\begin{definition}[Definition:Normal Subgroup/Definition 6]
Let $G$ be a group.
Let $N$ be a subgroup of $G$.
$N$ is a '''normal subgroup of $G$''' {{iff}}:
:$\forall g \in G: \paren {n \in N \iff g \circ n \circ g^{-1} \in N}$
:$\forall g \in G: \paren {n \in N \iff g^{-1} \circ n \circ g \in N}$
\end{definition}