Geodesic
On the Isomorphism of Integral Group Rings. II
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1182-1188

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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. 
Sehgal, Sudarshan K. On the Isomorphism of Integral Group Rings. II. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1182-1188. doi: 10.4153/CJM-1969-130-2
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[5] 5. Sehgal, S. K., On the isomorphism of integral group rings. I, Can. J. Math. 21 (1969), 410–413 Google Scholar

[6] 6. Whitcomb, A., The group ring problem, Ph.D. Thesis, University of Chicago, Chicago, Illinois, 1968. Google Scholar

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