Past exam of the mathematics course of the University of Cambridge / 2005 / ib / Paper 3 / 1C (Groups, Rings and Modules)

The conjugate group elements relation is $x\sim y$ when $y=gx{g}^{-1}$ for some $g\in G$. It is reflexive by taking the identity, symmetric because $x=g^{-1}yg$, and transitive because $y=gx{g}^{-1}$ and $z=hy{h}^{-1}$ imply $z=(hg)x(hg{)}^{-1}$. Hence it is an equivalence relation.

For a finite group, the centralizer $C_G(x)=\{g:gx=xg\}$ is a subgroup. The map $g\mapsto gx{g}^{-1}$ has equal values at $g,h$ exactly when $h^{-1}g\in C_G(x)$. Its fibers are therefore the left cosets of $C_G(x)$, givingBy Lagrange's theorem, this index divides $|G|$. This is also the orbit-stabilizer theorem for the conjugation action, with the centralizer as stabilizer.