On intervals in subgroup lattices of finite groups

Abstract

We investigate the question of which finite lattices L L are isomorphic to the lattice [ H , G ] [H,G] of all overgroups of a subgroup H H in a finite group G G . We show that the structure of G G is highly restricted if [ H , G ] [H,G] is disconnected. We define the notion of a “signalizer lattice" in H H and show for suitable disconnected lattices L L , if [ H , G ] [H,G] is minimal subject to being isomorphic to L L or its dual, then either G G is almost simple or H H admits a signalizer lattice isomorphic to L L or its dual. We use this theory to answer a question in functional analysis raised by Watatani.