Group theoretic and group ring theoretic determination of certain Sylow and Hall subgroups and the resolution of a question of R. Brauer
This is an extremely subtle and clever paper which tackles the problem of which properties of groups can be recognized by studying either their character table or the integral group ring. The two main results are: Theorem 1. The character table of a finite group \(G\) determines whether or not, for a given set of primes \(\pi\), \(G\) has nilpotent Hall \(\pi\)-subgroups. If the Hall \(\pi\)-subgroups are abelian, then they are determined up to isomorhism. Theorem 2. The integral group ring \(\mathbb{Z} G\) of a finite group \(G\) determines hamiltonian Hall subgroups up to isomorphism. The results are deduced by introducing the idea of a class correspondence between two finite groups \(G\) and \(G^*\). They show how this concept is related to groups which either have the same character table or isomorphic group rings. The relevant definitions are quite subtle and a short summary would not be helpful so it is better to look at the paper. The history of the topic goes back to \textit{R. Brauer} [Math. Z. 63, 406-444 (1956; Zbl 0093.03101)]\ and this was discussed by \textit{W. Feit} [Proc. Symp. Pure Math. 37, 405-411 (1980; Zbl 0448.20012)]\ at the 1979 conference in Santa Cruz where he asked if it was possible to discover whether a group has an abelian Sylow subgroup from the character table. The reviewer and \textit{M. Herzog} [Proc. Am. Math. Soc. 80, 533-535 (1980; Zbl 0447.20004)]\ answered the question affirmatively for the case of Sylow 2-subgroups. The paper contains a good bibliography of related work on this problem.
scientific article; zbMATH DE number 735303