Isomorphy Classes of Finite Order Automorphisms of SL(2, k)

TL;DR

This paper classifies automorphisms of SL(2,k) of finite order by characterizing their forms, establishing conditions for isomorphism, and analyzing the number of isomorphy classes over various fields.

Contribution

It provides a detailed characterization of finite order automorphisms of SL(2,k) and criteria for their isomorphism, extending understanding of their structure.

Findings

Characterization of automorphism forms based on eigenvalues and field properties

Conditions for isomorphism between automorphisms involving matrices A and B

Examples illustrating the classification over different fields

Abstract

In this paper, we consider the order m k-automorphisms of SL(2,k). We first characterize the forms that order m k-automorphisms of SL(2,k) take and then we simple conditions on matrices A and B, involving eigenvalues and the field that the entries of A and B lie in, that are equivalent to isomorphy between the order m k-automorphisms Inn_A and Inn_B. We examine the number of isomorphy classes and conclude with examples for selected fields.