Size of a conjugacy class is the index of the centraliser

Statement

Let $G$ be a group and let $a \in G$. Then $|\operatorname{Cl}(a)| = [G : C_G(a)]$, where $\operatorname{Cl}(a)$ is the conjugacy class of $a$ and $C_G(a)$ is its centraliser. In particular, if $G$ is finite, then $|\operatorname{Cl}(a)|$ divides $|G|$.

Remarks

This is the orbit-stabiliser theorem for the action by conjugation. Directly, $gag^{-1} = hah^{-1}$ if and only if $h^{-1}g \in C_G(a)$. For example, in $S_3$ the centraliser of $(1\ 2\ 3)$ is $A_3$, of index $2$, and the class of $(1\ 2\ 3)$ has $2$ elements.

Explanations

L1 human

Conjugates of $a$ by two elements are equal exactly when the two elements differ by something that commutes with $a$. So the conjugates of $a$ correspond to the cosets of the centraliser. An element that commutes with many elements has few conjugates.