---
tag: 0RG9SPQSWM
kind: definition
name: Subgroup generated by a subset
aliases: [generated-subgroup, subgroup-closure]
msc: [20A05]
defines: 0RP7VWQ0YN
uses: [0N0PXCTYA7, 0ZTGMDNVTB]
sources:
  - url: https://en.wikipedia.org/wiki/Generating_set_of_a_group
    locator: lead section
---

## Definition

Let $G$ be a [[group|G5S1M9]] and let $S$ be a subset of $G$.
The subgroup generated by $S$, written $\langle S \rangle$, is the intersection of all [[subgroups|0X1JXT47GA]] of $G$ that contain $S$.

## Remarks

By [[the intersection property|0ZTGMDNVTB]], $\langle S \rangle$ is a subgroup, and it is contained in each subgroup that contains $S$.
Its elements are the products $s_1^{\varepsilon_1} s_2^{\varepsilon_2} \cdots s_k^{\varepsilon_k}$ with $k \geq 0$, $s_i \in S$ and $\varepsilon_i \in \{1, -1\}$, where the empty product ($k = 0$) is the identity.
In particular, $\langle \emptyset \rangle = \{e\}$.

## Explanations

### L1 human

Start with some elements of a group, and add everything that you can make from them with products and inverses.
The result is the subgroup generated by these elements.
For example, in $(\mathbb{Z}, +)$, the subgroup generated by $\{4, 6\}$ is the set $2\mathbb{Z}$ of even integers.