The Cyclic Graph of a Z-group

TL;DR

This paper explores the structure of a graph defined on Z-groups, where vertices are non-identity elements and edges indicate cyclic subgroups generated by pairs, revealing properties of these groups through their associated graphs.

Contribution

It introduces and analyzes the graph $ riangle(G)$ for Z-groups, providing new insights into their subgroup structure and cyclic properties.

Findings

Characterization of $ riangle(G)$ for Z-groups

Connections between group properties and graph structure

Identification of special features of the graph in Z-groups

Abstract

For a group $G$, we define a graph $\Delta(G)$ by letting $G^{\#} = G \setminus \{ 1 \}$ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\#}$ if and only if the subgroup $\langle x,y\rangle$ is cyclic. Recall that a $Z$-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $\Delta(G)$ for a $Z$-group $G$.