The generating graph of the abelian groups

Abstract

For a group $G,$ let $\Gamma (G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Moreover let ${\Gamma }^{\ast }(G)$ be the subgraph of $\Gamma (G)$ that is induced by all the vertices of $\Gamma (G)$ that are not isolated. We prove that if $G$ is a 2-generated non-cyclic abelian group then ${\Gamma }^{\ast }(G)$ is connected. Moreover $\mathrm{diam}({\Gamma }^{\ast }(G))=2$ if the torsion subgroup of $G$ is non-trivial and $\mathrm{diam}({\Gamma }^{\ast }(G))=\infty$ otherwise. If $F$ is the free group of rank 2, then ${\Gamma }^{\ast }(F)$ is connected and we deduce from $\mathrm{diam}({\Gamma }^{\ast }(\mathbb{Z}\times \mathbb{Z}))=\infty$ that $\mathrm{diam}({\Gamma }^{\ast }(F))=\infty .$