Definition:Normal Subgroup

Definition
Let $G$ be a group and $H \le G$.

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


 * $\forall g \in G: g H = H g$

where $g H$ and $H g$ are the left and right cosets respectively of $g$ modulo $H$.

This is represented symbolically as $H \triangleleft G$.

Clearly, by mutiplying the above definition on either side by $g^{-1}$, this can be stated equivalently as:
 * $H \triangleleft G := \forall g \in G: g H g^{-1} = H = g^{-1} H g$

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

Some sources refer to a normal subgroup as an invariant subgroup or a self-conjugate subgroup.

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

One way to determine if a subgroup of $G$ is normal is by means of the normal subgroup test.

Alternative Definitions
There are several different ways of defining a normal subgroup. Each one is equivalent, and different sources will use whichever is most convenient as its base definition.