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Math Help - bijection - please help!

  1. #1
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    Sep 2008
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    bijection - please help!

    I'm having trouble with the following problem - I know how to show surjectivity, injectivity and thus bijectivity but I don't know how to apply it to this problem - help would be GREATLY appreciated!


    Given U={z∈C : |z| = 1 }, n>0, f:U->U defined by f(z) = z^n and ~ the equivalence relation associated with f.

    Show that f defines a bijection f-bar:U/~ -> U.


    I don't know how to type mathematical notation so i hope you can understand it ok!
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  2. #2
    Member
    Joined
    Aug 2008
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    Hello,

    I assume you know that U={e^{it}: 0 <= t < 2(pi)}, f(e^{it})=e^{int}.

    Show that U/~={e^{is}-bar: 0<= s < 2(pi)/n} where e^{is}-bar={e^{it}: t=s+2(pi)k/n for some integer k}.
    Then, f-bar(e^{is}-bar)=e^{ins} is a well-defined bijection.

    Bye.
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