Difference set

In combinatorics, a difference set is a subset of size of a group of order such that every non-identity element of can be expressed as a product of elements of in exactly ways. A difference set is said to be cyclic, abelian, non-abelian, etc., if the group has the corresponding property. A difference set with is sometimes called planar or simple. If is an abelian group written in additive notation, the defining condition is that every non-zero element of can be written as a difference of elements of in exactly ways. The term "difference set" arises in this way. A simple counting argument shows that there are exactly pairs of elements from that will yield nonidentity elements, so every difference set must satisfy the equation If is a difference set and then is also a difference set, and is called a translate of ( in additive notation). The complement of a -difference set is a -difference set. The set of all translates of a difference set forms a symmetric block design, called the development of and denoted by In such a design there are elements (usually called points) and blocks (subsets). Each block of the design consists of points, each point is contained in blocks. Any two blocks have exactly elements in common and any two points are simultaneously contained in exactly blocks. The group acts as an automorphism group of the design. It is sharply transitive on both points and blocks. In particular, if , then the difference set gives rise to a projective plane. An example of a (7,3,1) difference set in the group is the subset . The translates of this difference set form the Fano plane. Since every difference set gives a symmetric design, the parameter set must satisfy the Bruck–Ryser–Chowla theorem. Not every symmetric design gives a difference set. Two difference sets in group and in group are equivalent if there is a group isomorphism between and such that for some The two difference sets are isomorphic if the designs and are isomorphic as block designs.