The Cyclic Graph of a $2$-Frobenius Group

TL;DR

This paper investigates the structure of the cyclic graph of 2-Frobenius groups, revealing its disconnected nature and determining the number of connected components for these groups.

Contribution

It provides a detailed analysis of the cyclic graph of 2-Frobenius groups and explicitly calculates the number of connected components.

Findings

Cyclic graph of a 2-Frobenius group is disconnected.

Number of connected components of the cyclic graph is determined.

Insights into the subgroup structure of 2-Frobenius groups.

Abstract

The cyclic graph of a group $G$ is the graph whose vertices are the nonidentity elements of $G$ and whose edges connect distinct elements $x$ and $y$ if and only if the subgroup $\langle x,y\rangle$ is cyclic. We obtain information about the cyclic graph of $2$-Frobenius groups. The cyclic graph of a $2$-Frobenius group is disconnected. In this paper, we determine the number of connected components of the cyclic graph of any $2$-Frobenius group.