A p-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a one-to-one correspondence between connected p-compact groups and finite reflection groups over the p-adic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as p-compact groups by their Weyl groups seen as finite reflection groups over the p-adic integers. Our approach in fact gives a largely self-contained proof of the entire classification theorem for p odd.

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Under certain finiteness conditions, ^-completion commutes with the formation of a certain group of self-homotopy equivalences. Introduction.Let X be a pointed, 0-connected topological space, Aut(Z) the group of based homotopy classes of based self-homotopy equivalences of X, and Aut # (Z) the kernel of the obvious homomorphism Aut(Z)-> τiAutπi(X)where we further assume either that X is a Cΐf-complex of dimension d or that 7r*(Z)=0 for *>d, l^d

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