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

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about normal subgroup|Get more facts about amalgam-characteristic subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not amalgam-characteristic subgroup|View examples of subgroups satisfying property normal subgroup and amalgam-characteristic subgroup

## Contents

## Statement

### Verbal statement

A normal subgroup of a group need not be an amalgam-characteristic subgroup.

### Statement with symbols

Let be a group and be a normal subgroup of . Let . Then, it is not necessary that is characteristic in .

## Related facts

### Converse

### Similar facts

- Characteristic not implies amalgam-characteristic: This is a stronger fact, and examples of this also serve as examples of normal not implying amalgam-characteristic.
- Direct factor not implies amalgam-characteristic
- Cocentral 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. 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 .