The generating graph of the abelian groups

TL;DR

This paper studies the structure of generating graphs of abelian groups, proving connectivity and diameter properties for various classes, and extends results to free groups, revealing differences in their generating graph structures.

Contribution

It establishes new connectivity and diameter results for the generating graphs of 2-generated abelian groups and free groups, highlighting structural differences.

Findings

-generated non-cyclic abelian groups have connected generating graphs with diameter 2 or infinite.

The generating graph of the free group of rank 2 is connected with infinite diameter.

The diameter of the generating graph of  imes  is infinite.

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 $\Gamma^*(G)$ 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 $\Gamma^*(G)$ is connected. Moreover $\mathrm{diam}(\Gamma^*(G))=2$ if the torsion subgroup of $G$ is non-trivial and $\mathrm{diam}(\Gamma^*(G))=\infty$ otherwise. If $F$ is the free group of rank 2, then $\Gamma^*(F)$ is connected and we deduce from $\mathrm{diam}(\Gamma^*(\mathbb{Z}\times \mathbb{Z}))=\infty$ that $\mathrm{diam}(\Gamma^*(F))=\infty.$