# Cocentral not implies amalgam-characteristic

From Groupprops

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) neednotsatisfy the second subgroup property (i.e., amalgam-characteristic subgroup)

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## Contents

## Statement

It is possible to have a group and a cocentral subgroup of (i.e., ) such that is not a characteristic subgroup in the amalgamated free product .

## Related facts

### Similar facts

- Normal not implies amalgam-characteristic
- Direct factor not implies amalgam-characteristic
- Characteristic not implies amalgam-characteristic

### Opposite facts

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

## Proof

### Example of the free group

Let be a free group on two generators and be the group of integers. Let and be the embedded first direct factor. Note that since the second direct factor is central. So, is a cocentral subgroup. We have:

.

Thus, is a direct product of two copies of the free group on two generators, and moreover, the embedded subgroup in is simply , the first embedded direct factor. This is not a characteristic subgroup in , because there exists an exchange automorphism swapping the two direct factors of .