Abstract. We give a classification of the p-local stable homotopy type of BG, where G is a finite group, in purely algebraic terms. BG is determined by conjugacy classes of homomorphisms from p-groups into G. This classification greatly simplifies if G has a normal Sylow p-subgroup; the stable homotopy types then depends only on the Weyl group of the Sylow p-subgroup. If G is cyclic mod p then BG determines G up to isomorphism. The last class of groups is important because in an appropriate Grothendieck group BG can be written as a unique linear combination of BH 's, where H is cyclic mod p. 0. Introduction and statement of main results Let G be a finite group. In this note we give a classification of the stable homotopy type of BG in terms of G. Our analysis shows that for each prime number p, the p-local stable type of BG depends on the homomorphisms from p-groups Q into G
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