On the non-coprime k(GV) problem

TL;DR

This paper investigates bounds on the number of conjugacy classes in semidirect products of finite groups and modules, addressing the non-coprime k(GV)-problem with explicit linear bounds in module size.

Contribution

It provides explicit linear bounds in the size of the module for the number of conjugacy classes in non-coprime cases, advancing understanding of the k(GV)-problem.

Findings

Explicit linear bounds for k(GV) in terms of |V|

Bounds depend on the structure of the group G and module V

Results apply to various cases of the non-coprime k(GV)-problem

Abstract

Let V be a finite faithful completely reducible FG-module for a finite field F and a finite group G. In various cases explicit linear bounds in |V| are given for the numbers of conjugacy classes k(GV) and k(G) of the semidirect product GV and of the group G respectively. These results concern the so-called non-coprime k(GV)-problem.