# Closed subgroup of finite index implies open

## Contents

## Statement

### Statement for left-topological, right-topological, or semitopological groups

In a left-topological group *or* right-topological group, any closed subgroup of finite index (i.e., a closed subgroup that is also a subgroup of finite index) must be an open subgroup.

Note that a semitopological group is both a left-topological group and a right-topological group, so the result applies to semitopological groups.

### Statement for topological groups

In a topological group, any closed subgroup of finite index (i.e., a closed subgroup that is also a subgroup of finite index) must be an open subgroup.

Note that topological groups are semitopological groups, so the result applies to these.

## Related facts

- Open subgroup implies closed
- Connected implies no proper open subgroup
- Compact implies every open subgroup has finite index

## Proof

### Proof for left-topological groups

**Given**: A right-topological group , a closed subgroup of finite index in .

**To prove**: is an open subgroup of

**Proof**:

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

1 | For all , the map given by is a self-homeomorphism of . | Definition of left-topological group | is a left-topological group. | [SHOW MORE] | |

2 | Every left coset of in is a closed subset of . | Homeomorphisms take closed subsets to closed subsets | Step (1) | By Step (1), is a self-homeomorphism of , so it takes the closed subset to the closed subset . Thus, for any , is closed in . | |

3 | The union of all the left cosets of other than itself is closed in | Union of finitely many closed subsets is closed | has finite index in | Step (2) | Step-fact combination direct. |

4 | is open in | A subset is open iff its set-theoretic complement is closed. | Step (3) | The set-theoretic complement of in is precisely the union of all the left cosets other than itself, and by Step (3), this is closed. Hence, is open. |

### Proof for right-topological groups

The proof is analogous to the proof for left-topological groups, except that we use right cosets instead of left cosets.