Definition:Generated Subgroup/Definition 1

Definition
Let $G$ be a group.

Let $S \subset G$ be a subset.

The subgroup generated by $S$ is the smallest subgroup containing $S$, that is, the intersection of all subgroups of $G$ containing $S$.

Also see

 * Intersection of Subgroups is Subgroup
 * Equivalence of Definitions of Generated Subgroup