Some applications of the first cohomology group
type: Item
instance of: scholarly article
title: Some applications of the first cohomology group
zbMATH Open document ID: 0554.20017
DOI: 10.1016/0021-8693(84)90183-2
author: Michael Aschbacher
author: Robert Guralnick
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: The object of the article is \(H^ 1(G,V)\), the first cohomology group of a finite group G and a finite \({\mathbb{Z}}G\)-module V. It is shown that \(H^ 1(G,V)\) is not too big in many cases. Theorem A. If V is a simple faithful \({\mathbb{Z}}G\)-module, then \(| H^ 1(G,V)| <| V|.\) The concept of the proof is as follows. Reduction to the case when G is simple. Theorem A follows then from generation properties of simple groups. It is the main part of the article to establish suitable (in particular with respect to Chevalley groups) generation properties. Among other it is shown that the sporadic simple groups are generated by an involution and another element. This is not only relevant for Theorem A. By the classification of the finite simple groups this completes the proof of the following well-known conjecture. Theorem B. Every finite simple group can be generated by two elements. The bound on \(H^ 1\) has several applications, e.g. it follows that minimal relation modules of finite simple groups are unique [cf. \textit{J. Williams} and the reviewer, Arch. Math. 42, 214-223 (1984; Zbl 0553.20003)]. Another application is a computation of the minimum number of generators of a finite group G with an abelian minimal normal subgroup A by the knowledge of the minimum number of generators of G/A. Finally the authors give reasons for their conjecture that the number of irreducible characters of G bounds the number of conjugacy classes of maximal subgroups of G.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D08
zbMATH DE Number: 3882637
zbMATH Keywords: first cohomology group
zbMATH Keywords: Chevalley groups
zbMATH Keywords: sporadic simple groups
zbMATH Keywords: minimal relation modules
zbMATH Keywords: finite simple groups
zbMATH Keywords: minimum number of generators
zbMATH Keywords: number of conjugacy classes of maximal subgroups
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90183-2
OpenAlex ID: W2046433533
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description: scientific article; zbMATH DE number 3882637