Let $G$ be a group, $H,K$ be its normal subgroups, let $I = H \cap K$. Obviously $I$ is a normal subgroup of $G$.
Show that $(HK/I) \cong (H/I) \times (K/I)$
I have defined a map $$\alpha: (H/I) \times (K/I) \longrightarrow (HK/I)$$ $$\alpha(I_h,I_k) = I_{hk}$$
I have shown that this map is a well defined homomorphism. But how do I show that the inverse is also a homomorphism?