Definition:Normal Subgroup/Definition 6

Definition
Let $G$ be a group.

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

$N$ is a normal subgroup of $G$ :
 * $\forall g \in G: \paren {n \in N \iff g \circ n \circ g^{-1} \in N}$
 * $\forall g \in G: \paren {n \in N \iff g^{-1} \circ n \circ g \in N}$

Also see

 * Equivalence of Definitions of Normal Subgroup