# Characteristic equals strictly characteristic in Hopfian

From Groupprops

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.

View all subgroup property implications in Hopfian groups View all subgroup property non-implications in Hopfian groups View all subgroup property implications View all subgroup property non-implications

## 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.