On symmetric digraphs of the congruence

We assign to each pair of positive integers n and k ≥ 2 a digraph G ( n , k ) whose set of vertices is H = { 0 , 1 , … , n − 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a k ≡ b ( mod n ) . The digraph G ( n , k ) is symmetric of order M if its set of components can be partitioned into subsets of size M with each subset containing M isomorphic components. We generalize earlier theorems by Szalay, Carlip, and Mincheva on symmetric digraphs G ( n , 2 ) of order 2 to symmetric digraphs G ( n , k ) of order M when k ≥ 2 is arbitrary.