Subnormal subgroups of group ring units

Abstract

Let G G be an arbitrary group. If a ∈ Z G a\in \mathbb {Z}G satisfies a 2 = 0 a^{2}=0 , a ≠ 0 a\ne 0 , then the units 1 + a 1+a , 1 + a ∗ 1+a^{*} generate a nonabelian free subgroup of units. As an application we show that if G G is contained in an almost subnormal subgroup V V of units in Z G \mathbb {Z}G then either V V contains a nonabelian free subgroup or all finite subgroups of G G are normal. This was known before to be true for finite groups G G only.