We prove that if G is a finite group whose order is not divisible
by 3 , and G has n conjugacy classes, then the
congruence |G| = n mod 3 holds.
This is easy to prove using representation theory of finite groups,
but we give an elementary proof, building on a technique of Poonen .
If G is a finite group such that every prime divisor, p ,
of its order satisfies p ≡ 1 mod m , then we find
and prove the strongest possible congruence between |G| and