Definition:Normal Subgroup/Also known as

Definition
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:

which is another way of stating that $N$ is normal $N$ stays the same under all inner automorphisms of $G$.

See Inner Automorphism Maps Subgroup to Itself iff Normal.

Some sources use distinguished subgroup.