Dedekind not implies abelian

Statement
The group property of being a Dedekind group (i.e., a group where every subgroup is normal) does not imply the group property of being an Abelian group.

Proof
Consider the quaternion group. This is a group of order eight, where every subgroup is normal. However, the group is not Abelian: the elements $$i$$ and $$j$$, for instance, do not commute.

In fact, the quaternion group is in some sense the only counterexample: any non-Abelian Dedekind group is a direct product of the quaternion group and an Abelian group with the Abelian group satisfying certain conditions.