The order complex of $PG{L}_2(p^{2^n})$ is contractible when $p$ is odd

Abstract

Given a group $G$, its lattice of subgroups $\mathcal{L}(G)$ can be viewed as a simplicial complex in a natural way. The inclusion of $1_G,G\in \mathcal{L}(G)$ implies that $\mathcal{L}(G)$ is contractible, and so we study the topology of the order complex $\widehat{\mathcal{L}(G)}:=\mathcal{L}(G)\setminus \{1_G,G\}$. In this short note we consider the homotopy type of $\widehat{\mathcal{L}(G)}$ where $G\cong PG{L}_2(p^{2^n})$, $p\ge 3$, $n\ge 1$ and show that $\widehat{\mathcal{L}(G)}$ is contractible. This is consistent with a conjecture of Shareshian on the homotopy type of order complexes of finite groups.

3 pages