How to find (describe) all groups which have 3 conjugacy classes?
Thanks in advance!
How to find (describe) all groups which have 3 conjugacy classes?
Thanks in advance!
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