Subgroups Determined by Certain Products of Augmentation Ideals

Abstract

Let G be a group, ZG the integral group ring of G, and I(G) its augmentation ideal. Let H be a subgroup of G. It is proved that the subgroup of G determined by the product I(H)I(G)I(H) equals γ3(H), i.e., the third term in the lower central series of H. Also, the subgroup determined by I(H)I(G)Iⁿ(H) (resp., Iⁿ(H)I(G)I(H)) for n > 1 equals D_n+2(H), the (n + 2)th dimension subgroup of H.