Rendiconti del Seminario Matematico della Università di Padova


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Volume 129, 2013, pp. 35–46
DOI: 10.4171/RSMUP/129-3

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

Massimiliano Patassini[1]

(1) Vidor (TV), Italy

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 $m\in \{3,\ldots,35\}$ 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.

Keywords: Probabilistic zeta functions; simplicial complexes; order complexes; contractibility; coset posets; alternating groups; Brown conjecture

Patassini Massimiliano: On the (non-)Contractibility of the Order Complex of the Coset Poset of an Alternating Group. Rend. Sem. Mat. Univ. Padova 129 (2013), 35-46. doi: 10.4171/RSMUP/129-3