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 ${\pi }_1(A_3(A_{10}))$ is not a free group (here $A_{10}$ 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, ${\pi }_1(A_p(G))={\pi }_1(A_p(S_G))\ast F$, where $F$ is a free group and ${\pi }_1(A_p(S_G))$ is free if $S_G$ is not almost simple. Here $S_G={\Omega }_1(G)/O_{p^{\prime }}({\Omega }_1(G))$. 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.

Small corrections, exposition improved, added acknowledgements, updated references