---
tag: 0N0PXCTYA7
kind: definition
name: Subgroup
msc: [20A05]
defines: 0X1JXT47GA
uses: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Subgroup
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Subgroup
    locator: Basic properties of subgroups
---

## Definition

Let $G$ be a [[group|G5S1M9]] with identity element $e$.
A subgroup of $G$ is a subset $H \subseteq G$ with these three properties.
The identity $e$ is in $H$.
For all $a, b \in H$, the product $ab$ is in $H$.
For each $a \in H$, the inverse $a^{-1}$ is in $H$.
The notation $H \leq G$ means that $H$ is a subgroup of $G$.

## Remarks

A subgroup $H$ is a group with the operation of $G$ restricted to $H$, and its identity is the identity of $G$.
Each group $G$ has the subgroups $\{e\}$ and $G$.
The even integers $2\mathbb{Z}$ form a subgroup of $(\mathbb{Z}, +)$.
The natural numbers do not form a subgroup of $(\mathbb{Z}, +)$, because $-1$ is not a natural number.

## Explanations

### L1 human

A subgroup is a smaller group that lives inside a bigger group and uses the same operation.
To check that a subset is a subgroup, you check that it contains the identity and that it is closed under products and inverses.
For example, the rotations of a square form a subgroup of all the symmetries of the square.