Definition:Normal Subgroup

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

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

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

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

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.