# Dedekind not implies abelian

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property need not satisfy the second subgroup property
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EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property {{{stronger}}} but not {{{weaker}}}
• Property "Satisfies property" (as page type) with input value "{{{stronger}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
• Property "Dissatisfies property" (as page type) with input value "{{{weaker}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
|View examples of subgroups satisfying property {{{stronger}}} and {{{weaker}}}
• Property "Satisfies property" (as page type) with input value "{{{stronger}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
• Property "Satisfies property" (as page type) with input value "{{{weaker}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.

## 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.