Definition:Generated Normal Subgroup/Definition 1

Definition
Let $G$ be a group.

Let $S \subseteq G$ be a subset.

The normal subgroup generated by $S$, denoted $\left\langle{S^G}\right\rangle$, is the intersection of all normal subgroups of $G$ containing $S$.

Also see

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