The inspiration of my question comes from Henry B. Mann's 1942 paper The Construction of Orthogonal Latin Squares. My goal is to add rigor to Mann's explanation that two Latin squares $P = (P_1,P_2,...,P_m)$ and $Q = (Q_1,Q_2,...,Q_m)$ are orthogonal if and only if $P^{-1}Q = (P_1^{-1}Q_1,P_2^{-1}Q_2,...,P_m^{-1}Q_m)$ is a Latin square, which is as follows:
$P_i^{-1}P_j$ is the transformation which transforms the $i$th row of $(P_1,P_2,...,P_m)$ into the $i$th row of $(Q_1,Q_2,...,Q_m)$. Since every pair of variables occurs exactly once if the second square is imposed upon the first, the square $(P_1^{-1}Q_1,P_2^{-1}Q_2,...,P_m^{-1}Q_m)$ contains for every $i$ and $k$ a permutation which transforms $x_i$ into $x_k$. But then it can not contain two permutations which transform $x_i$ into $x_k$. This argument can be reversed and it follows that $(P_1,P_2,...,P_m)$ and $(Q_1,Q_2,...,Q_m)$ are orthogonal if and only if $(P_1^{-1}Q_1,P_2^{-1}Q_2,...,P_m^{-1}Q_m)$ is a Latin square.
My Questions: What does Mann mean by "transformations"? I feel like this wording is slightly vague and it is unclear to me what this means from a technical viewpoint. Also, why is it that if we assume $P^{-1}Q$ is not a Latin square, then this implies that $P^{-1}Q$ contains two permutations which transform $x_i$ into $x_k$?
If anyone knows of a more rigorous proof than the one Mann provided, that would answer all of my questions. Thanks for reading and for any responses!
Mann, Henry B., The construction of orthogonal Latin squares, Ann. Math. Stat. 13, 418-423 (1942). ZBL0060.02706.