Isomorphisms of Integral Group Rings of Infinite Groups

AbstractThis paper deals with the isomorphism problem for integral group rings of infinite groups. In the first part we answer a question of Mazur by giving conditions for the isomorphism problem to be true for integral group rings of groups that are a direct product of a finite group and a finitely generated free abelian group. It is also shown that the isomorphism problem for infinite groups is strongly related to the normalizer conjecture. Next we show that the automorphism conjecture holds for infinite finitely generated abelian groups G if and only if ZG has only trivial units. In the second part we partially answer a problem of Sehgal. It is shown that the class of a finitely generated nilpotent group G is determined by its integral group ring provided G has only odd torsion. When G has nilpotency class two then the finitely generated restriction is not needed. This, together with a result of Ritter and Sehgal, settles the isomorphism problem for finitely generated nilpotency class two groups. A link is pointed out between this problem and the dimension subgroup problem