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

In summary, the conversation discusses how to show that R^x/<-1> is isomorphic to the group of positive real numbers under multiplication. The participants mention the need for a homomorphism and for it to be one-to-one and onto in order for the groups to be isomorphic. They also discuss using functions such as f(x) = x^2 or |x|, but note that 1-1 does not work. The conversation also brings up the kernel of f and the use of isomorphism theorems.
  • #1
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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  • #2


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?
 
  • #3


I was trying to use f(x)=x^2
 
  • #4


my problem is I can figure out onto, homomorphism, but 1-1 doesn't work.
 
  • #5


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?
 

What is isomorphism?

Isomorphism is a mathematical concept that describes a relationship between two structures that preserves their inherent properties and operations.

What is R^x/<-1>?

R^x/<-1> is the quotient group of the real numbers, denoted by R^x, and the subgroup generated by -1. Essentially, it is the set of all real numbers excluding -1.

What are positive real numbers?

Positive real numbers are the set of all real numbers greater than 0, including both rational and irrational numbers.

Why is proving isomorphism important?

Proving isomorphism is important because it helps us understand the structure and relationships between mathematical objects. It also allows us to transfer knowledge and properties from one structure to another.

What is the process for proving isomorphism of R^x/<-1> and positive real numbers?

The process for proving isomorphism involves showing that there exists a bijective homomorphism between the two structures, meaning that the map preserves the operations and properties of the structures. This can be done through various methods such as defining an explicit map or using properties of the structures to show the isomorphism.

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