I am trying to understand the following proof: 
I understand most of it just fine, except for the last line. I don't really understand why the projection will show us that $\langle xy \rangle \neq \langle xy^{-1} \rangle$. More precisely in the case of $p=2$: We are supposed to project onto $C_4 \times C_4$ but if i'm not mistaken, every element of $C_4$ generates the same subgroup as its inverse, so i don't see how $xy$ and $xy^{-1}$ would generate something different from eachother. Perhaps i misunderstood the maps at work here, i'm assuming that $x$ gets mapped to $(x,1)$ , $y$ to $(1,y)$ and that $xy = (x,y)$? and that the projection is by modding out $p$ (or 4 in the case of $p=2$)?
Edit: I believe to have somewhat of a better grip on it now: so $xy$ gets mapped to $(x,y)$ and thus the group generated by this is of the form $(x^l,y^l)$ while $(x,y^{-1})$ gives us a group with elements of the form $(x^l, y^{(n-l)})$. I believe to then look at the projection is to assure/show that there is for example no $k$ such that $(x^k,y^k) = (x, y^{n-1})$. However, i don't really understand how this proves that or how looking at the projection allows us to make claims about $\langle xy \rangle \neq \langle xy^{-1} \rangle$ in general.