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On the (non-)contractibility of the order complex of the coset poset of an alternating group. (English) Zbl 1281.20027
The coset poset of a finite group \(G\) consists of all cosets of all proper subgroups of \(G\), ordered by inclusion, and the order complex of any poset \(P\) is the simplicial complex whose \(k\)-dimensional faces are the ordered subsets of \(P\) that have size \(k+1\). Brown conjectured that the order complex of the coset poset is never contractible, and in fact that its reduced Euler characteristic is never zero. This fascinating conjecture remains open. Some help to deal with this question comes from the study of the Dirichlet polynomial \(P_G(s)\) associated to \(G\). Indeed \(P_G(-1)\) coincides with the reduced Euler characteristic of the order complex of the coset poset of \(G\).
In a previous paper [J. Algebra 343, No. 1, 37-77 (2011; Zbl 1242.20026)] the author used this approach to prove that the order complex of the coset poset of \(G\) is non-contractible if \(G\) is an almost simple finite classical group that does not contain a graph automorphism of its socle.
In the present paper he deals with the alternating groups, proving that the order complex of the coset poset of \(\text{Alt}(k)\) 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 \(p>k/2\) is a prime. Moreover he extends the result to some monolithic primitive groups whose socle is a direct product of alternating groups.

MSC:
20D30 Series and lattices of subgroups
20P05 Probabilistic methods in group theory
11M41 Other Dirichlet series and zeta functions
20D06 Simple groups: alternating groups and groups of Lie type
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