# The diameter of the generating graph of a finite soluble group

## TL;DR

This paper investigates the diameter of the generating graph of finite soluble groups, establishing conditions under which certain generating pairs exist and providing a counterexample for a specific case.

## Contribution

It proves new conditions for the existence of common generators in finite soluble groups and constructs a counterexample demonstrating limitations of these conditions.

## Key findings

- Existence of a common generator under certain conditions
- Counterexample of a 2-generated soluble group where the condition fails
- Weaker universal result for all finite soluble groups

## Abstract

Let $G$ be a finite 2-generated soluble group and suppose that $\langle a_1,b_1\rangle=\langle a_2,b_2\rangle=G$. If either $G^\prime$ is of odd order or $G^\prime$ is nilpotent, then there exists $b \in G$ with $\langle a_1,b\rangle=\langle a_2,b\rangle=G.$ We construct a soluble 2-generated group $G$ of order $2^{10}\cdot 3^2$ for which the previous result does not hold. However a weaker result is true for every finite soluble group: if $\langle a_1,b_1\rangle=\langle a_2,b_2\rangle=G$, then there exist $c_1, c_2$ such that $\langle a_1, c_1\rangle = \langle c_1, c_2\rangle =\langle c_2, a_2\rangle=G.$