text: It was proved by \textit{G. R. Robinson} [J. Algebra 114, No. 1, 53-57 (1988; Zbl 0656.20004)] that \((\alpha_{N_K(P)})^{N_G(P)}\in\text{Irr}(N_G(P))\), where \(P\) is a Sylow \(p\)-subgroup of \(K\), \(\alpha\in\text{Char}(K)\), \(\alpha^G\in\text{Irr}(G)\), \(\alpha(1)\leq p/2\). In this paper a generalization is shown in the sense that the same result follows from the condition ``\(P\text{Ker}(\alpha)\vartriangleleft K\)'', but with ``\(\alpha(1)\leq p/2\)'' dropped. As a corollary it follows that if \(K/\ker(\alpha)\) is a \(p'\)-group, \((\alpha_{N_K(P)})^{N_G(P)}\) is a \(p\)-defect zero character of \(N_G(P)/P\). In order to achieve all this, three elementary lemmas with proofs are stated. The following is also shown in this paper; its proof turned out to be somewhat hard. Theorem. Suppose \(\chi=\alpha^G\in\text{Irr}(G)\), where \(K\leq G\), \(\alpha\in\text{Irr}(K)\) and \(K/\ker(\alpha)\) is a \(p'\)-group. Suppose \(G\) has a normal \(p\)-complement \(M\). Then \(MK\) is the stabilizer in \(G\) of the character \((\alpha^{MK})_M\in\text{Irr}(M)\).
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 1744068
zbMATH Keywords: irreducibility criteria
zbMATH Keywords: finite groups
zbMATH Keywords: induction
zbMATH Keywords: restiction
zbMATH Keywords: characters
zbMATH Keywords: local subgroups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: normal complements
zbMATH Keywords: stabilizers
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
OpenAlex ID: W2093979744
cites work: Groups which have a faithful representation of degree less than \((p-1)/2\)
cites work: Hall Subgroup Normalizers and Character Correspondences in M-Groups
cites work: Q4871631
cites work: Character correspondences and irreducible induction and restriction
cites work: Modules with small source
DOI: 10.1216/RMJM/1020171679
description: scientific article; zbMATH DE number 1744068