\begin{definition}[Definition:Generator of Subgroup]
Let $\struct {G, \circ}$ be a group.
Let $S \subseteq G$.
Let $H$ be the subgroup generated by $S$.
Then '''$S$ is a generator of $H$''', denoted $H = \gen S$, {{iff}} $H$ is the subgroup generated by $S$.
\end{definition}