Generated Subgroup

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 $S$.

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 $\{1,-1\}$.