It may happen that a finite group \(G\) turns out to be the projective limit of certain quotient groups \(G_ j\). For example, if \(G\) is solvable, then \(G\) is such a limit in a canonical way. The corresponding projective limit \(\Gamma G\) of the integral group rings \(\mathbb{Z} G_ j\) is a quotient of \(\mathbb{Z} G\), and it seems quite natural to ask which consequences equalities \(\mathbb{Z} G = \mathbb{Z} H\), \(\Gamma G = \Gamma H\) for two solvable groups \(G\), \(H\) would have. This is discussed with respect to the isomorphism problem and certain variations of the Zassenhaus conjecture. It is shown that a Čech style cohomology set yields obstructions for these conjectures to be true.
Some of the results are: 1. If \(\mathbb{Z} G = \mathbb{Z} H\), and if the augmentation ideal of \(\mathbb{Z} G\) decomposes, then \(G\simeq H\). K. W. Gruenberg and K. W. Roggenkamp have given a list of all solvable groups \(G\) whose augmentation ideals decompose [Proc. Lond. Math. Soc., III. Ser. 31, 149-166 (1975; Zbl 0313.20004), J. Pure Appl. Algebra 6, 165-176 (1975; Zbl 0313.20003), J. Lond. Math. Soc., II. Ser. 12, 262-266 (1976; Zbl 0325.20015)]. 2. If \(G\) modulo its Fitting subgroup is abelian, then the \(p\)-version of the Zassenhaus conjecture holds simultaneously for all primes \(p\) for \(\Gamma G\) and for \(\mathbb{Z} G\). 3. If \(G\) is a Frobenius group, then the \(p\)-version of the Zassenhaus conjecture holds true. 4. There are two non-isomorphic groups \(G\), \(H\) such that the \(\Gamma G\) and \(\Gamma H\) are semi-locally isomorphic.