Cohomology and induction from elementary Abelian subgroups

In this paper we present a general view of the role of elementary Abelian s ubgroups in the representation theory of a finite group G. We show that any G-module M is a direct summand of a module that has a filtration by module s induced from elementary Abelian subgroups. This implies that if we are gi ven any sequence of cohomology elements, such that the product of the eleme nts in the sequence males sense and that the elements restrict to zero on t he elementary Abelian subgroups, then the product of the elements is zero p rovided the sequence has sufficient length. If the coefficient ring is a fi eld of finite characteristic p, then only the elementary Abelian p-subgroup s are relevant. In that case, the theorems of Quillen on the dimension of t he mod-p cohomology ring and of Alperin-Evens on the complexity of modules are easy consequences of our results.