# Cocentral not implies amalgam-characteristic

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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 $L := G *_H G$.

## 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 $L$.