The order complex of $PGL_2(p^{2^n})$ is contractible when $p$ is odd

Given a group $G$ , its lattice of subgroups $L (G)$ can be viewed as a simplicial complex in a natural way. The inclusion of $1_G, G \in $L$ (G) $i m p l i e s t h a t$ $L$ (G)$ is contractible, and so we study the topology of the order complex $\widehat{ $L$ (G)} := $L$ (G) \setminus {1_G,G} $. I n t h i s s h o r t n o t e w e c o n s i d e r t h e h o m o t o p y t y p e o f$ \widehat{ $L$ (G)} $w h e r e$ G \cong PGL_2(p^{2^n}) $,$ p \geq 3 $,$ n \geq 1 $a n d s h o w t h a t$ \widehat{ $L$ (G)}$ is contractible. This is consistent with a conjecture of Shareshian on the homotopy type of order complexes of finite groups.

The order complex of $P G L_{2} (p^{2^{n}})$ is contractible when $p$ is odd

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The order complex of $P G L_{2} (p^{2^{n}})$ is contractible when $p$ is odd

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The order complex of PGL2(p2n)PGL2​(p2n) is contractible when pp is odd

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The order complex of PGL2(p2n)PGL2​(p2n) is contractible when pp is odd

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The order complex of $P G L_{2} (p^{2^{n}})$ is contractible when $p$ is odd

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The order complex of $P G L_{2} (p^{2^{n}})$ is contractible when $p$ is odd

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