How to form a row complete Latin squares of order odd?

Question

Hello dear mathematicians How to form a row complete latin square of order odd. I found some answers for Latin squares of order even. Regards

Answer 1

Perhaps the best known way of constructing row complete latin squares is via sequencable groups. A group is sequencable if its elements can be ordered, say as $g_0,g_1,\ldots,g_{n-1}$, so that the map $g_i \mapsto g_i^{-1}g_{i+1}$ (for $0 \leq i \leq n-1$) is injective. We call such an ordering of the elements of a group a directed terrace.

Given a directed terrace $g_0,g_1,\ldots,g_{n-1}$ the latin square $L$ defined by $L_{i,j} = g_i^{-1}g_{i+1}$ is row complete (for a proof of this fact see the introduction to M.A. Ollis' dynamic survey on sequencable groups).

But now we need to ask what we know about sequencable groups. Keedwell conjectured that every nonabelian group of order at least 10 is sequencable. The conjecture has been confirmed for all nonabelian groups of odd order at most 555.

Which leaves us with our final question: for which odd orders do we know there exists a nonabelian group? The answer to this question is nontrivial, and is given in this Mathoverflow question/answer.