Subgroup generated by a subset

Definition

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

Remarks

By the intersection property, $\langle S \rangle$ 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.