# Second isomorphism theorem

This is copied from a Facebook chat message I had with someone a few weeks ago, with wordpress latex applied over the math:
A couple weeks ago, I learned the statement of the second isomorphism theorem, which states that given a subgroup $S$ and normal subgroup $N$ of $G$, $SN$ is a subgroup of $G$ and $S \cap N$ is a normal subgroup of $S$, with $SN / N$ isomorphic to $S / (S \cap N)$.
Any element of $SN / N$ can be represented as $anN = aN$ for $a \in S$, where the $n$ on the LHS is in $N$. A similar statement of representation via $a(S \cap N)$, $a \in S$ holds for $S / (S \cap N)$. Define $\phi: SN/N \to S / (S \cap N)$ with $\phi(aN) = a(S \cap N)$, which is bijective. By normality, $\phi(abN) = ab(S \cap N) = a(S \cap N)b(S \cap N) = \phi(aN)\phi(bN)$. Thus, $\phi$ is an isomorphism. QED.