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Math Help - Isomorphism Question

  1. #1
    Junior Member
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    Isomorphism Question

    Show that U(10)-->Z4 where f(3)=2. Show it's either an isomorphism or it isn't.

    U(10) = {1,3,7,9} under multiplication and <3> with order 4.

    Z4 is 0,1,2,3 under addition mod 4.

    I know from a proposition in the book that U(10)-->Z4 is an isomorphism, but I can't quite figure out how f(3)=2 changes that.

    Thanks.
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  2. #2
    MHF Contributor

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    Tejas
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    Re: Isomorphism Question

    you know that 3 is a generator for U(10), good. that means if f:U(10)-->Z4 is given by f(3) = 2, you should be able to calculate the other images of f by taking images of powers of 3:

    f(9) = f(3^2) = f(3) + f(3) = ?
    f(7) = f(3^3) = f(3) + f(3) + f(3) = ?
    f(1) = f(3^4) = f(3) + f(3) + f(3) = ?

    is f bijective?
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