# The use of Isomorphic Theorems

I'm trying to figure out when and which isomorphism theorems are used in the following proof, as well as if there is another way of proving that relies more on the theorems:

Let $$A\triangleleft G$$ and $$B\triangleleft H$$. Show that $$(A\times B)\triangleleft (G\times H)$$ and $$(G\times H)/(A\times B)\cong (G/A)\times (H/B).$$

Proof:

Define the homomorphism $$\varphi: G\times H \rightarrow (G/A)\times (H/B)$$ such that $$\varphi((g,h))=(gA,hB).$$ This is a homomorphism because for $$g_1,g_2\in G, h_1,h_2\in H$$ we have $$\varphi ((g_1,h_1),(g_2,h_2))=\varphi((g_1g_2,h_1h_2))=(g_1g_2A,h_1h_2B)=(g_1G,h_1B)(g_2A,h_2B)=\varphi((g_1,h_1))\varphi((g_2,h_2)).$$ The kernal of $$\varphi$$ is any element $$(g,h)$$ such that $$g\in A$$ and $$h\in B$$. This yields $$A\times B$$ therefore, $$(A\times B) \triangleleft (G\times H$$) and ($$G\times H)/(A\times B)\cong (G/A)\times (H/B).$$

$$G/\operatorname{ker f} \cong f(G)$$
for a group morphism $$f:G \to H$$.