A partial analogue of Borel's fixed point theorem for finite \(p\)-groups.
type: Item
instance of: scholarly article
title: A partial analogue of Borel's fixed point theorem for finite \(p\)-groups.
zbMATH Open document ID: 1339.20016
author: George Glauberman
published in: Journal of Algebra
publication date: 2016-01-29T00:00:00Z
review text: This very interesting paper deals with analogues of Borel's Fixed Point Theorem. This states that if \(G\) is a connected, solvable algebraic group that acts morphically on a non-empty complete variety, then \(V\) has a fixed point under \(G\) [p.~134 of \textit{A. Borel}, Linear algebraic groups. 2nd enlarged ed. New York: Springer-Verlag (1991; Zbl 0726.20030)]. When \(V\) is a family of subgroups of \(G\), on which \(G\) acts by conjugation, then a fixed point is a normal subgroup of \(G\). \textit{B. Kostant} used a variation of the latter situation to prove that if a unipotent complex algebraic group \(G\) has an abelian algebraic subgroup of dimension \(k\), for some \(k\), then \(G\) has a normal abelian algebraic subgroup of dimension \(k\) [Topology 3, Suppl. 2, 147-159 (1965; Zbl 0134.03504)]. \textit{J. L. Alperin} and the author of the paper under review obtained an analogue of Kostant's result for a finite \(p\)-group, and a suitable family of elementary abelian subgroups [J. Algebra 203, No. 2, 533-566 (1998; Zbl 0964.20011)]. These results were further extended by the author, allowing the subgroups to have arbitrary exponent, provided they have small nilpotence class [J. Algebra 272, No. 1, 128-153 (2004; Zbl 1048.20004)]. In the paper under review, the results of the latter paper are used to obtain further generalisations of Kostant's result, some of them without restriction on the nilpotence class of the subgroups. Among the wealth of results, some of which have inevitably rather technical statements, Corollary 1 states that if \(S\) is a finite \(p\)-group which is the product of normal subgroups, each of class at most \(p-1\), and if \(B\) is a subgroup of \(S\), then there is a normal subgroup \(B^*\) of \(S\) which has the same order as \(B\), and nilpotence class at most that of \(B\). There is such a subgroup \(B^*\) within the normal closure of \(B\) in \(S\).
reviewed by: Andrea Caranti
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D25
zbMATH DE Number: 6537420
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: Borel fixed point theorem
zbMATH Keywords: action by conjugation
zbMATH Keywords: normal subgroups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2015.11.012
OpenAlex ID: W2253226774
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DOI: 10.1016/J.JALGEBRA.2015.11.012
description: scientific article; zbMATH DE number 6537420