# 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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## Statement

Any amalgam-characteristic subgroup of a group is a potentially characteristic subgroup.

## Definitions used

### Amalgam-characteristic subgroup

`Further information: Amalgam-characteristic subgroup`

A subgroup of a group is termed amalgam-characteristic in if is a characteristic subgroup in the amalgam .

### 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 .

## Related facts

### Converse

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.

## Proof

**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 .