Projective limits of group rings

Abstract

A finite group G may be written as a projective limit of certain quotients G i . Denote by Γ the corresponding projective limit of the integral group rings Z G i . The basic topic of the paper is the question whether Γ may be a replacement of Z G. In particular, this is studied in connection with the isomorphism problem of integral group rings and with the conjecture of Zassenhaus that different group bases of Z G are conjugate within Q G. Using such projective limits, a Čech style cohomology set yields obstructions for these conjectures to be true, if G is soluble. This is used to construct two non-isomorphic groups as projective limits such that the projective limits of the corresponding group rings are semi-locally isomorphic. On the other hand, it is shown that for special classes of groups certain p-versions of the Zassenhaus conjecture hold. These p-versions are weaker than the conjecture but still provide a strong positive answer to the Isomorphism problem. In particular, such p-versions hold when G has a nilpotent commutator subgroup or when G is a Frobenius or a 2-Frobenius group.