Direct factor not implies amalgam-characteristic

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., direct factor) need not satisfy the second subgroup property (i.e., amalgam-characteristic subgroup)
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Statement

It is possible to have a group $G$ and a direct factor $H$ of $G$ such that $H$ is not a characteristic subgroup in the amalgamated free product $L:=G*_{H}G$.

Related facts

Similar facts

• Normal not implies amalgam-characteristic
• Cocentral not implies amalgam-characteristic
• Characteristic not implies amalgam-characteristic

Opposite facts

• Central implies amalgam-characteristic
• Normal subgroup contained in the hypercenter is amalgam-characteristic
• Finite normal implies amalgam-characteristic
• Periodic normal implies amalgam-characteristic

Proof

Example of the free group

Let $F$ be a free group on two generators and $\mathbb {Z}$ be the group of integers. Let $G=F\times \mathbb {Z}$ and $H=F\times \{0\}$ be the embedded first direct factor. We have:

$L=(F\times \mathbb {Z} )*_{F\times \{0\}}(F\times \mathbb {Z} )=F\times (\mathbb {Z} *\mathbb {Z} )\cong F\times F$.

Thus, $L$ is a direct product of two copies of the free group on two generators, and moreover, the embedded subgroup $H$ in $L$ is simply $F\times \{e\}$, the first embedded direct factor. This is not a characteristic subgroup in $L$, because there exists an exchange automorphism swapping the two direct factors of $L$.