# Closed subgroup of finite index

From Groupprops

This article defines a property that can be evaluated for a subgroup of a semitopological group

## Definition

A subgroup of a topological group (or more generally any of the variations of topological group that involve a group structure and a topological space structure, including left-topological group, right-topological group, semitopological group, quasitopological group, or paratopological group) is termed a **closed subgroup of finite index** or **open subgroup of finite index** if it satisfies the following equivalent conditions:

- It is a closed subgroup that is also a subgroup of finite index in the whole group
- It is an open subgroup that is also a subgroup of finite index in the whole group

### Equivalence of definitions

- (1) implies (2): closed subgroup of finite index implies open
- (2) implies (1): follows from open subgroup implies closed