Let Z(G) denote the integral group ring of a group G. Let be the class of groups G with the property that for any isomorphism θ: Z(G) → Z(H), we have θ(g) = ±h, h ∈ H, for all g ∈ G. We study this class in § 2 and establish that it contains classes of torsion-free abelian groups, torsion abelian groups, and ordered groups.
In § 4, we prove the following result.
THEOREM. Let G be a group which contains a normal abelian subgroup A such that
. Suppose that θ: Z(G) → Z(H) is an isomorphism such that θ(Δ(G, A)) = Δ(H, B) for a suitable normal subgroup B of H. Then G ≃ H. (Here Δ(G, A) is the kernel of the natural map Z(G) → Z(G/A).)
Jackson (3) and Whitcomb (6) proved the special case of this theorem when G is supposed to be finite metabelian. The lemmas needed are given in §3.