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Exercise (A Subgroup Is Normal If And Only If It Is The Kernel Of Some Homomorphism)
Prove that $N$ is a normal subgroup of $G$ if and only if $N$ is the kernel of some homomorphism $f:G\to H$ from $G$ to another group $H$.
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We already know that if $N$ is the kernel of a homomorphism, then it is normal. So one direction of this if and only if proof is complete.
If we know $N$ is normal, then we can let $H=G/N$ and define $f:G\to H$ by $f(g)=Ng$. This map is a homomorphism because we already showed that $f(ab)=N(ab)=(Na)(Nb) = f(a)f(b)$ when we showed that the set product is a binary operation on cosets of a normal subgroup. The kernel of this map is the set of elements $g\in G$ such that $f(g)=N$. This is precise the set of $g\in G$ such that $Ng=N$ which is true if and only if $g\in N$ by the properties of cosets. Hence the kernel of this map is $N$.