  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.
