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 that π1(A3(A10)) is not a free group (here A10 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, π1(Ap(G))=π1(Ap(SG))∗F, where F is a free group and π1(Ap(SG)) is free if SG is not almost simple. Here SG=Ω1(G)/Op′(Ω1(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. [ABSTRACT FROM AUTHOR]