# The fundamental group of the $p$-subgroup complex

## TL;DR

This paper investigates the fundamental group of the $p$-subgroup complex in finite groups, providing the first example of a non-free fundamental group and reducing the problem to the almost simple case.

## Contribution

It demonstrates that the fundamental group can be non-free and establishes a reduction of the problem to almost simple groups under a conjecture.

## Key findings

- First concrete example of a non-free fundamental group in a $p$-subgroup complex
- Reduction of the fundamental group study to almost simple groups
- Identification of families with free fundamental groups

## 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, $\pi_1(A_p(G)) = \pi_1(A_p(S_G)) * F$, where $F$ is a free group and $\pi_1(A_p(S_G))$ is free if $S_G$ is not almost simple. Here $S_G = \Omega_1(G)/O_{p'}(\Omega_1(G))$. 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.