Quotient graphs for power graphs

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 P-0(G) of a finite group G, finding a formula for the number of its components which is particularly illuminative when G <= S-n is a fusion controlled permutation group. We make use of the proper quotient power graph (P) over tilde (0)(G), the proper order graph O-0(G) and the proper type graph T-0(G). All those graphs are quotient of 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 P(S-n) is 2-connected if and only if the type graph T(S-n) is 2-connected, if and only if the order graph O(S-n) is 2-connected, that is, if and only if either n = 2 or none of n; n - 1 is a prime.