Topology of order complexes of intervals in subgroup lattices

Abstract

We conjecture that the order complex of an open interval in the subgroup lattice of a finite group has the homotopy type of a wedge of spheres and prove that if ( H, G) is a minimal counterexample to this conjecture then either G is almost simple or G= HN, where N is the unique minimal normal subgroup of G, N is non-Abelian and H∩ N=1.