Some applications of the first cohomology group
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.
number of conjugacy classes of maximal subgroups
scientific article; zbMATH DE number 3882637
scientific article