Difference sets in abelian groups ofp-rank two

Abstract

Under a technical assumption that pertains to the so-called “self-conjugacy”, we prove: if an abelian groupG ofp-rank two,p a prime, admits a (nontrivial) (v, k, λ) difference setD, then for each $$x D,x.C_p \subseteq D$$ for some subgroupC p ofG of orderp. Consequently,k≤(p=1)λ, with equality only ifF=1/p D σ, whereD σ is the image ofD under the canonical homomorphism fromG ontoG/E (E being the unique elementary abelian subgroup ofG of orderp 2), is a (v/p 2,k/p, λ) difference set inG/E. As applications, we establish the nonexistence of (i) (96, 20, 4) difference sets in ℤ4 x ℤ8 x ℤ3, (ii) (640, 72, 8) difference sets in ℤ8 x ℤ16 x ℤ5 and (iii) (320, 88, 24) difference sets in ℤ8 x ℤ8 x ℤ5. The first one fills a missing entry in Lander's table [6] and the other two in Kopilovich's table [5] (all with the answer ‘no’). We also point out the connection of the parameter sets in (i) above with the Turyn-type bounds [10] for the McFarland difference sets [9].