Non-commutative generalized Latin squares of order 5 and their embeddings in finite groups

Let n be a positive integer. A generalized Latin square of order n is an n x n matrix such that the elements in each row and each column are distinct. In this paper, we investigate classes of non-commutative generalized Latin squares of order 5 with 5, 24, and 25 distinct elements. We shall divide the squares into equivalence classes and determine completely the squares which are embeddable in groups. We also show that given any integer m where 5 <= m <= 25, there exists a non-commutative generalized Latin square of order 5 with m distinct elements which is embeddable in a finite group.