Quotient graphs for power graphs

الأصل · EN

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 P₀(G) of a finite group G, finding a formula for the number c(P₀(G)) of its components which is particularly illuminative when G≤ Sₙ is a fusion controlled permutation group. We make use of the proper quotient power graph P₀(G), the proper order graph O₀(G) and the proper type graph T₀(G). We show that all those graphs are quotient of P₀(G) and demonstrate a strong link between them dealing with G=Sₙ. We find simultaneously c(P₀(Sₙ)) as well as the number of components of P₀(Sₙ), O₀(Sₙ) and T₀(Sₙ).