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 one of its quotient graphs. Here we apply that procedure to the proper power graph ${\mathcal{P}}_0(G)$ of a finite group $G$, finding a formula for the number 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 ${\mathcal{T}}_0(G)$. All those graphs are quotient of ${\mathcal{P}}_0(G)$. We emphasize the strong link between them determining number and typology of the components of the above graphs for $G=S_n$. In particular, we prove that the power graph $\mathcal{P}(S_n)$ is $2$-connected if and only if the type graph $\mathcal{T}(S_n)$ is $2$-connected, if and only if the order graph $\mathcal{O}(S_n)$ is $2$-connected, that is, if and only if either $n=2$ or none of $n,n-1$ is a prime.