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

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Homework Help Overview

The discussion revolves around proving that the group of nonzero real numbers under multiplication, modulo the subgroup generated by -1, is isomorphic to the group of positive real numbers under multiplication.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the need for a suitable function mapping and explore the properties required for isomorphism, including homomorphism, one-to-one, and onto characteristics. There is a specific inquiry about functions that can map both -r and r to r.

Discussion Status

Some participants have proposed functions such as f(x) = x^2 and |x|, noting that these functions are onto and homomorphisms. However, there is recognition that these functions are not one-to-one, leading to further exploration of the implications of this property in the context of the isomorphism question.

Contextual Notes

Participants are considering the implications of the kernel of the proposed functions and referencing isomorphism theorems, indicating a deeper exploration of group theory concepts. There is a mention of the specific characteristics of the groups involved, such as the presence of an element of order 2 in R^x that is not present in the positive real numbers.

kathrynag
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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.
 
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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?
 


I was trying to use f(x)=x^2
 


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


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?
 

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