The Number of Conjugacy Classes

The Number of Conjugacy Classes, by Michael Reid
American Mathematical Monthly 105 (1998), no. 4, pp. 359-361.
[DOI] [Zentralblatt]
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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 .
[1] Bjorn Poonen, Congruences Relating the Order of a Group to the Number of Conjugacy Classes, American Mathematical Monthly 102 (1995), no. 5, pp. 440-442. [Zentralblatt]
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Updated January 8, 2008.