Size of conjugacy class is bounded by order of derived subgroup

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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 Derived subgroup (?) of G.

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

Facts used

  1. Every conjugacy class is contained in a coset of the derived subgroup
  2. 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.