# Connected implies no proper closed subgroup of finite index

From Groupprops

## Contents

## Statement

In a connected topological group, there cannot be any proper closed subgroup of finite index (i.e., a closed subgroup that is also a subgroup of finite index).

## Related facts

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

## Facts used

## Proof

The proof is direct from Fact (1), and the observation that the existence of a *proper* subgroup (and hence a proper nonempty subset) that is both closed and open means that the group is not connected.