Dedekind implies ACIC

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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

Property-theoretic statement

The group property of being a Dedekind group is stronger than the group property of being an ACIC-group.

Definitions used

Dedekind group

Further information: Dedekind group

A group is said to be Dedekind if every subgroup is normal.

ACIC-group

Further information: ACIC-group

A group is said to be ACIC if every automorph-conjugate subgroup is normal, or equivalently, every automorph-conjugate subgroup is characteristic.

Proof

Proof using subgroup property collapse

In a Dedekind group, every subgroup is normal. In particular, every automorph-conjugate subgroup is normal, and hence, the group is an ACIC-group.