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., $H Z (G) = G$ ) 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 $Z$ be the group of integers. Let $G = F \times Z$ and $H = F \times {0}$ be the embedded first direct factor. Note that $H Z (G) = G$ since the second direct factor is central. So, $H$ is a cocentral subgroup. We have:

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

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 $L$ .