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On minimal relation modules and 1-cohomology of finite groups. (English) Zbl 0553.20003
The study of presentations of finite groups is very difficult; hence the first approximation is the abelianized problem: i.e. if R is the normal subgroup generated by the relations in a minimal free resolution of the finite group G, then one studies \(\bar R=R^{ab}\) as a \({\mathbb{Z}}G\)-lattice, \(\bar R\) is called a minimal relation module. A prime question is when minimal relation modules are unique; and this is the question the paper deals with. The question of uniqueness of minimal relation modules is closely related to the question of where \(\bar R\) is a generator. So let \({\mathcal C}\) be the class of finite groups for which \(\bar R\) is a generator - for at least one \(\bar R\) (and hence for all).
The results of this paper: I) A finite non abelian simple group lies in \({\mathcal C}\), and hence minimal relation modules are unique. II) Let \(G\) be finite non-cyclic. Then \(G\) is in \({\mathcal C}\) if and only if for every split abelian chief factor \(A\) of \(G\) \(d(A\supset G)=d(G)\). (\(d\) the minimal number of generators.) The authors also draw consequences for the presentation rank of \(G\).
Reviewer: K.W.Roggenkamp

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
20J06 Cohomology of groups
20J05 Homological methods in group theory
20C10 Integral representations of finite groups
20C20 Modular representations and characters
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