Quotient graphs for power graphs

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 $G\le S_n$ is a fusion controlled permutation group. We make use of the proper quotient power graph ${\tilde{\mathcal{P}}}_0(G)$, 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 ${\tilde{\mathcal{P}}}_0(S_n)$, ${\mathcal{O}}_0(S_n)$ and ${\mathcal{T}}_0(S_n)$.

Cited by in corpus (1)

• The complement of proper power graphs of finite groups