Conjugacy class

A conjugacy class of an element $x$ from a group $(G,\cdot)$ is the set $\{gxg^{-1}, g\in G\}$ .

It can also be viewed as the orbit of $x$ under the action $(g,x)\in G\times X\longmapsto gxg^{-1}\in X$ of $G$ over $X:=G$ .

Practice this concept with exercises

Let $G$ be a group with only one conjugacy class. What can $G$ be ?

Open exercise

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