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
Propertytheoretic statement
The group property of being a Dedekind group is stronger than the group property of being an ACICgroup.
Definitions used
Dedekind group
Further information: Dedekind group
A group is said to be Dedekind if every subgroup is normal.
ACICgroup
Further information: ACICgroup
A group is said to be ACIC if every automorphconjugate subgroup is normal, or equivalently, every automorphconjugate subgroup is characteristic.
Proof
Proof using subgroup property collapse
In a Dedekind group, every subgroup is normal. In particular, every automorphconjugate subgroup is normal, and hence, the group is an ACICgroup.