On conjugacy classes of GL(n,q) and SL(n,q)

TL;DR

This paper investigates the structure of conjugacy class products in the groups GL(n,q) and SL(n,q), establishing lower bounds on the number of conjugacy classes in their products.

Contribution

It provides new lower bounds on the number of conjugacy classes resulting from the product of two non-central classes in GL(n,q) and SL(n,q).

Findings

Product of two non-central classes in GL(n,q) is at least q-1 classes.

Product of two non-central classes in SL(n,q) is at least ceiling(q/2) classes.

Results deepen understanding of conjugacy class interactions in these groups.

Abstract

Let GL(n,q) be the group of nxn invertible matrices over a field with q elements, and SL(n,q) be the group of nxn matrices with determinant 1 over a field with q elements. We prove that the product of any two non-central conjugacy classes in GL(n,q) is the union of at least q-1 distinct conjugacy classes, and that the product of any two non-central conjugacy classes in SL(n,q) is the union of at least $\lceil\frac{q}{2} \rceil$ distinct conjugacy classes.