# 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 $H$ of a group $G$ is termed amalgam-characteristic in $G$ if $H$ is a characteristic subgroup in the amalgam $L := G *_H G$.

### Potentially characteristic subgroup

Further information: Potentially characteristic subgroup

A subgroup $H$ of a group $G$ is termed potentially characteristic in $G$ if there exists a group $L$ with an injective map $\alpha: G \to L$ such that the image $\alpha(H)$ is a characteristic subgroup of $K$.

## 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 $G$ with a subgroup $H$ that is characteristic in the amalgam $L = G*_H G$.

To prove: $H$ is a potentially characteristic subgroup of $G$: there exists a group with an injective map from $G$ to that group such that the image of $H$ is characteristic in that group.

Proof: We claim that the group is, in fact, $L$ itself.

Observe that we can take the injective map $\alpha:G \to L$ as the embedding of the first amalgamated factor $G$. Under this embedding $\alpha(H)$ is the same as the amalgamated $H$, which by assumption is characteristic in $L$. Thus, we have an injective map from $G$ to $L$ under which the image of $H$ is characteristic in $L$.