Definition:Normal Closure

Definition
Let $\left({G, \circ}\right)$ be a group.

Let $S$ be a subset of $G$.

The normal closure of $S$ in $G$ is the smallest normal subgroup of $G$ that contains $S$ as a subset:
 * $S^G := \left\{{x \circ S \circ x^{-1}: x \in G}\right\}$

Also see

 * Definition:Normal Subgroup
 * Definition:Contranormal Subgroup


 * Definition:Conjugate Closure