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Math Help - proving existence

  1. #1
    No one in Particular VonNemo19's Avatar
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    proving existence

    Hi, how would I show that for any group table that for each row there exists every element of the group at least once. That is, every row of the table lists every element of the group.

    OK, So I have proven uniqueness:

    Assume that x occurs twice in the same row, call it row a, then there must be columns y_1 and y_2 such that ay_1=x=ay_2. But this implies that y_1=y_2. Contradiction.

    Now, for the existence...
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Re: proving existence

    Quote Originally Posted by VonNemo19 View Post
    Hi, how would I show that for any group table that for each row there exists every element of the group at least once. That is, every row of the table lists every element of the group.

    OK, So I have proven uniqueness:

    Assume that x occurs twice in the same row, call it row a, then there must be columns y_1 and y_2 such that ay_1=x=ay_2. But this implies that y_1=y_2. Contradiction.

    Now, for the existence...
    a*(a^{-1}b)=b (or similar depending on how you write your rows and columns).
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