The Cyclic Graph of a Z-group

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 $⟨x,y⟩$ 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 $G$.

6 pages, 3 figures