# Local finiteness is extension-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., extension-closed group property)

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## Statement

### Statement with symbols

Suppose is a group with a normal subgroup such that both and are locally finite groups: any finitely generated subgroup of is finite, and any finitely generated subgroup of is finite. Then, is itself locally finite.

## Related facts

## Facts used

- Schreier's lemma: In its factual form, this states that any subgroup of finite index in a finitely generated group is again finitely generated.
- Finiteness is extension-closed
- First isomorphism theorem

## Proof

**Given**: A group with a normal subgroup such that both and are locally finite. Let be the quotient map.A finite subset of with .

**To prove**: is finite.

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | is a finite subgroup of | is locally finite | Since is finite, is finite. By local finiteness of , is a finite subgroup of . We have , so is finite. | ||

2 | is a normal subgroup of finite index in | Fact (3) | We have . Since the left side is finite, we conclude that has finite index in . | ||

3 | is finitely generated | Fact (1) | is finite | Step (2) | is finite, so is finitely generated, and by step (2), has finite index in . Thus, by fact (1), is also a finitely generated group. |

4 | is finite | is locally finite | Step (3) | is a finitely generated subgroup of . Local finiteness of forces to be finite. | |

5 | is finite | Fact (2) | Steps (2), (4) | By step (4), is finite, and by step (2), is finite. Thus, by fact (2), is finite. |

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