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How to find (describe) all groups which have 3 conjugacy classes?

Thanks in advance!

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    http://math.stackexchange.com/questions/52350/finite-groups-with-exactly-n-conjugacy-classes-n-2-3?rq=12012-12-26

1 Answers 1

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Well, the unit element is always a conjugacy class by itself, just as any element is in any abelian group, so you need other two classes...for example, the (cyclic) abelian group of order $\,3\,$, but also the permutation group $\,S_3\,$, which only has transpositions and $\,3-$cycles.

It could be now a nice exercise to show the above are the only finite groups, up to isomorphism, with three conjugacy classes.

About infinite groups I don't know: constructions like HNN show there can be very funny groups all the non-unit elements of which are conjugated, so this may require lots of care

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    I'd rather write the above as $|G|=1+\frac{|G|}{|C_2|}+\frac{|G|}{|C_3|}\Longleftrightarrow 1=\frac{1}{|G|}+\frac{1}{|C_2|}+\frac{1}{|C_3|}$ Observe that we can assume \,|G|>3\, and non abelian, so it can't be that $\,|C_2|\geq 4\,$ and etc. Try to finish this.2012-12-27