Quotient graphs for power graphs (Q1700659)

Summary: 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 \(G\leq S_n\) 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 \(\mathcal{T}_0(G)\). 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.