# Size of conjugacy class is bounded by order of derived subgroup

From Groupprops

## Statement

Suppose is a group and is a conjugacy class in . Then, the size of is bounded by the order of the Derived subgroup (?) of .

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

## Facts used

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