Embeddings of generalized Latin squares in finite groups

Abstract

Let n be a positive integer. A generalized Latin square of order n is an $n\times n$ matrix such that the elements in each row and each column are distinct. The square is said to be commutative if the $n\times n$ matrix is symmetric. Given $n\ge 2$, we show that for any $m\in \left\{ n, n+1, \ldots , n(n+1)/2\right\}$, there exists a commutative generalized Latin square of order n with m distinct elements which is embeddable in a finite group. We also show that for $(n, m)=(2, 3), (2, 4)$ and for any $m\in \{n, n+1, \ldots , n^2\}$ where $n\ge 3$, there exists a non-commutative generalized Latin square of order n with m distinct elements which is embeddable in a finite group.