Let F be non trivial group homomorphism F: Z -> Q*. Want to prove that either Ker(F)={0} or Ker(F)= 2Z.
Okay here is what i did;
since i know that if there is homomorphism between two groups then there should be and isomorphism T such that, T: Z/Ker(F) -> Image(F).
so i took Z/{0}=Z implies Z is isomorphic to Image(F). and i took Z/2Z={2Z, 1+2Z} is isomorphic to Image(F). and i take Z/3Z to be isomorphic to Image(F). Then since all three are isomorphic to Image(F) then they should be isomorphic to each other i.e. Z/3Z should be isomorphic to Z/2Z. (hence we arrive to a contradiction!).
Is my method correct? If not then i need some directions. And once I'm done answering this question, can i deduce from it that (Q*, x) is not cyclic? Because to show that we need to show isomorphism of Q* with Z..