Free Unit Groups in Group Algebras,

Let K[G] denote the group algebra of a finite group G over a field K. If either char K = 0 and G is nonabelian, or K is a nonabsolute field of characteristic π > 0 and G/ π(G) is nonabelian, then it is well known that the group of units U(K[G]) contains a nonabelian free group. For the most part, this follows from the fact that GL2(K) contains such a free subgroup. In this paper, we refine the above result by showing that there are two cyclic subgroups X and Y of G of prime power order, and two units uX U(K[X]) and uY U(K[Y]), such that uX, uY contains a nonabelian free group. Indeed, we obtain a rather precise description of these units by using an aspect of Titsʹ theorem on free subgroups in linear groups.