Find isomorphism for an operation

Question

I was trying to solve this problem, but am having trouble seeing why it is an isomorphism. To map from R* to G, I think that the phi function would be Phi(x)=x/2 but that doesn't work. This phi function does not preserve the operation because if we take Phi(ab)=ab/2, but Phi(a)=a/2 and Phi(b)=b/2. So the operation is not preserved.

Answer 1

$f(x)=2x$ should work here. $f(x)*f(y)=(2x)*(2y)=\frac{2x2y}{2}=2xy=f(xy)$

Answer 2

Try the function $$ \phi: x \mapsto 2\,x $$ Now, show that $\phi(xy) = \phi(x)*\phi(y)$.