Existence of generalized Latin squares which are not embeddable in any group

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. In this paper, we show that for any integer \(n\ge 6\) and any integer m where \(m\in \left\{ n, n+1, \dots , \frac{n(n+1)}{2}-2\right\} \), there exists a commutative generalized Latin square of order n with m distinct elements which is not embeddable in any group. In addition, we show that for any integer \(r\ge 3\) and any integer s where \(s\in \{ r, r+1, \dots , r^2-2\}\), there exists a non-commutative generalized Latin square of order r with s distinct elements which is not embeddable in any group.