Definition:Normal Subgroup/Definition 5

Definition
Let $G$ be a group.

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

$N$ is a normal subgroup of $G$ iff:
 * $\forall g \in G: g \circ N \circ g^{-1} = N$
 * $\forall g \in G: g^{-1} \circ N \circ g = N$

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

Notation
To use the notation introduced in the definition of the congugate:
 * $N \lhd G \iff \forall g \in G: N^g \in N$

Also see

 * Equivalence of Definitions of Normal Subgroup