A classification of the stable type of 𝐵𝐺

Martino, John; Priddy, Stewart

doi:10.1090/S0273-0979-1992-00300-2

A classification of the stable type of $BG$
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by John Martino and Stewart Priddy PDF

Bull. Amer. Math. Soc. 27 (1992), 165-170 Request permission

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 ${\bmod \;p}$ 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 ${\bmod \;p}$.

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