Past exam of the options II course of the University of Oxford / 2012 / B / B2 / c / Solution

Let $H=Z(x)$. The map $G/H\to C(x)$ sending $gH$ to $gx{g}^{-1}$ is well defined: multiplying $g$ on the right by an element commuting with $x$ leaves its conjugate unchanged. It is onto by the orbit definition. If $gx{g}^{-1}=hx{h}^{-1}$, then $h^{-1}g$ commutes with $x$, so $gH=hH$, proving injectivity.

Every coset has $|H|$ elements, so $|C(x)|=|G|/|H|$. ThereforeThis is the orbit-stabilizer theorem specialized to conjugation.