Abstract
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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.