The form of the second isomorphism theorem that I learned (for $G$ a group with $S$ a subgroup and $N$ a normal subgroup) is: $$SN \ / \ N \cong S \ / \ (S \cap N)$$ I'm seeing it used in a proof, however, as: $$SN \ / \ S \cong N \ / \ (S \cap N)$$ i.e. it looks as if the quotient groups in the equation are being cross-multiplied as if it were $\frac{a}{b} = \frac{c}{d} \implies \frac{a}{c} = \frac{b}{d}$.
Is this some general property of groups or am I misunderstanding something?