Connectivity of Intersection Graphs of Finite Groups

TL;DR

This paper classifies finite solvable and nilpotent groups based on the connectivity properties of their intersection graphs, providing elementary methods for the classification.

Contribution

It offers a classification of finite solvable and nilpotent groups with specific intersection graph connectivity properties, using elementary techniques.

Findings

Finite solvable groups with non-2-connected intersection graphs identified.

Finite nilpotent groups with non-3-connected intersection graphs characterized.

Elementary methods used for classification.

Abstract

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.