On \(\pi\)-block induction in a \(\pi\)-separable group
type: Item
instance of: scholarly article
title: On \(\pi\)-block induction in a \(\pi\)-separable group
zbMATH Open document ID: 0974.20011
author: Yixin Zhu
published in: Journal of Algebra
publication date: 2001-09-04T00:00:00Z
review text: Let \(\pi\) be a set of primes, and let \(G\) be a \(\pi\)-separable group. A nonempty set \(B\) of irreducible characters of \(G\) is called a \(\pi\)-block of \(G\) if it is a union of \(p\)-blocks for each \(p\in\pi\), and is minimal with respect to this property. Let \(D\) be a \(\pi\)-subgroup of \(G\), and let \(H\) be a subgroup of \(G\) such that \[ DC_G(D)\subseteq H\subseteq N_G(D).\tag{*} \] \textit{M. C. Slattery} [J. Algebra 124, No. 1, 236-269 (1989; Zbl 0684.20008)] defined a block induction map \(b\mapsto bi(H,G)\) from \(\pi\)-blocks of \(H\) to those of \(G\). Assuming always that (*) holds, the author proves two main results. First, for any \(\xi\in b\), there is \(\chi\in bi(H,G)\) with \([\chi,\xi^G]\neq 0\), and given \(\chi\in bi(H,G)\), there is \(\xi\in b\) such that \([\chi,\xi^G]\neq 0\). For the second result, assume that \(G\) is \(p\)-solvable, for some \(p\in\pi\), and that \(D\) has a \(p\)-subgroup \(P\) with \(C_G(P)\subseteq H\subseteq N_G(P)\). If \(b\) is a \(\pi\)-block of \(H\) and \(\mu\) is a \(p\)-block of \(H\) that is contained in \(b\), then \(\mu^G\) is contained in \(bi(H,G)\).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 1563188
zbMATH Keywords: \(\pi\)-separable groups
zbMATH Keywords: block induction
zbMATH Keywords: \(p\)-solvable groups
reviewed by: William H. Gustafson
MaRDI profile type: Publication
full work available at URL: jabr.2000.8476
OpenAlex ID: W2001753844
cites work: Character correspondences and irreducible induction and restriction
cites work: Q3826736
cites work: pi-blocks of pi-separable groups. I
cites work: pi-blocks of pi-separable groups. II
cites work: Character correspondences in solvable groups
DOI: 10.1006/JABR.2000.8476
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description: scientific article; zbMATH DE number 1563188