Conjugacy classes of completely reducible cyclic subgroups of GL$(2, q)$

TL;DR

This paper determines the number of conjugacy classes of completely reducible cyclic subgroups of a given order in the group GL(2, q), expanding understanding of subgroup classifications in linear algebraic groups.

Contribution

It provides a formula for counting conjugacy classes of such subgroups in GL(2, q) for orders coprime to the characteristic, a new classification result.

Findings

Derived explicit count of conjugacy classes for given order m

Extended subgroup classification in GL(2, q)

Enhanced understanding of cyclic subgroup structures

Abstract

Let $m$ be a positive integer such that $p$ does not divide $m$ where $p$ is prime. In this paper we find the number of conjugacy classes of completely reducible cyclic subgroups in GL$(2, q)$ of order $m$, where $q$ is a power of $p$.