Difference sets in abelian groups ofp-rank two

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, ) difference setD, then for each for some subgroupC_p ofG of orderp. Consequently,k(p=1), with equality only ifF=1/p D, whereD is the image ofD under the canonical homomorphism fromG ontoG/E (E being the unique elementary abelian subgroup ofG of orderp²), is a (v/p²,k/p, ) difference set inG/E. As applications, we establish the nonexistence of (i) (96, 20, 4) difference sets in ₄ x ₈ x ₃, (ii) (640, 72, 8) difference sets in ₈ x ₁₆ x ₅ and (iii) (320, 88, 24) difference sets in ₈ x ₈ x ₅. 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].Research partially supported by NSA Grant #904-92-H-3057 and by NSF Grant # NCR-9200265.