# Normal subgroup

A normal subgroup $$N$$ 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, $$hNh^{-1} = N$$.

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.)

knows-requisite(category theory equaliser): 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.

why are they interesting

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• Group

The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.