All difference family structures arise from groups

Abstract

Planar nearrings have been used to define classes of 2-designs since Ferrero's work in 1970. These 2-designs are a class of difference families. Recent work from Pianta has generalised Ferrero and Clay's work with planar nearrings to investigate planar nearrings with nonassociative additive structure. Thus we are led to the question of nonassociative difference families.

Difference families are traditionally built using groups as their basis. This paper looks at what sort of generalized difference family constructions could be made, using the standard basis of translation and difference.

We determine minimal axioms for a difference family structure to give a 2-design. Using these minimal axioms we show that  we obtain quasigroups. These quasigroups are shown to be isotopic to groups and the derived 2-designs from the nonassociative difference family are identical to the 2-designs from the isotopic groups. Thus all difference families arise from groups.

This result will be of interest to those using nonstandard algebras as bases for defining 2-designs.