---
tag: 4K9RYSNPG3
kind: definition
name: Open subgroup
msc: [22A05]
defines: QYCFH68RD1
uses: [92700QAB5Y, 0X1JXT47GA, 02BH3MCBQB, 0ZCY0Z80B6]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_group
---

## Definition

Let $G$ be a [[topological group|92700QAB5Y]].
An *open subgroup* of $G$ is a [[subgroup|0X1JXT47GA]] $H$ of $G$ that is an [[open set|02BH3MCBQB]] of $G$.

## Remarks

Example: in $\R \times \Z$ with the product topology, where $\Z$ is discrete, the subgroup $\R \times \set{0}$ is open, because $\set{0}$ is open in $\Z$.
Non-example: $\Z$ is a subgroup of $(\R, +)$ but it is not open, since no interval around $0$ lies in $\Z$.
A subgroup $H$ is open if and only if it is a [[neighbourhood|0ZCY0Z80B6]] of the identity $e$.
Indeed, if $U \subseteq H$ is a neighbourhood of $e$, then $hU \subseteq H$ is a neighbourhood of each $h \in H$.