Free unit groups in group algebras

[摘要] 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 pi > 0 and G/O-pi(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 GL(2)(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 u(X) is an element of U(K[X]) and u(y) is an element of U(K[Y]), such that (u(x), u(y)) contains a nonabelian free group. Indeed, we obtain a rather precise description of these units by using an aspect of Tits' theorem on free subgroups in linear groups. (C) 2001 Elsevier Science.

[发布日期] 2001-12-01

[关键词] group algebra;unit group;free subgroup