The diameter of the generating graph of a finite soluble group

Abstract

Let $G$ be a finite 2-generated soluble group and suppose that $⟨a_1,b_1⟩=⟨a_2,b_2⟩=G$. If either $G^{\prime }$ is of odd order or $G^{\prime }$ is nilpotent, then there exists $b\in G$ with $⟨a_1,b⟩=⟨a_2,b⟩=G.$ 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 $⟨a_1,b_1⟩=⟨a_2,b_2⟩=G$, then there exist $c_1,c_2$ such that $⟨a_1,c_1⟩=⟨c_1,c_2⟩=⟨c_2,a_2⟩=G.$