Characteristic not implies strictly characteristic

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., strictly characteristic subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
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EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property characteristic subgroup but not strictly characteristic subgroup|View examples of subgroups satisfying property characteristic subgroup and strictly characteristic subgroup

Statement

A characteristic subgroup of a group need not be a strictly characteristic subgroup.

Partial truth

It is true that for any finite group and more generally for any Hopfian group, characteristic subgroups are always strictly characteristic. This is because for Hopfian groups, surjective endomorphisms are the same thing as automorphisms. For full proof, refer: Characteristic equals strictly characteristic in Hopfian

Proof

Example constructed by Baer

This example is somewhat complex.

References

Journal references

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