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Math Help - well-defined function (cartesian product)

  1. #1
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    well-defined function (cartesian product)

    Define \phi: A x B ---> A/C x B/D by \phi((a,b))=(aC,bD)

    I have to show this map is well-defined, here's what I did:

    Let (a_1, b_1), (a_2, b_2) \in A x B, and let \phi(a_1,b_1)=\phi(a_2,b_2). Then (a_1C,b_1D)=(a_2C,b_2D) \implies (a_1, b_1)=(a_2, b_2). Is this implication clear?
    Can I write (a_1C, b_1D) = (a_1, b_1)(C x D)?

    Please hlep, thank you.
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    Define \phi: A x B ---> A/C x B/D by \phi((a,b))=(aC,bD)

    I have to show this map is well-defined, here's what I did:

    Let (a_1, b_1), (a_2, b_2) \in A x B, and let \phi(a_1,b_1)=\phi(a_2,b_2). Then (a_1C,b_1D)=(a_2C,b_2D) \implies (a_1, b_1)=(a_2, b_2).
    You need to give us more information. For a quotient map of the form \phi:A\times B\to A/C\times B/D to be well-defined, it is necessary that \phi should be constant on the cosets of C\times D. You haven't said anything about the nature of C\times D or \phi, and you haven't actually proved anything.
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  3. #3
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    Quote Originally Posted by Opalg View Post
    You need to give us more information. For a quotient map of the form \phi:A\times B\to A/C\times B/D to be well-defined, it is necessary that \phi should be constant on the cosets of C\times D. You haven't said anything about the nature of C\times D or \phi, and you haven't actually proved anything.
    I am given that C is a normal subgroup of A and D is a normal subgroup of B. What else is needed to show \phi is well-defined?
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  4. #4
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    In that case, there isn't much to prove. You want to show that \phi(a,b) depends only on the cosets aC and bD, and that is immediately obvious from the way that \phi is defined.
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