1. ## 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...

2. ## Re: proving existence

Originally Posted by VonNemo19
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).