Classification of finite groups with birdcage-shaped Hasse diagrams

Abstract

The subgroup posets of finite groups are illustrated by graphs (Hasse diagrams). The properties of these graphs have been studied by many researchers — initiated by K. Brown and D. Quillen for $p$-subgroups. We consider the opposite direction, that is, a realization problem: Given a graph, when does a finite group with its Hasse diagram being the graph exist?, and if any, classify all such finite groups. The Hasse diagram of a group is not arbitrary — it has the top and the bottom vertices, and they are connected by paths of edges. We divide such graphs into two types ``branched'' and “unbranched”, where unbranched graphs are birdcage-shaped, and finite groups with their Hasse diagrams being such graphs are called birdcage groups. We completely classify the unbranched case: A birdcage group is either a cyclic group of prime power order or a semidirect product of two cyclic groups of prime orders (). In the former, the Hasse diagram is a straight line (a birdcage with a single bar) and in the latter, a birdcage with all bars being of length 2.