Realizing regular representations of finite groups

Abstract

Given a regular representation of a finite group G and a positive integer number n, we construct a (finite) topological space X such that its group of homotopy classes of self-homotopy equivalences $E\left(X\right)$ and its group of homeomorphisms $Aut\left(X\right)$ are isomorphic to G, and the action of G on the n-th homology group $H_n\left(X\right)$ is the regular representation. We also discuss other representations.