Abelian difference set

Let be a group of order and with . Then is called a -difference set of order in if every element in has exactly different representations with , see [a1]. For instance, is a -difference set in the cyclic group of order . If is Abelian (cyclic, non-Abelian), the difference set is called Abelian (cyclic, non-Abelian). Two difference sets and in are equivalent if there is a group automorphism such that . The existence of a -difference set is equivalent to the existence of a symmetric -design with acting as a regular automorphism group (cf. also Difference set). If two difference sets correspond to isomorphic designs, the difference sets are called isomorphic. Given certain parameters , and and a group , the problem is to construct a difference set with those parameters or prove non-existence. To prove non-existence of Abelian difference sets, results from algebraic number theory are required: The existence of the difference set implies the existence of an algebraic integer of absolute value in some cyclotomic field. In several cases one can prove that no such element exists, see [a5]. Another approach for non-existence results uses multipliers: A multiplier of an Abelian difference set in is an automorphism of such that . A statement that certain group automorphisms have to be multipliers of putative difference sets is called a multiplier theorem. It is known, for instance, that the mapping is a multiplier of an Abelian difference set provided that: i) divides the order ; ii) is relatively prime to ; and iii) . Several generalizations of this theorem are known, see [a1].

On the existence side, some families of Abelian difference sets are known, see [a3].

Examples.

The most popular examples are as follows.

Cyclic -difference sets, a prime power. The classical construction of these difference sets (elements in the multiplicative group of whose trace is ) corresponds to the classical point-hyperplane designs of a finite projective space. For non-equivalent cyclic examples with the same parameters, see [a5].

Quadratic residues in , (Paley difference sets). Some other cyclotomic classes yield difference sets too, see [a1].

-difference sets, , where is a product of odd prime numbers (Hadamard difference sets, [a2]). If , it is known that an Abelian Hadamard difference set exists if and only if the exponent of is at most , see [a4].

-difference sets, where ( an odd prime power) or or (generalized Hadamard difference sets, [a2]).

-difference sets, a prime power (McFarland difference sets).

References

[a1]	T. Beth, D. Jungnickel, H. Lenz, "Design theory" , Cambridge Univ. Press (1986)
[a2]	Y.Q. Chen, "On the existence of abelian Hadamard difference sets and generalized Hadamard difference sets" Finite Fields and Appl. (to appear)
[a3]	D. Jungnickel, A. Pott, "Difference sets: Abelian" Ch.J. Colbourn (ed.) J.H. Dinitz (ed.) , CRC Handbook of Combinatorial Designs , CRC (1996) pp. 297–307
[a4]	R.G. Kraemer, "Proof of a conjecture on Hadamard -groups" J. Combinatorial Th. A , 63 (1993) pp. 1–10
[a5]	A. Pott, "Finite geometry and character theory" , Lecture Notes in Mathematics , 1601 , Springer (1995)