## The order of a conjugacy class is bounded by the index of the center

Let $G$ be a group. If $Z(G)$ has index $n$, prove that every conjugacy class has at most $n$ elements.

Under the action of $G$ on itself by conjugacy, note that $Z(G) \leq \mathsf{stab}(x)$ for all $x \in G$. Now $[G:\mathsf{stab}(x)][\mathsf{stab}(x):Z(G)] = [G:Z(G)]$, so that by the Orbit-Stabilizer Theorem, $[G : \mathsf{stab}(x)] = |G \cdot x|$ divides $n$. In particular, the number of elements in each conjugacy class is at most $n$.