Group theoretic and group ring theoretic determination of certain Sylow and Hall subgroups and the resolution of a question of R. Brauer
type: Item
instance of: scholarly article
title: Group theoretic and group ring theoretic determination of certain Sylow and Hall subgroups and the resolution of a question of R. Brauer
zbMATH Open document ID: 0840.20006
author: Robert Sandling
author: Wolfgang Kimmerle
published in: Journal of Algebra
publication date: 1996-06-30T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 735303
zbMATH Keywords: character tables
zbMATH Keywords: integral group rings
zbMATH Keywords: finite groups
zbMATH Keywords: nilpotent Hall \(\pi\)-subgroups
zbMATH Keywords: Hamiltonian Hall subgroups
zbMATH Keywords: isomorphic group rings
zbMATH Keywords: Abelian Sylow subgroups
reviewed by: A. R. Camina
MaRDI profile type: Publication
full work available at URL: jabr.1995.1014
OpenAlex ID: W1983743171
DOI: 10.1006/JABR.1995.1014
description: scientific article; zbMATH DE number 735303