Latin square and groups

Question

From what I can gather, table of each group present latin square. But I wonder, how can I give an example of latin square which isn't a result of the operations on group?

Answer 1

\begin{array} & &2 &1 &3 \\ &1 &3 &2 \\ &3 &2 &1 \\ \end{array}

This cannot represent a group because there is no identity element. $1*1=2$, $2*2=3$, and $3*3=1$.

Answer 2

The answer is yes; there are just a few (up to relabeling) $3\times 3$ latin squares. One is the group multiplication table for the cyclic group of order 3. The other will be the multiplication table of what is called a quasigroup. These are algebraic objects interesting in their own right and even have their own representation theory. In this case, the idea is to try and make a latin square where no element acts as an identity element. You will probably get a right (or left) identity, though.