The order complex of $PGL_2(p^{2^n})$ is contractible when $p$ is odd
2020
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 $G \cong PGL_2(p^{2^n})$, $p \geq 3$, $n \geq 1$ 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.
Keywords:
- Correction
- Source
- Cite
- Save
- Machine Reading By IdeaReader
9
References
0
Citations
NaN
KQI