Journal of the
Korean Mathematical Society JKMS

The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$ where $1$ denotes the trivial subgroup of $G.$ In this paper we characterize all finite groups whose intersection graphs are planar. 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 \cite{boh,schm2,schm,starr}.

Keywords: finite groups, subgroup, intersection graph, planar

MSC numbers: Primary 20D99; Secondary 20D15, 20D25, 05C25