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.

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