I am basically wondering why the two groups I have marked in red are isomorphic. I will explain something after the picture:
Let's assume that we accept that $G\simeq Z\times Z \times Z\times Z$ as they state. Then we can say that $2G \simeq 2Z\times 2Z\times 2Z\times 2Z$? But why are the the cosets groups also isomorphic?
I mean lets assume that C is normal in A, D is normal in B, and we have that $A \simeq B$, and $C \simeq D$. Then do we have that $A/C \simeq B/D$? I tried proving this, but I was not able to. Can someone please help?