Difference sets in Abelian groups

Abstract

1. Difference sets in finite groups have been studied at great lengtll in a paper by R. H. Bruck [I]. Marshall Hall, Jr.'s notion of a multiplier of a difference set has been extended by Bruck to difference sets in groups. The latter has also extended a theorem of Hall on multipliers [3] to abelian groups. Hall has later obtained a generalization of his theorem. The object of this paper is, in the first instance, to extend this generalized theorem of Hall on multipliers to difference sets in abelian groups. A new feature that has been introduced here is the application of the theory of characters to the discussion of the problem. By this method I have also been able to obtain a new class of different sets. The later part of the paper is concerned with the discussion of this new class. Let G be an abelian group of order v and let D be a subset of G containing k elements such that every element of G other than the identity can be expressed exactly X times in the form ab-' where a, b are elements of D. Then D is called a difference set of type (v, k, X). The numbers v, k, X satisfy the relation