A normal subgroup of group is one which is closed under conjugation: for all , it is the case that . In shorter form, .
Since conjugacy is equivalent to "changing the worldview", a normal subgroup is one which "looks the same from the point of view of every element of ".
A subgroup of is normal if and only if it is the kernel of some group homomorphism from to some group . (Proof.)
From a category-theoretic point of view, the kernel of is an equaliser of an arrow with the zero arrow; it is therefore universal such that composition with yields zero.