Quotient graphs for power graphs

TL;DR

This paper develops a formula for counting components of the proper power graph of finite groups, especially symmetric groups, using quotient graph techniques and explores their interrelations.

Contribution

It introduces a novel application of quotient graph procedures to the proper power graph of finite groups, providing explicit formulas and insights for symmetric groups.

Findings

Derived a formula for the number of components in the proper power graph of G

Established connections between various quotient graphs of the power graph

Computed the number of components for symmetric groups S_n

Abstract

In a previous paper of the first author a procedure was developed for counting the components of a graph through the knowledge of the components of its quotient graphs. We apply here that procedure to the proper power graph $\mathcal{P}_0(G)$ of a finite group $G$, finding a formula for the number $c(\mathcal{P}_0(G))$ of its components which is particularly illuminative when $G\leq S_n$ is a fusion controlled permutation group. We make use of the proper quotient power graph $\widetilde{\mathcal{P}}_0(G)$, the proper order graph $\mathcal{O}_0(G)$ and the proper type graph $\mathcal{T}_0(G)$. We show that all those graphs are quotient of $\mathcal{P}_0(G)$ and demonstrate a strong link between them dealing with $G=S_n$. We find simultaneously $c(\mathcal{P}_0(S_n))$ as well as the number of components of $\widetilde{\mathcal{P}}_0(S_n)$, $\mathcal{O}_0(S_n)$ and $\mathcal{T}_0(S_n)$.