Using the Fundamental Theorem of Homomorphism

[iceybloop]

G = Q* [the rational numbers other than zero]
H = a/b, a and b both odd integers

Show G/H isomorphic to <Z, +> (the integers under addition)

I know the problem is essentially done if I can construct some mapping where H itself is the kernel (usually called Greek letter Phi in my particular class). What I'm having trouble doing is constructing such a Phi.

 

[iceybloop]

G = Q* [the rational numbers other than zero]
H = a/b, a and b both odd integers

Show G/H isomorphic to <Z, +> (the integers under addition)

I know the problem is essentially done if I can construct some mapping where H itself is the kernel (usually called Greek letter Phi in my particular class). What I'm having trouble doing is constructing such a Phi.

I have a lot of details to fill in, but I think I have the gist of the Phi I need to construct.

All a/b can be written (a₀/b₀)2^k

If a and b are both odd, k will be zero. If you set the phi(a/b)=k, it should prove to be an onto homomorphism.