Totally lost again!
That's algebra!
Therefore is a normal subgroup of .
is surjective by definition of , a group homomorphism because and injective because .
We have found an isomorphism between and .
Consider a,b)\mapsto b" alt="\phi: G\rightarrow G_2a,b)\mapsto b" /> .
, so is an homomorphism, which is surjective by definition of .
Moreover, .
Hence , and we can conclude .