Size of conjugacy class is bounded by order of derived subgroup

Statement
Suppose $$G$$ is a group and $$c$$ is a conjugacy class in $$G$$. Then, the size of $$c$$ is bounded by the order of the fact about::derived subgroup of $$G$$.

This in particular imposes a constraint on the conjugacy class size statistics of a finite group.

Facts used

 * 1) uses::Every conjugacy class is contained in a coset of the derived subgroup
 * 2) uses::Left cosets are in bijection via left multiplication

Proof
By Facts (1) and (2), every conjugacy class is contained in a set whose size equals that of the derived subgroup, completing the proof.