# Dedekind not implies abelian

From Groupprops

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 (i.e., Dedekind group) neednotsatisfy the second group property (i.e., Abelian group)

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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.