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Math Help - Inverse of a homomorphic function

  1. #1
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    Inverse of a homomorphic function

    Let f(A,)-->(B,) be homomorphic. Prove that if f1 is a function then it is homomorphic
    Inorder for
    f1 to be a function then f must be a bijection
    Last edited by mathrld; August 17th 2012 at 06:41 AM.
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  2. #2
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    Re: Inverse of a homomorphic function

    First, in order for f^{-1} to exist, f must be "onto". That is, for every y in B there must be x in A such that f(x)= y. Otherwise we could not define f^{-1}(y). If f were not "one-to-one", there would be some y in B such that for both x_1 and x_2 in A, f(x_1)= y= f(x_2). In that case we would have both f^{-1}(y)= x_1 and f^{-1}(y)= x_2, contradicting the fact that f^{-1} is a function.

    f being a homomorphism means f(x_1*x_2)= f(x_1).f(x_2) for all x_1 and x_2 in A. To show that f^{-1} is a homorphism, you must use that to show that f^{-1}(y_1.y_2)= f^{-1}(y_1)*f^{-1}(y_2) for all y_1 and y_2 in B. I suggest you start by defining, for given y_1 and y_2 in B, x_1= f^{-1}(y_1) and x_2= f^{-1}(y_2).
    Thanks from mathrld
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    Re: Inverse of a homomorphic function

    I cant figure out how to use latex on this site? how is it done here?
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    Re: Inverse of a homomorphic function

    Start with [ t e x ] and end with [ / t e x ]. A tutorial on using LaTeX is at
    LaTeX Help
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    Re: Inverse of a homomorphic function

    Quote Originally Posted by mathrld View Post
    I cant figure out how to use latex on this site? how is it done here?
    On the tool bar is a \Sigma icon. That gives [TEX][/TEX] LaTeX wrap.
    [TEX]\sqrt{x^4+1}[/TEX] gives \sqrt{x^4+1}.
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    Re: Inverse of a homomorphic function

    Thanks DDD
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