On the shellability of the order complex of the subgroup lattice of a finite group (Q2719043)

Given a finite group \(G\), it determines a subgroup lattice \(L(G)\) which reflects properties of \(G\). In turn a poset \(P\) determines a simplicial order complex \(\Delta(P)\) whose simplexes are maximal chains with faces of lesser dimension subchains of such maximal chains. Thus, if \(P=L (G)\), then \(G\) determines a simplicial complex in a natural but nevertheless quite complicated way. It may therefore seem quite surprising to find that as shown in this rather interesting paper a very important property of groups, i.e., solvability, shall be matched with an equally important property of simplicial complexes \(\Delta\), i.e., (nonpure) shellability, in the form of a theorem: A finite group \(G\) is solvable if \(\Delta(L(G))\) is shellable (nonpure) as a simplicial complex, thus generalizing another equally interesting theorem: a finite group \(G\) is supersolvable iff \(\Delta(L(G))\) is pure shellable as a simplicial complex. Considering such pairs of properties of \(G\) and \(\Delta (L(G))\) as ``matched'' (somehow), it seems proper to consider the question of existence of similar such matchings as of interest and these results as invitations to further exploration in this direction. As noted by the author and a consequence of his method of proof he determines the homology types of the order complexes of the subgroup lattices of many minimal simple groups in settling the issue that nonsolvable groups do not have shellable (order complexes associated with) subgroup lattices. For those seeking salvation in isolation of subjects in all purity, this is an excellent counterexample demonstrating once more the mysterious one-ness an aspect of which is so nicely exhibited here.