Definition 1. Let $G$ be a group and $S \subseteq G$ be a subset. The subgroup generated by $S$ is the minimal subgroup that contains $S$:
\[\langle S \rangle = \bigcap \{H \leq G : S \subseteq H\}.\]If $H$ is a subgroup and $\langle S \rangle = H$, we say $S$ generates $H$. The elements of $S$ are then called generators.