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.
Let $G$ be a group with neutral element $e$.
- \[\langle\varnothing\rangle = \{e\}.\]
- \[\langle\{e\}\rangle = G.\]