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119–137; MR1356164 (96i:55025)] says that, if two finite groups have homotopy equivalent p-completed classifying spaces, then there is an isomorphism of corresponding Sylow p-subgroups which preserves p-fusion. Approaching from this point of view, the authors demonstrate the equivalence of p-fusion systems between many pairs of finite groups of Lie type (in differing characteristics both prime to p) by showing that they have the same p-completed classifying space. Let G be a connected reductive group scheme over Z, K an algebraically closed field of characteristic prime to p, ψ = ψ q a field automorphism of G(K), τ any graph automorphism of G, and H = τ G(q) the fixed subgroup of τψ q on G(K). By a theorem of E. M. Friedlander [Étale homotop

Year: 2013

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