Definition:Normal Subgroup

Definition
Let $G$ be a group and let $N \le G$. That is, let $N$ be a subgroup of $G$.

Then the subgroup $N$ is called a normal subgroup of $G$ iff:

Definition 2
The statement that $N$ is a normal subgroup of $G$ is represented symbolically as $N \triangleleft G$.

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.

Also see

 * Equivalence of Normal Subgroup Definitions


 * Subgroup equals Conjugate iff Normal, where it is shown that:
 * $H \triangleleft G := \forall g \in G: g H g^{-1} = H = g^{-1} H g$

or, to use the notation introduced in the definition of the congugate:
 * $H \triangleleft G := H^g = H $

This is used by many authors as the definition of a normal subgroup.


 * Normal Subgroup Equivalent Definitions
 * Normal Subgroup Test