Amalgam-characteristic implies potentially characteristic
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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., amalgam-characteristic subgroup) must also satisfy the second subgroup property (i.e., potentially characteristic subgroup)
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Further information: Amalgam-characteristic subgroup
Potentially characteristic subgroup
Further information: Potentially characteristic subgroup
A subgroup of a group is termed potentially characteristic in if there exists a group with an injective map such that the image is a characteristic subgroup of .
The converse is not true. This follows from the fact that characteristic not implies amalgam-characteristic and characteristic implies potentially characteristic. In other words, there are characteristic subgroups that are not amalgam-characteristic. Since any characteristic subgroup is potentially characteristic, we obtain examples of potentially characteristic subgroups that are not amalgam-characteristic.
Given: A group with a subgroup that is characteristic in the amalgam .
To prove: is a potentially characteristic subgroup of : there exists a group with an injective map from to that group such that the image of is characteristic in that group.
Proof: We claim that the group is, in fact, itself.
Observe that we can take the injective map as the embedding of the first amalgamated factor . Under this embedding is the same as the amalgamated , which by assumption is characteristic in . Thus, we have an injective map from to under which the image of is characteristic in .