The number of homomorphisms from finite groups to classical groups.
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.
scientific article; zbMATH DE number 5144427