# number of groups with fixed number of conjugacy classes [duplicate]

why is there only finite number of (finite or infinite)groups with a fixed number of conjugacy classes? I know this is classical ,so plz give me a reference if you have. thank you

## marked as duplicate by Mikko Korhonen, Dietrich Burde, Ali Caglayan, Aditya Hase, UserXNov 27 '14 at 17:58

• I am not sure that there are finitely many isomorphism types with a given finite number of conjugacy classes if you allow infinite groups. I think there are infinite groups of prime exponent $p$ with $p$ conjugacy classes, for example, and I think there are infinite groups with two conjugacy classes. – Geoff Robinson Nov 27 '14 at 16:43
In 1903 Edmund Landau proved that, for any positive integer $k$, there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes. I think the paper is to be found in the Math. Annalen 56, in German (Über die Klassenzahl der binären quadratischen Formen von negativer Discriminante). See also here.