Enumeration of conjugacy classes in affine groups

TL;DR

This paper analyzes the conjugacy classes of classical affine groups, deriving generating functions and bounds, which are crucial for understanding complex cases in the non-coprime k(GV) problem.

Contribution

It introduces new generating functions and bounds for conjugacy classes in affine groups, advancing the understanding of their structure.

Findings

Derived generating functions for conjugacy classes

Provided upper bounds for the number of classes

Identified applications to the non-coprime k(GV) problem

Abstract

We study the conjugacy classes of the classical affine groups. We derive generating functions for the number of classes analogous to formulas of Wall and the authors for the classical groups. We use these to get good upper bounds for the number of classes. These naturally come up as difficult cases in the study of the non-coprime k(GV) problem of Brauer.