On the converse of singer's theorem

A construction is given in which the nonzero elements of a planar difference set give rise to a totally symmetric quasi-group. Examples are provided which suggest that the quasi-group is essentially the additive group of the field. The evidence supports the conjecture that the converse of Singer's theorem holds. The Multiplier Theorem is used to characterize when the totally symmetric quasi-groups are totally symmetric loops. The results extend to Abelian group difference sets (λ = 1).