The connectivity of the normalising and permuting graph of a finite soluble group

TL;DR

This paper investigates the connectivity properties of the normalising and permuting graphs of finite soluble groups, establishing bounds on their diameter and classifying groups with disconnected graphs.

Contribution

It introduces the normalising graph of a group and provides a classification of finite soluble groups based on the connectivity of these graphs.

Findings

Connected normalising graph has diameter at most 6.

The bound on diameter is tight.

Connectivity properties of the permuting graph are derived as a corollary.

Abstract

We introduce the normalising graph of a group and study the connectivity of the normalising and permuting graphs of a group when the group is finite and soluble. In particular, we classify finite soluble groups with disconnected normalising graph. The main results shows that if a finite soluble group has connected normalising graph then this graph has diameter at most 6. Furthermore, this bound is tight. A corollary then presents the connectivity properties of the permuting graph.