Definition:Normal Subgroup

Definition
Let $G$ be a group.

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

Notation
The statement that $N$ is a normal subgroup of $G$ is represented symbolically as $N \lhd G$.

A normal subgroup is often represented by the letter $N$, as opposed to $H$ (which is used for a general subgroup which may or may not be normal).

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:

which is another way of stating that $N$ is normal iff $N$ is invariant under all inner automorphisms of $G$.

Also see

 * Equivalence of Definitions of Normal Subgroup