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.

The normal closure is equal to the conjugate closure $\left\langle S^G \right\rangle := \left\langle \left\{{x \circ s \circ x^{-1}: x \in G, s \in S}\right\}\right\rangle$

which is the subgroup generated by the set of all elements of $G$ that are conjugate to an element of $S$.

Also see

 * Definition:Normal Subgroup
 * Definition:Contranormal Subgroup


 * Definition:Conjugate Closure