Isomorphisms of integral group rings of infinite groups

This paper deals with the isomorphism problem for integral group rings of i nfinite groups. Tn the first part we answer a question of Mazur by giving c onditions for the isomorphism problem to be true for integral group rings o f groups that are a direct product of a finite group and a finitely generat ed free abelian group. It is also shown that the isomorphism problem for in finite groups is strongly related to the normalizer conjecture. Next we sho w that the automorphism conjecture holds for infinite finitely generated ab elian 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 f initely generated nilpotent group G is determined by its integral group rin g 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 generat ed nilpotency class two groups. A link is pointed out between this problem and the dimension subgroup problem. (C) 2000 Academic Press.