Third Isomorphism Theorem/Groups/Corollary 1

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Theorem

Let $G$ be a group.

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

Let $q: G \to \dfrac G N$ be the quotient epimorphism from $G$ to the quotient group $\dfrac G N$.

Let $K$ be the kernel of $q$.


Then:

$\dfrac G N \cong \dfrac {G / K} {N / K}$


Proof

From Kernel is Normal Subgroup of Domain we have that $K$ is a normal subgroup of $G$.

Thus the Third Isomorphism Theorem can be applied directly.

$\blacksquare$


Sources