Which alternating and symmetric groups are unit groups?

TL;DR

This paper classifies which symmetric and alternating groups can be realized as the unit group of a ring, proving non-existence for most cases and providing explicit examples for some small groups.

Contribution

It establishes new non-existence results for rings with unit groups isomorphic to S_n and A_n for certain n, and constructs examples for specific small groups.

Findings

No rings have unit group isomorphic to S_n for n ≥ 5.

No rings have unit group isomorphic to A_n for n ≥ 5, n ≠ 8.

Explicit examples of rings with unit groups isomorphic to small symmetric and alternating groups.

Abstract

We prove there is no ring with unit group isomorphic to S_n for n \geq 5 and that there is no ring with unit group isomorphic to A_n for n \geq 5, n \neq 8. We give examples of rings with unit groups isomorphic to S_1, S_2, S_3, S_4, A_1, A_2, A_3, A_4, and A_8. We expect our methods to work similarly for other groups with trivial center; in particular, we plan to consider other simple groups in later work.