Dedekind implies class two

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., Dedekind group) must also satisfy the second group property (i.e., group of nilpotency class two)
View all group property implications | View all group property non-implications
Get more facts about Dedekind group|Get more facts about group of nilpotency 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. 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.