G(a, m)

V = {0, 1, …, m−1}, edges b → b + a (mod m)

m (vertices) 12

a (step) 3

What to look for

Every vertex has out-degree 1, so G(a, m) is a disjoint union of directed cycles.

Let d = gcd(a, m). Then G(a, m) has exactly d cycles, each of length m/d.

Toggle to Disjoint cycles to see each coset b + aℤ drawn as its own regular polygon, with vertices in the order b → b+a → b+2a → ….

G(a, m)