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Thread: [SOLVED] addition and multiplication tables

  1. #1
    Mar 2008

    Lightbulb [SOLVED] addition and multiplication tables

    I have these two Cayley tables. The addition table is complete and the multiplication table is not. The instructions say to use the distributive laws, but I can't seem to find the solutions using the distributive laws. I can only find the solution using the definition of a group.

    Can someone help me?

    ring R = {a, b, c}

    + a b c
    a a b c
    b b c a
    c c a b

    x a b c
    a a a a
    b a c
    c a

    Thanks a bunch!
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  2. #2
    Senior Member TheAbstractionist's Avatar
    Apr 2009
    The additiion table shows that \langle R+\rangle is a group with identity element a. This is the zero element of R.

    To complete the table for multiplication, take for example bc. From the addition table, c=b+b. Hence, bc=b(b+b)=bb+bb=c+c=b. (You know bb=c from the partially completed table.)

    Do the same for cb and cc. (Hint: It will turn out that c is the multiplicative identity, and the ring is a field with 3 elements.)
    Last edited by TheAbstractionist; Jun 4th 2009 at 01:20 AM.
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