Proving that real with multiplaication is not isomorphic to real with addition

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 .

So:

Finally:

(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 ...)

Re: Proving that real with multiplaication is not isomorphic to real with addition

Quote:

Originally Posted by

**Stormey** 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.

Suppose that

is an isomorphism (I'll try to derive a contradiction).

Then it must hold that

If you have posted what you actually meant then your mapping is mistaken.

so what you have is it must hold that .

Did you mean

Re: Proving that real with multiplaication is not isomorphic to real with addition

yes of course. thank you.

Re: Proving that real with multiplaication is not isomorphic to real with addition

Stormey,

I'm afraid your computations are totally garbled.

Since is onto, there is

So

As you pointed out, and since is one to one, 2a=0 or a=0

Contradiction --

Aside - the group of __positive__ reals under multiplication is isomorphic to the additive group of the reals; any logarithm function shows this.

Re: Proving that real with multiplaication is not isomorphic to real with addition

Thank you.

Actually that was the contradiction I was looking for in the first place - showing that can't be one-to-one.