Intersection graphs of finite groups

Let $G$ be a group. The intersection graph $\Gamma(G)$ of $G$ 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 $X$ and $Y$ if and only if $X\cap Y\neq 1$ where $1$ denotes the trivial subgroup of $G$. The purpose of this thesis is to study the intersection graphs of finite groups. Particular emphasis was put on the graph theoretical invariants of those objects. In general, two non-isomorphic groups may have isomorphic intersection graphs. However, finite abelian groups can almost be distinguished by their intersection graphs. We prove that for any two abelian groups $A$ and $B$, their intersection graphs are isomorphic if and only if (i) the product of the non-cyclic Sylow subgroups of $A$ is isomorphic to the product of the non-cyclic Sylow subgroups of $B$, and (ii) exponents of the orders of the cyclic Sylow subgroups of $A$ and of $B$ are equal up to a permutation. We classified all finite groups whose intersection graphs are planar. There are a few abelian groups with planar intersection graphs and the only non-abelian nilpotent groups with planar intersection graph are the dihedral group $D_8$ of order eight and the quaternion group $Q_8$. The rest of the list consists of some semi-direct products. In particular, there is no non-solvable group whose intersection graphs is planar. By Kuratowski's Theorem a graph is planar if and only if it does not contain the complete graph $K_5$ over five vertices and the complete bipartite graph $K_{3,3}$ as a minor. We further determine the finite groups whose intersection graphs contains a $K_5$ but not $K_{3,3}$ as a subgraph. We studied the connectivity of intersection graphs of finite groups. Intuitively, intersection graphs should be highly connected graphs and if there are some examples of such graphs with `low' connectivity, they must be exceptional. We classified finite solvable groups whose intersection graphs are not $2$-connected and finite nilpotent groups whose intersection graphs are not $3$-connected.