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

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