Intersections of matrix algebras and permutation representation of \(\text{PSL}(n,q)\)
type: Item
instance of: scholarly article
title: Intersections of matrix algebras and permutation representation of \(\text{PSL}(n,q)\)
zbMATH Open document ID: 0957.20013
author: Johannes Siemons
author: A. Zalesski
published in: Journal of Algebra
publication date: 2001-01-16T00:00:00Z
review text: A question that has many applications in finite group theory is the following. Let \(G\) be a finite group that acts transitively on a \(G\)-set \(\Omega\). Let \(H\) be some special type of subgroup of \(G\). Under what conditions does \(H\) have a regular orbit on \(\Omega\)? There is also a version of this problem for module actions. Suppose that \(K\) is a finite field and \(M\) is a faithful \(KG\)-module. Under what conditions is the regular module \(KH\) a submodule of \(M\). This type of problem was already significant in the Hall-Higman paper of 1956, when \(K\) was a field of characteristic \(p\), \(G\) a \(p\)-solvable group, and \(H\) a cyclic \(p\)-subgroup. The authors of the paper under review address the regular orbit problem in the case that \(G\) is a subgroup of the projective general linear group \(\text{PGL}(n,q)\) containing the projective special linear group \(\text{PSL}(n,q)\), where \(q\) is an integral power of the prime \(p\) and \(H\) is a cyclic subgroup of \(G\). The \(G\)-set \(\Omega\) is assumed to be non-trivial, in the sense that \(\text{PSL}(n,q)\) does not act trivially on it. The main result of the authors is that \(H\) has a regular orbit on \(\Omega\) except in the three cases that one might have anticipated to be problematic, namely, when \(n=2\) and \(q=2\) or \(3\), or when \(n=4\), \(q=2\), in which cases there are exceptional actions corresponding to isomorphisms with the symmetric group \(S_3\) or the alternating groups \(A_4\) and \(A_8\). The authors also consider a module-theoretic version of their theorem. In this case, let \(G\) be a subgroup of the general linear group \(\text{GL}(n,q)\) containing the special linear group \(\text{SL}(n,q)\) and let \(H\) be an Abelian subgroup of \(G\) whose Sylow \(p\)-subgroup is cyclic. Let \(K\) be a field of characteristic \(0\) or a prime that does not divide \(|G|\). Let \(M\) be a non-trivial permutation module for \(KG\). Let \(H_0\) be the subgroup of \(H\) that acts tr…
reviewed by: Rod Gow
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 1441818
zbMATH Keywords: permutation representations
zbMATH Keywords: projective special linear groups
zbMATH Keywords: regular orbits
zbMATH Keywords: regular modules
zbMATH Keywords: actions
zbMATH Keywords: permutation modules
zbMATH Keywords: matrix algebras
MaRDI profile type: Publication
full work available at URL: jabr.1999.8201
OpenAlex ID: W2072310470
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DOI: 10.1006/JABR.1999.8201
description: scientific article; zbMATH DE number 1441818