A note on the Glauberman correspondence of \(p\)-blocks of finite \(p\)-solvable groups
type: Item
instance of: scholarly article
title: A note on the Glauberman correspondence of \(p\)-blocks of finite \(p\)-solvable groups
zbMATH Open document ID: 1006.20009
author: Hiroshi Horimoto
published in: Hokkaido Mathematical Journal
publication date: 2002-12-10T00:00:00Z
review text: Let \(p\) be a prime, and let \(S\) be a finite solvable group acting on a finite group \(G\) such that \((|G|,|S|)=1\). Let \(B\) be an \(S\)-invariant \(p\)-block of \(G\) with defect group \(D\leq C_G(S)\). \textit{A. Watanabe} has shown [J. Algebra 216, No. 2, 548-565 (1999; Zbl 0936.20006)] that all irreducible characters in \(B\) are \(S\)-invariant and that their Glauberman correspondents form a \(p\)-block \(B^*\) of \(C_G(S)\). The author proves that \(B\) and \(B^*\) are Morita equivalent whenever \(G\) is \(p\)-solvable. This fact, which generalizes a result by \textit{S. Koshitani} and \textit{G. O. Michler} [J. Algebra 243, No. 2, 504-517 (2001; Zbl 1002.20009)], was also observed by \textit{M. E. Harris} and \textit{M. Linckelmann} [Trans. Am. Math. Soc. 354, No. 9, 3435-3453 (2002; Zbl 1011.20008)].
reviewed by: Burkhard Külshammer
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 1751381
zbMATH Keywords: blocks
zbMATH Keywords: Morita equivalences
zbMATH Keywords: Glauberman correspondence
zbMATH Keywords: finite solvable groups
zbMATH Keywords: defect groups
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1350911779
OpenAlex ID: W2062008948
DOI: 10.14492/HOKMJ/1350911779
description: scientific article; zbMATH DE number 1751381