symmetric latin square of order 5

Question

My textbook said if a latin square of order 5 is idempotent and had a 2 in the (1,3) entry, it could not be completed as a symmetric square. But isn't this one such square?

Or do I have the definition of "symmetric" wrong?

1   4   2   5   3

4   2   5   3   1

2   5   3   1   4

5   3   1   4   2

3   1   4   2   5

Answer 1

Your definition of a symmetric latin square is used in several papers. However, it's possible they actually mean the latin square is the multiplication table of a (totally) symmetric quasigroup. These are ones such that $ab=c \implies ba=c, ca=b$. Also, idempotent meaning $i$ is in the entry $(i,i)$ becomes idempotent meaning $xx=x$ for all $x$, which means your current square is still idempotent iff the row and column number of an entry are the elements whose product lies at that entry, i.e. $2$ is at $(2,2)$ because $2=2*2$.