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.

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