Homotopy Equivalences of p-Completed Classifying Spaces of Finite Groups

We study homotopy equivalences of p-completions of classifying spaces of finite groups. To each finite group G and each prime p, we associate a finite category L-p(c)(G) with the following properties. Two p-completed classifying spaces BG((p) over cap) and BG'(p) over cap have the same homotopy type if and only if the associated categories L-p(c)(G) and L-p(c)(G') are equivalent. Furthermore, the topological monoid Aut(BG((p) over cap)) of self equivalences is determined by the self equivalences of the associated category L-p(c)(G).