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Math Help - isomorphisms external direct product

  1. #1
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    isomorphisms external direct product

    Suppose we have some isomorphism φ which maps Z(subscript 3) + Z(subscript 5)
    (ie the external direct product between Z3 and Z5..sorry dont know how to make the symbols) to Z(subscript 15)

    If φ(2,3) = 2 which element in the external direct product maps to 1 in Z15


    This is screaming chinese remainder thm to me but it doesnt work. I thought for sure the mapping would be something along the lines of 2*3 + 3*5 = ___ (mod 15)
    but clearly doesnt work for the given example. ive tried numerous other different combinations of the same type of idea I just feel like im playing a "guess and check" type of game, any thoughts on a method?
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  2. #2
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    Quote Originally Posted by stumped765 View Post
    Suppose we have some isomorphism φ which maps Z(subscript 3) + Z(subscript 5)
    (ie the external direct product between Z3 and Z5..sorry dont know how to make the symbols) to Z(subscript 15)

    If φ(2,3) = 2 which element in the external direct product maps to 1 in Z15


    This is screaming chinese remainder thm to me but it doesnt work. I thought for sure the mapping would be something along the lines of 2*3 + 3*5 = ___ (mod 15)
    but clearly doesnt work for the given example. ive tried numerous other different combinations of the same type of idea I just feel like im playing a "guess and check" type of game, any thoughts on a method?


    As 2^4=1\!\!\pmod{15} , we get that 1=2^4=\phi(2,3)^4=\phi(1,1) ...which is hardly surprising since

    a homom. of groups (or unitary rings) must map the unit (the multiplicative unit) to the unit...

    Tonio
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