Show that $$G =(\mathbb{Q}/\{-1\}{,∗), where ∀a,b∈\mathbb{Z}}:a∗b=a+b+ab $$ forms a group and show that it is isomorphic to the group group $M=(\mathbb{Q}/\{0\},.)$ where it is the opertaion of normal multiplication.
I've already proven that G is a group but i'm having trouble to show that it is an isomorphism, because i don't seem to understand is what happens to an element from G that it transforms to M. i don't understand what happens to a single element, let's say f(x) where $x\in G$ and f is the isomorphism. wouldn't it remain the same?, or is it that th echange occurs when i do the operation of one group. but wouldn't that be hard to prove f(x*y)=f(x)·f(y)?
I would appreciate any tips