Which finite simple groups are unit groups?

TL;DR

This paper classifies finite simple groups that can occur as the group of units of a ring, showing they are limited to specific cyclic groups and certain projective special linear groups, with all such groups actually realizable.

Contribution

It provides a complete classification of finite simple groups that are unit groups of rings, identifying exactly which groups can occur.

Findings

Finite simple groups as unit groups are cyclic of order 2 or prime Mersenne numbers, or PSL groups over F_2.

All identified groups are realizable as unit groups.

The classification extends to groups with no non-trivial normal 2-subgroup.

Abstract

We prove that if $G$ is a finite simple group which is the unit group of a ring, then $G$ is isomorphic to either (a) a cyclic group of order 2; (b) a cyclic group of prime order $2^k -1$ for some $k$; or (c) a projective special linear group $PSL_n(\mathbb{F}_2)$ for some $n \geq 3$. Moreover, these groups do (trivially) all occur as unit groups. We deduce this classification from a more general result, which holds for groups $G$ with no non-trivial normal 2-subgroup.