Finite groups with planar subgroup lattices

Abstract

It is natural to ask when a group has a planar Hasse lattice or more generally when its subgroup graph is planar. In this paper, we completely answer this question for finite groups. We analyze abelian groups, p-groups, solvable groups, and nonsolvable groups in turn. We find seven infinite families (four depending on two parameters, one on three, two on four), and three "sporadic" groups. In particular, we show that no nonabelian group whose order has three distinct prime factors can be planar.

Finite Groups Whose Intersection Graphs Are Planar

“…It is clear that the subgroup lattice determines the intersection graph, but not conversely. Moreover, comparing our main result with the main results of [2,11] we see that there are groups whose subgroup lattices are planar but the intersection graphs are not planar, and vice versa.…”

“…There are interesting graphs constructed from algebraic objects such as the subgroup lattice and the subgroup graph of a group. Planarity of the subgroup lattice and the subgroup graph of a group were studied by Bohanon and Reid in [2] and by Schmidt in [10,11] and by Starr and Turner III in [12], and planarity of the intersection graph of a module over any ring was studied in [13].…”

“…Our methods are elementary. Among the graphs similar to the intersection graphs, we may count the subgroup lattice and the subgroup graph of a group, each of whose planarity was already considered before in [2,10,11,12]. …”

The stable subgroup graph

“…In this paper we consider a graph with vertex set of all subgroups of a group so let us name some graphs which has a connection with the graph that is defined here, for instance subgroup graphs, subgroup lattice and intersection graphs. The subgroup graph of a group is the graph whose vertices are the subgroups of the group and two vertices, H 1 and H 2 , are connected by an edge if and only if H 1 ≤ H 2 and there is no subgroup K such that H 1 < K < H 2 (see [1]). Starr and Turner [11] were the first to study groups G with planar subgroup graph and classified all planar abelian groups.…”

“…Starr and Turner [11] were the first to study groups G with planar subgroup graph and classified all planar abelian groups. Also, Schmidt [9,10], Bohanon and Reid [1] simultaneously classified all finite planar groups.…”