Characteristic equals strictly characteristic in Hopfian

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a Hopfian group. That is, it states that in a Hopfian group (?), every subgroup satisfying the first subgroup property (i.e., Characteristic subgroup (?)) must also satisfy the second subgroup property (i.e., Strictly characteristic subgroup (?)). In other words, every characteristic subgroup of Hopfian group is a strictly characteristic subgroup of Hopfian group.
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Statement

In a Hopfian group (i.e., a group that is not isomorphic to any proper quotient of itself), the property of being a characteristic subgroup is equivalent to the property of being a strictly characteristic subgroup. In other words, every characteristic subgroup is strictly characteristic, and every strictly characteristic subgroup is characteristic.