Conjectures on intervals in subgroup lattices of finite groups
by
John Shareshian
Washington University, St. Louis

The question "Is every finite lattice isomorphic to an interval in the lattice of subgroups of a finite group?" is open. There has been significant progress towards showing that the answer to this question is "no", obtained by examining lattices of height two. I will present three conjectures, a positive answer to any of these providing a negative answer to the original question. The strongest of these is that the order complex of the proper part of every interval in the subgroup lattice any finite group has the homotopy type of a wedge of spheres. I will show that if [H,G] is a minimal counterexample to any one of my conjectures, then G has a unique minimal normal subgroup N, N is nonabelian, and either N is simple or H is a complement to N in G.

Date received: March 30, 2005