Rigidity of $p$-completed classifying spaces of alternating groups and classical groups over a finite field

Abstract:

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 $(B\pi )_p^ \wedge$ have no nontrivial retracts. As an application, it is shown that $(B{A_n})_p^ \wedge \simeq (B{\Sigma _n})_p^ \wedge$ if and only if $n\not \equiv 0,1,\;\bmod p$. It is also shown that $(BSL(n,{\mathbb {F}_q}))_p^ \wedge \simeq (BGL(n,{\mathbb {F}_q}))_p^ \wedge$ where $q$ is a power of $p$ if and only if $(n,q - 1) = 1$.