# Finite normal implies amalgam-characteristic

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., finite normal subgroup) must also satisfy the second subgroup property (i.e., amalgam-characteristic subgroup)

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finite group) must also satisfy the second group property (i.e., always amalgam-characteristic group)

View all group property implications | View all group property non-implications

Get more facts about finite group|Get more facts about always amalgam-characteristic group

## Contents

## Statement

### Verbal statement

Any finite normal subgroup (i.e., a normal subgroup that is finite as a group) is an amalgam-characteristic subgroup.

Equivalently, a finite group is an always amalgam-characteristic group: it is an amalgam-characteristic subgroup in any group in which it is a normal subgroup.

### Statement with symbols

Suppose is a finite normal subgroup of a group . Then, is a characteristic subgroup inside the amalgam .

## Related facts

### Stronger facts

### Similar facts

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

### Opposite facts

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

### Applications

- Finite normal implies potentially characteristic
- Finite normal implies image-potentially characteristic

## Facts used

- Quotient of amalgamated free product by amalgamated normal subgroup equals free product of quotient groups
- Free product of nontrivial groups has no nontrivial finite normal subgroup
- Normality is upper join-closed
- Normality satisfies image condition

## Proof

**Given**: A group , a finite normal subgroup of . .

**To prove**: is a characteristic subgroup in .

**Proof**:

- By fact (1), .
- has no nontrivial finite normal subgroup: By fact (2), if is a proper subgroup of , then has no nontrivial finite normal subgroup. If , then is trivial and hence has no nontrivial finite normal subgroup.
- is a finite normal subgroup of : Since is normal in both copies of , it is normal in . Also, is finite.
- is the unique largest finite normal subgroup of : Suppose is a finite normal subgroup of . Then, by fact (4), the image of in the quotient map is a normal subgroup of . Also, since is finite, its image is finite. In step (2), we concluded that has no nontrivial finite normal subgroup. Thus, the image of is trivial, so .
- is characteristic in : This follows since it is the unique largest finite normal subgroup of .