RIGIDITY OF P-COMPLETED CLASSIFYING-SPACES OF ALTERNATING GROUPS AND CLASSICAL-GROUPS OVER A FINITE-FIELD

A p-adic rigid structure of the classifying spaces of certain finite groups pi, including alternating groups A(n) and finite classical groups, is shown in terms of the maps into the p-completed classifying spaces of compact Lie groups. The spaces (B-pi)p AND have no nontrivial retracts. As an application, it is shown that (BA(n))p AND congruent-to (B-SIGMA(n))p AND if and only if n not-equal-to 0, 1, mod p. It is also shown that (BSL(n, F(q))p AND congruent-to (BGL(n, F(q))p AND where q is a power of p if and only if (n, q - 1) = 1.