\begin{definition}[Definition:Normal Subgroup/Also known as]
It is usual to describe a '''normal subgroup of $G$''' as '''normal in $G$'''.
Some sources refer to a '''normal subgroup''' as an '''invariant subgroup''' or a '''self-conjugate subgroup'''.
This arises from Definition 6:
{{:Definition:Normal Subgroup/Definition 6}}
which is another way of stating that $N$ is '''normal''' {{iff}} $N$ stays the same under all inner automorphisms of $G$.
See Inner Automorphism Maps Subgroup to Itself iff Normal.
Some sources use '''distinguished subgroup'''.
\end{definition}