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If $G$ is an infinite group with finitely many conjugacy classes, what can be said about $G$? (Should $G$ be simple/ solvable/....?) For $n\geq 2$, does there exists a (infinite) group $G$ with exactly $n$ conjugacy classes, which is periodic also?

I couldn't find any information on these questions. One may provide links also for details. Thanks in advance.

Beginner
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    Well, since any torsion free group can be embedded in a group with only two conjugation classes (this imo is astonishing. It follows from the HNN extension theorems), we already see that not much can be said... – DonAntonio Nov 19 '13 at 05:32
  • @DonAntonio I never knew that. Neat! – zibadawa timmy Nov 19 '13 at 06:26
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    For you second question, consider the group $Z/nZ$. – Moishe Kohan Nov 19 '13 at 08:31
  • @studious: this is correct; but I had mistake in question; the groups I was expecting were infinite. – Beginner Nov 19 '13 at 08:59
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    @zibadawatimmy, you can read about the HNN extensions in the books by Rotman and by Robinson. The former also brings some topological applications, with van Kampen and stuff. Very interesting. – DonAntonio Nov 19 '13 at 13:36

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