Definition:Normal Subgroup/Definition 1

Definition
Let $G$ be a group.

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

$N$ is a normal subgroup of $G$ iff:
 * $\forall g \in G: g \circ N = N \circ g$

where $g \circ N$ denotes the subset product of $g$ with $N$.

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

Also see

 * Equivalence of Definitions of Normal Subgroup