Some remarks on the \(k(GV)\) problem
Let \(G\) be a finite group, \(p\) be a prime, \(V\) be a \(\text{GF}(p)G\)- module. The number of conjugacy classes of the semi-direct product \(GV\) is denoted by \(k(GV)\). The author proves the following: Theorem 1. Let \(G\) be a finite group of order prime to \(p\), and let \(V\) be a faithful \(\text{GF}(p)G\)-module. Suppose that there is some \(v\) in \(V\) such that \(V\cong V^*\) as a \(\text{GF}(p)C_G(v)\)-module. Then (i) \(k(GV)\leq|V|\). (ii) If \(p\) is odd and \(C_G(v)\) contains elements of determinant \(-1\) in their action on \(V\), we have \(k(GV)\leq(p+3)|V|/2p\). If, in this case, no element of \(C_G (v)\) acts as a reflection on \(V\), we have \(k(GV)\leq(p+1)|V|/2p\). The next result gives a criterion for the sum of the irreducible constituents of \(\text{Res}^G_{C_G(v)}(\chi)\) to be essential, where \(v\in V\), \(\chi\) is a complex irreducible constituent of the Brauer character of \(G\) afforded by \(V\).
scientific article; zbMATH DE number 763914