On the symmetric digraphs from powers modulo

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