The Third Isomorphism Theorem

Relevant Classes: Math 493

The Third Isomorphism Theorem

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

1. There is a natural one-to-one correspondence between subgroups of $G/N$ and subgroups of $G$ containing $N$. The correspondence matches a subgroup $K$ of $G$ containing $N$ with a subgroup $K/N$ of $G/N$.
2. This correspondence preserves normality: the normal subgroups of $G/N$ correspond to the normal subgroups of $G$ containing $N$. That is, the subgroup $K$ of $G$ containing $N$ is normal in $G$ if and only if $K/N$ is normal in $G/N$.
3. The correspondence preserves quotients: if $K$ is a normal subgroup of $G$ containing $N$, then $(G/N)/(K/N)\simeq G/K$

Proof: