American Mathematical Monthly
(1998), no. 4, pp. 359-361.

The Number of Conjugacy Classes,
by Michael Reid

American Mathematical Monthly
(1998), no. 4, pp. 359-361.

Abstract

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 [1].
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
*n* .

Reference

[1]
Bjorn Poonen,
Congruences Relating the Order of a Group to the Number of Conjugacy Classes,
American Mathematical Monthly
(1995), no. 5, pp. 440-442.
[Zentralblatt]

Updated January 8, 2008.