Alternative stable homotopy classification of

We give an alternative to the stable classification of p-completed homotopy types of classifying spaces of finite groups offered by Martino–Priddy. For a finite group G with Sylow subgroup S, we regard the stable p-completed classifying space Σ ∞ BG p ∧ as an object under Σ ∞ BS via the canonical inclusion map. Thus we get a classification in terms of induced fusion systems. Applying Oliverʹs solution to the Martino–Priddy conjecture, we obtain the surprising result that the unstable homotopy type of BG p ∧ is determined by the map Σ ∞ BS → Σ ∞ BG p ∧ , but not by the homotopy type of Σ ∞ BG p ∧ .