A normal subgroup of group $G$ is one which is closed under conjugation: for all $h \in G$ , it is the case that ${h n h^{- 1} : n \in N} = N$ . In shorter form, $h N h^{- 1} = N$ .

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 $G$ ".

A subgroup of $G$ is normal if and only if it is the kernel of some group homomorphism from $G$ to some group $H$ . (Proof.)

From a category-theoretic point of view, the kernel of $f$ is an equaliser of an arrow $f$ with the zero arrow; it is therefore universal such that composition with $f$ yields zero.

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