The fundamental group of the p-subgroup complex

Abstract

We study the fundamental group of the p-subgroup complex of a finite group G. We show first that pi1(A_3(Alt_{10})) is not a free group (here Alt_{10} is the alternating group on 10 letters). This is the first concrete example in the literature of a p-subgroup complex with non-free fundamental group. We prove that, modulo a well-known conjecture of M. Aschbacher, pi1(A_p(G)) = pi1(A_p(S_G)) * F, where F is a free group and pi1(A_p(S_G)) is free if S_G is not almost simple. Here S_G = Omega_1(G)/O_{p´}(Omega_1(G)). This result essentially reduces the study of the fundamental group of p-subgroup complexes to the almost simple case. We also exhibit various families of almost simple groups whose p-subgroup complexes have free fundamental group.