I'm trying to prove that is not isomorphic to (the notation means ) but I'm having difficulties when I get to the final conclusion.
This is my work so far (from now on, I'll drop the and notations for conveniency):
Suppose that is an isomorphism (I'll try to derive a contradiction).
Then it must hold that .
Let , then it has its inverse .
(properties of multiplication in )
Now, since we know that can be either or ... what does that tell me? that is not well defined (and therefor cannot be isomorphism)? ( is sent to two different ...)
I'm afraid your computations are totally garbled.
Since is onto, there is
As you pointed out, and since is one to one, 2a=0 or a=0
Aside - the group of positive reals under multiplication is isomorphic to the additive group of the reals; any logarithm function shows this.