Topology of subgroup lattices of symmetric and alternating groups.
When \(G\) is a non-solvable finite group its order complex \(\Delta(G)\) is mostly an unknown entity other than what certain forays, as executed in this interesting paper, have managed to demonstrate. By looking at \(\Delta(S_n)\) and \(\Delta(A_n)\) when \(n\) is prime or for \(\Delta(S_n)\) when \(n\) is a power of 2, more detailed information about these complexes shows them as wedges of spheres of (not always the same) certain dimensions particularizing the information in the \(\Delta(S_n)\) case for \(n>1\) contained in Theorem 1.1, with the partition lattice playing an important role and noting that \(\dim(\widetilde H_{n-3} (\Delta(S_n)))\geq\tfrac{n!}2\) for all \(n>1\) as a consequence of Theorem 1.1 as well. Conjecture 1.3 is then also one of natural expectation as well as curiosity which points ahead at directions to follow using methods and techniques which have become if not all standard at least recognizable when first encountered by the somewhat cognoscenti in this area.
scientific article; zbMATH DE number 2022048