Connectivity Of Intersection Graphs Of Finite Groups

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 classify finite solvable groups whose intersection graphs are not $2$-connected and finite nilpotent groups whose intersection graphs are not $3$-connected. Our methods are elementary.
Corrected according to the referee's comments. In particular, statements and proofs of Lemma 2 and Lemma 12 are changed. Also the proof of Theorem A in this version is due to I. M. Isaacs