Dedekind implies class two

Statement
Any Dedekind group (i.e., a group in which every subgroup is normal) must be a group of class (at most) two, i.e., the derived subgroup is contained in the center.

Facts used

 * 1) uses::Second center contains Baer norm: This in turn follows from Cooper's theorem.

Proof
The proof follows from fact (1) and the observations that a group is Dedekind iff it equals its Baer norm, and a group is of nilpotency class (at most) two iff it equals its second center.