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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

##### MSC:
 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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