The non-two-primes graph of a finite group

TL;DR

This paper introduces a graph associated with finite groups based on subgroup order divisibility by at least three primes, proving the connectivity and diameter bounds of its non-isolated vertices.

Contribution

It defines a new graph structure for finite groups and establishes its connectivity and diameter properties, advancing understanding of subgroup interactions.

Findings

The induced subgraph of non-isolated vertices is connected.

The diameter of this subgraph is at most 5.

The graph's structure reveals new insights into subgroup divisibility properties.

Abstract

To any finite group $G$, we may associate a graph whose vertices are the elements of $G$ and where two distinct vertices $x$ and $y$ are adjacent if and only if the order of the subgroup $\langle x, y\rangle$ is divisible by at least 3 distinct primes. We prove that the subgraph of this graph induced by the non-isolated vertices is connected and has diameter at most 5.