Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 7 / a (Conjugation and its orbits)

Write $g\cdot x=gx{g}^{-1}$. If $e$ is the identity element of $G$, then $e\cdot x=ex{e}^{-1}=x$. Also, for $g,h,x\in G$,The middle equality uses associativity and the inverse element rule $(gh{)}^{-1}=h^{-1}{g}^{-1}$. Thus this is a left group action, the conjugation action of $G$ on itself.

The orbit of $x$ under conjugation is its conjugacy classIt contains $x$ by using $g=e$, so the conjugacy classes cover $G$. If $z=gx{g}^{-1}$, then $x=g^{-1}zg$, and the group action identity shows that conjugating either element further gives the same orbit; hence $C(z)=C(x)$. If $C(x)$ and $C(y)$ intersect at $z$, this gives $C(x)=C(z)=C(y)$. Thus distinct conjugacy classes are disjoint and the conjugacy classes form a partition of a set $G$.