Cocentral 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., cocentral subgroup) 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 cocentral subgroup $H$ of $G$ (i.e., $HZ(G)=G$ ) such that $H$ is not a characteristic subgroup in the amalgamated free product $L:=G*_{H}G$ .

Related facts

Similar facts

Opposite facts

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. Note that $HZ(G)=G$ since the second direct factor is central. So, $H$ is a cocentral subgroup. 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 $L$ .