Latin Squares Based on Groups

Abstract

Latin squares and orthogonal Latin squares have been used in the construction of many classes of designs: nets, affine planes, projective planes, and transversal designs, in particular. When orthogonal sets of Latin squares are obtained from the multiplication table of a finite group by permuting columns, each square is determined by its first row, which is a permutation of the elements of the group. This enables us to describe and study orthogonality from a purely algebraic point of view, using difference matrices, complete mappings, and orthomorphisms. The nets, affine planes, projective planes, and transversal designs constructed in this way are characterized by the action of the group on these designs. We introduce these concepts in this chapter.