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A group where every two elements different than 1 are conjugate has order 1 or 2.
I have to show this result:
If G is a finite group with exactly two conjugacy classes, then G is isomorphic to $\mathbb{Z}_2$
I have tried to show that $|G|=2$ but without success. I'm not supposed to use Sylow's theorems on this one. I have tried to make $g$ act upon itself through conjugation, but that does not lead me to any conclusion upon its order.
Can someone help me?