The fundamental group of the p ‐subgroup complex

We study the fundamental group of the p ‐subgroup complex of a finite group G . We show first thatπ 1 ( A 3 ( A 10 ) )is not a free group (here A 10 is the alternating group on ten 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 Aschbacher,π 1 ( A p ( G ) ) = π 1 ( A p ( S G ) ) ∗ F , where F is a free group andπ 1 ( A p ( S G ) )is free if S G is not almost simple. HereS G = Ω 1 ( G ) / O p ′( Ω 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.