Projective limits of group rings

AbstractA finite group G may be written as a projective limit of certain quotients Gi. Denote by Γ the corresponding projective limit of the integral group rings ZGi. The basic topic of the paper is the question whether Γ may be a replacement of ZG. 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 ZG are conjugate within QG.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