Definition:Normal Subgroup/Definition 2

Definition
Let $G$ be a group.

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

$N$ is a normal subgroup of $G$ iff:
 * Every right coset of $N$ in $G$ is a left coset

that is:
 * The right coset space of $N$ in $G$ equals its left coset space.

Also see

 * Equivalence of Definitions of Normal Subgroup