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Find the elements of the direct product

  1. Nov 16, 2008 #1
    1. The problem statement, all variables and given/known data

    Compute the direct sum Z_12 (+) U(10)

    Z_24 is the group Z under addition modulo 12
    U(10) is the group Z under multiplication modulo 10

    3. The attempt at a solution
    I have computed direct sums of Z_n groups before:

    For example: Z_2 (+) Z_3 = {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)}

    From this I would think I would follow a similar process but my textbook has the example:
    U(8) (+) U(10) = {(1,1),(1,3),(1,7),(1,9),(3,1),(3,3),(3,7),(3,9),(5,1),(5,3),(5,7),(5,9),(7,1),(7,3),(7,7),(7,9)}

    What's going on here?
    Last edited: Nov 16, 2008
  2. jcsd
  3. Nov 16, 2008 #2


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    Maybe what you're missing is that U(10) is actually the group of invertibles in Z_10 under multiplication. Can you write down what the elements of U(10) are?
  4. Nov 16, 2008 #3
    ok.. so U(10) = {1,3,7,9} under multiplication mod 10
    and Z_12 = {0,1,2,3,4,5,6,7,8,9,10,11} under addition mod 12
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