Dedekind not implies abelian
This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property need not satisfy the second group property
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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 and , 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.