On the symmetric digraphs from powers modulo nn

For any positive integers nn and kk, let G(n,k)G(n,k) denote the digraph whose set of vertices is H={0,1,2,…,n−1}H={0,1,2,…,n−1} and there is a directed edge from a∈Ha∈H to b∈Hb∈H if ak≡b(modn). The digraph G(n,k)G(n,k) is called symmetric of order MM if its set of connected components can be partitioned into subsets of size MM with each subset containing MM isomorphic components. In this paper, we establish a necessary and sufficient condition for G(n,k)G(n,k) to be symmetric of order MM, where MM has an odd prime divisor.