The number of homomorphisms from finite groups to classical groups.
type: Item
instance of: scholarly article
title: The number of homomorphisms from finite groups to classical groups.
zbMATH Open document ID: 1123.20019
author: Michael Bate
published in: Journal of Algebra
publication date: 2007-04-18T00:00:00Z
review text: Let \(\Gamma\) be a finite group and \(k\) be a finite field. Let \(H\) be a general classical group over \(k\) (if \(H\) is symplectic or orthogonal then the characteristic of \(k\) is odd). Let \(\Hom_{cr}(\Gamma,H)\) denote the set of completely reducible homomorphisms from \(\Gamma\) to \(H\); i.e., the set of homomorphisms \(\rho\colon\Gamma\to H\) such that the natural module for \(H\) is a completely reducible \(k\Gamma\)-module via \(\rho\). In this paper, the author provides upper and lower bounds for the size of \(\Hom_{cr}(\Gamma,H)\). These bounds are expressed in terms of invariants associated to the order of \(H\). The approach is based on the paper by \textit{M. W. Liebeck} and \textit{A. Shalev} [Commun. Algebra 32, No. 2, 657-661 (2004; Zbl 1074.20016)], where the case \(H=\text{GL}_n(k)\) is considered.
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 05A16
zbMATH DE Number: 5144427
zbMATH Keywords: finite groups
zbMATH Keywords: classical groups over finite fields
zbMATH Keywords: completely reducible representations
zbMATH Keywords: estimates
zbMATH Keywords: numbers of homomorphisms
reviewed by: Anatoli Kondrat'ev
MaRDI profile type: Publication
OpenAlex ID: W1990666004
cites work: Q3682663
cites work: Q3684278
cites work: Q5813937
cites work: Q4670625
cites work: On the subgroup structure of classical groups
cites work: The Number of Homomorphisms from a Finite Group to a General Linear Group
cites work: Q5662587
DOI: 10.1016/J.JALGEBRA.2006.09.003
description: scientific article; zbMATH DE number 5144427