Definition:Generated Normal Subgroup/Definition 2

Definition
Let $G$ be a group.

Let $S \subseteq G$ be a subset.

The normal subgroup generated by $S$, denoted $\gen {S^G}$, is the subgroup generated by the set of conjugates of $S$:
 * $\gen {S^G} = \set {g^{−1}sg: g \in G, s \in S}$

Also see

 * Equivalence of Definitions of Generated Normal Subgroup