\begin{definition}[Definition:Normal Subgroup/Definition 5]
Let $G$ be a group.
Let $N$ be a subgroup of $G$.
$N$ is a '''normal subgroup of $G$''' {{iff}}:
:$\forall g \in G: g \circ N \circ g^{-1} = N$
:$\forall g \in G: g^{-1} \circ N \circ g = N$
where $g \circ N$ etc. denotes the subset product of $g$ with $N$.
That is:
:$\forall g \in G: \map {\kappa_g} N = N$
where $\map {\kappa_g} N$ denotes the inner automorphism of $N$ by $g$.
\end{definition}