Definition:Normal Subgroup

Definitions
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:
 * $$H \triangleleft G \ \stackrel {\mathbf {def}} {=\!=} \ \forall g \in G: g H g^{-1} = H = g^{-1} H g$$

or, to use the notation introduced in the definition of the congugate:
 * $$H \triangleleft G \ \stackrel {\mathbf {def}} {=\!=} \ H^g = H $$

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.