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) and BG'_∧~p 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) of self equivalences is determined by the self equivalences of the associated category L_p~c(G).
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