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.