# Connected implies no proper open subgroup

From Groupprops

## Contents

## Statement

### Statement for semitopological groups

A connected left-topological group has no proper open subgroup. Similarly, a connected right-topological group has no proper open subgroup.

Since semitopological groups are both left-topological and right-topologica, this tells us that a connected semitopological group has no proper open subgroup.

### Statement for topological groups

A connected topological group has no proper open subgroup.

## Related facts

### Similar facts

- Open subgroup implies closed
- Closed subgroup of finite index implies open
- Compact implies every open subgroup has finite index

### Converse

The converse is not true for all groups. See no proper open subgroup not implies connected.

However, the converse is true in some contexts:

- It is true for algebraic groups, i.e.,it is true if the topology is a Zariski topology. See equivalence of definitions of connected algebraic group.
- It is true for all locally connected topological groups. In particular, it is true for Lie groups. See equivalence of definitions of connected Lie group.

## Facts used

- Open subgroup implies closed (this is true in both left-topological groups and right-topological groups)

## Proof

By Fact (1), a proper open subgroup is a nonempty subset that is both open and closed (note that it is nonempty because it is a subgroup). The existence of such a subset contradicts connectedness.