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

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

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



    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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    Quote Originally Posted by kathrynmath View Post
    Show that R^x/<-1> is isomorphic to the group of positive real numbers under multiplication.



    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.


    An idea: check the function \phi :\mathbb{R}^{*}\rightarrow \mathbb{R}^+ \,,\,\,\phi(x):=|x| ...

    Tonio
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    I'm curious as to how you chose that function?
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    Quote Originally Posted by kathrynmath View Post
    I'm curious as to how you chose that function?

    Well, you need a multiplicative function, i.e. \phi \,\,s.t.\,\,\phi(xy)=\phi(x)\phi(y) and I don't know that many, so....

    Tonio
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  5. #5
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    So you can just choose any multiplicative function?
    so assume f(x1)=f(x2)
    |x1|=|x2|
    x1=x2
    1-1
    onto because a maps to |a|
    Now f(x)f(y)=|x||y|=|xy|=f(xy) homomorphism
    Thus isomorphic.
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    Quote Originally Posted by kathrynmath View Post
    So you can just choose any multiplicative function?


    No, not any. Say, f(x):=1 doesn't work. Do you know any other multiplicative functions for our problem?

    Tonio



    so assume f(x1)=f(x2)
    |x1|=|x2|
    x1=x2
    1-1
    onto because a maps to |a|
    Now f(x)f(y)=|x||y|=|xy|=f(xy) homomorphism
    Thus isomorphic.
    .
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  7. #7
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    Quote Originally Posted by kathrynmath View Post
    So you can just choose any multiplicative function?
    so assume f(x1)=f(x2)
    |x1|=|x2|
    x1=x2
    1-1
    onto because a maps to |a|
    Now f(x)f(y)=|x||y|=|xy|=f(xy) homomorphism
    Thus isomorphic.

    No. My function is NOT an isomorphism since it is not 1-1, but you can apply the first isomorphism theorem for groups...

    Tonio
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    If G is a group, N is a normal subgroup of G, and H is any subgroup of G. Then HN is a subgroup of G, H intersect N is a normal subgroup of H and HN/N isomorphic to H/(H intersectN) by the first isomorphism theorem.

    I'm not quite sure how to use that.
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  9. #9
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    Quote Originally Posted by kathrynmath View Post
    If G is a group, N is a normal subgroup of G, and H is any subgroup of G. Then HN is a subgroup of G, H intersect N is a normal subgroup of H and HN/N isomorphic to H/(H intersectN) by the first isomorphism theorem.

    I'm not quite sure how to use that.

    That is not the standard first isomorphism theorem for groups, which you can read in one of the following sites:

    Isomorphism theorem - Wikipedia, the free encyclopedia

    http://www.tjsullivan.org.uk/math/morphthm.pdf (pages 5-6)

    http://marauder.millersville.edu/~bi...o/firstiso.pdf

    If you look at the above carefully then you'll understand why the absolute value function solves your problem.

    Tonio
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