On the (non-)Contractibility of the Order Complex of the Coset Poset of an Alternating Group

Abstract

Let Alt ${}_k$ be the alternating group of degree $k$. In this paper we prove that the order complex of the coset poset of Alt ${}_k$ is non-contractible for a big family of $k\in \mathbb{N}$ , including the numbers of the form $k=p+m$ where $m\in \{3,\dots ,35\}$ and $p>k/2$. In order to prove this result, we show that $P_G(-1)$ does not vanish, where $P_G(s)$ is the Dirichlet polynomial associated to the group $G$. Moreover, we extend the result to some monolithic primitive groups whose socle is a direct product of alternating groups.