Invariable generation of finite simple groups and rational homology of coset posets (Q6612182)

Let \(G\) be a finite group, then two subsets \(S, T \subseteq G\) generate \(G\) invariably if \(G=\langle S^{g}, T^{h} \rangle\) for all \(g, h \in G\).\N\NIn the paper under review, the authors using the classification of finite simple groups prove Theorem 1.1: Every finite simple group is generated invariably by a Sylow subgroup and a cyclic subgroup.\N\NIf \(\mathcal{P}\) is a finite poset, then the order complex \(\Delta \mathcal{P}\) is the abstract simplicial complex whose \(d\)-dimensional faces are the chains of length \(d\) from \(\mathcal{P}\). If \(G\) is a finite group, let \(\mathcal{C}(G)\) be the set of all cosets of all proper subgroups of \(G\), ordered by inclusion. Thanks to Theorem 1.1, the authors also prove Theorem 1.2: If \(G\) is a finite group, then \(\Delta \mathcal{C}(G)\) has non-trivial reduced rational homology.