DIVISIBILITY GRAPH FOR SYMMETRIC AND ALTERNATING GROUPS

Let X be a nonempty set of positive integers and X* = X\{1}. The divisibility graph D(X) has X* as the vertex set, and there is an edge connecting a and b with a, b is an element of X* whenever a divides b or b divides a. Let X = cs(G) be the set of conjugacy class sizes of a group G. In this case, we denote D(cs(G)) by D(G). In this paper, we will find the number of connected components of D(G) where G is the symmetric group S-n or is the alternating group A(n).