Proving Isomorphism of R^x/<-1> and Positive Real Numbers

[kathrynag]

Homework Statement

Show that R^x/<-1> is isomorphic to the group of positive real numbers under multiplication.

Homework Equations

The Attempt at a Solution

I know I need to show we have a homomorphism, and is one - to one and onto in order to be isomorphic. I know all that, but I don't know what function mapping to use.

 

[jbunniii]

I assume by R^x you mean the multiplicative group of nonzero real numbers?

If so, what's a function which maps both -r and r (r > 0) to r?

 

[kathrynag]

I was trying to use f(x)=x^2

 

[kathrynag]

my problem is I can figure out onto, homomorphism, but 1-1 doesn't work.

 

[jbunniii]

f(x) = x^2 is fine, |x| would work equally well. Both maps are onto and homomorphisms.

You should not expect f(x) to be 1-1. If so, then you would have proved that R^x is isomorphic to the positive real numbers, but that is not what the question asked. [In fact, they are not isomorphic: R^x has an element (-1) of order 2, but the positive real numbers have no such element.]

What is the kernel of f, and what isomorphism theorems do you know?