Abelian implies every subgroup is normal

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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 must also satisfy the second group property
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Statement

Verbal statement

Every subgroup of an Abelian group is a normal subgroup.

Property-theoretic statement

The group property of being an Abelian group is stronger than the group property of being a Dedekind group (a group where every subgroup is normal).

Definitions used

Abelian group

Normal subgroup

Dedekind group

Proof

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