Generated Subgroup - Imported Knowlpedia Content

Let $G$ be a group and let $S\subseteq G$ be a subset. The subgroup generated by $S$ , denoted $\langle S\rangle$ , is defined by

\langle S\rangle \;=\; \bigcap\{\,H\le G : S\subseteq H\,\},

the intersection of all subgroups of $G$ that contain $S$ .

Equivalent characterizations

Equivalently, $\langle S\rangle$ is the set of all finite products of elements of $S$ and their inverses (i.e. all "words" in $S\cup S^{-1}$ ). When $S=\{g\}$ is a singleton, $\langle S\rangle$ is a cyclic subgroup.

Examples

In $(\mathbb{Z},+)$ , $\langle 6,15\rangle = 3\mathbb{Z}$ .
In $S_3$ , the set $\{(12),(123)\}$ generates all of $S_3$ .
In $\mathbb{R}^{\times}$ , the subset $\{-1\}$ generates $\{1,-1\}$ .