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Math Help - Isomorphism

  1. #1
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    Isomorphism

    Let G be the group of all real-valued functions on the unit interval [0,1], where we define for f,g elements of G, addition by (f + g)(x) = f(x) + g(x) for every x elements of [0,1]. If N = { f element of G,f(1/4) = 0 }, prove that G/N is ismorphic to real numbers under addition
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  2. #2
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    Quote Originally Posted by Godisgood View Post
    Let G be the group of all real-valued functions on the unit interval [0,1], where we define for f,g elements of G, addition by (f + g)(x) = f(x) + g(x) for every x elements of [0,1]. If N = { f element of G,f(1/4) = 0 }, prove that G/N is ismorphic to real numbers under addition

    What about H: G --> R+ defined by H(f):= f(1/4)? Can you prove this is a homom. of groups and it is onto? Then use the first isomorphism theorem.

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    What about H: G --> R+ defined by H(f):= f(1/4)? Can you prove this is a homom. of groups and it is onto? Then use the first isomorphism theorem.

    Tonio
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    Last edited by Godisgood; October 15th 2009 at 10:12 AM.
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  4. #4
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    Quote Originally Posted by Godisgood View Post
    Thanks but am still not able to prove is homomorphism and onto?Can u pls show me how to dat that..

    Hmm....to study this stuff you must have taken before at least an introductory calculuis course and even a little more, and linear algebra and stuff...

    Is is true that H(f + g) = H(f) + H(g)? Or what is the same, is it true that (f + g)(1/4) = f(1/4) + g(1/4)? To answer this just check CAREFULLY the definition of addition in G...

    About onto: is it true that for any real number r we can find a real valued function defined on [0,1] s.t. f(1/4) = r?

    Please do think a little about these questions and make an effort towards their solution, and THEN if you're still stuck write back.

    Tonio
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