Given an element $~$g$~$ of a Group $~$G$~$, the conjugacy class of $~$g$~$ in $~$G$~$ is $~$\{ x g x^{-1} : x \in G \}$~$. It is the collection of elements to which $~$g$~$ is conjugate.

[todo: examples] [todo: class equation] [todo: it is the stabiliser of a certain action, which we can show conditionally on the right requisites]