Definition:Normal Subgroup/Definition 6

Definition
Let $G$ be a group.

Let $N$ be a subgroup of $G$.

$N$ is a normal subgroup of $G$ iff:
 * $\forall g, n \in G: \left({n \in N \iff g \circ x \circ g^{-1} \in N}\right)$
 * $\forall g, n \in G: \left({n \in N \iff g^{-1} \circ n \circ g \in N}\right)$

Also see

 * Equivalence of Definitions of Normal Subgroup