The interplay of designs and difference sets

Abstract

It is well known that a (divisible) design with a regular automorphism group (Singer group) is equivalent to a (relative) difference set in that group. Therefore, the results and tools in designs and difference sets sometimes can be transferred to each other. In this dissertation, we shall discuss three problems to illustrate how the two theories interplay with each other. The first problem is about the construction of relative difference sets. The fascinating point is that one can see through it how various algebraic tools can be applied to combinatorial problems. There are many results on the construction of relative difference sets, see [7],[10],[29],[30],[35],[37]. Unfortunately, most of the constructions work for abelian groups, but few for non-abelian ones, since algebraic tools in the latter case are limited. By investigating the elements in affine general linear groups, which are also automorphisms of some classical divisible designs, we obtain a new construction of infinite families of (p^{a},p^{b},p^{a},p^{a-b})-relative difference sets. This new construction shows that (p^{a},p^{b},p^{a},p^{a-b})-relative difference sets exist in many non-abelian groups which were not covered by previous constructions.