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
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group is not necessarily finite in my case – joda Nov 27 '14 at 16:32
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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
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are u sure? can u give me a reference plz? – joda Nov 27 '14 at 16:47
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You can embed any torsion free group in a group with only two conjugacy classes. This is a standard application of HNN extensions. – Derek Holt Nov 27 '14 at 20:43
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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.
Nicky Hekster
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i cant read german but it seems it is in finite case at the end. has it proven for all groups in this? – joda Nov 27 '14 at 16:48
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Yes, for finite groups. See the link in my answer: for infinite groups the situation is different. – Nicky Hekster Nov 27 '14 at 19:02