# Local finiteness is quotient-closed

From Groupprops

This article gives the statement, and possibly proof, of a group property (i.e., locally finite group) satisfying a group metaproperty (i.e., quotient-closed group property)

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Get more facts about locally finite group |Get facts that use property satisfaction of locally finite group | Get facts that use property satisfaction of locally finite group|Get more facts about quotient-closed group property

## Statement

Any quotient group of a locally finite group is also locally finite. In other words, if is a surjective homomorphism, and is locally finite, so is .

## Definitions used

### Locally finite group

`Further information: Locally finite group`

A group is termed **locally finite** if every finitely generated subgroup of it is finite.

## Related facts

## Proof

**Given**: A locally finite group , a surjective homomorphism .

**To prove**: If is a finite subset of , is finite.

**Proof**:

- Construction of a finite set such that : Since is surjective, we can pick, for each , an element such that . Making such a choice for each , we get a finite subset of such that .
- : This follows from the fact that is a homomorphism.
- is a finite group (
**Given data used**: is locally finite): Since is locally finite, and is a finite subset, is a finite group. - is finite: By steps (3) and (4), is the image of a finite group, and hence, is finite.