New spence difference sets

Abstract Spence 9 constructed (3^d+1 (3^{d+1-1) }2, 3ᵈ (3^d+1+1) 2, 3ᵈ (3ᵈ+1) 2) 3 d + 1 (3 d + 1 - 1) 2, 3 d (3 d + 1 + 1) 2, 3 d (3 d + 1) 2 -difference sets in groups K C₃^d+1 K × C 3 d + 1 for d any positive integer and K any group of order 3^d+1-12 3 d + 1 - 1 2. Smith and Webster 8 have exhaustively studied the d=1 d = 1 case without requiring that the group have the form listed above and found many constructions. Among these, one intriguing example constructs Spence difference sets in A₄ C₃ A 4 × C 3 by using (3, 3, 3, 1) -relative difference sets in a non-normal subgroup isomorphic to C₃² C 3 2. Drisko 3 has a note implying that his techniques allow constructions of Spence difference sets in groups with a noncentral normal subgroup isomorphic to C₃^d+1 C 3 d + 1 as long as 3^d+1-12 3 d + 1 - 1 2 is a prime power. We generalize this result by constructing Spence difference sets in similar families of groups, but we drop the requirement that 3^d+1-12 3 d + 1 - 1 2 is a prime power. We conjecture that any group of order 3^d+1 (3^{d+1-1) }2 3 d + 1 (3 d + 1 - 1) 2 with a normal subgroup isomorphic to C₃^d+1 C 3 d + 1 will have a Spence difference set (this is analogous to Dillon’s conjecture in 2-groups, and that result was proved in Drisko’s work). Finally, we present the first known example of a Spence difference set in a group where the Sylow 3-subgroup is nonabelian and has exponent bigger than 3. This new construction, found by computing the full automorphism group Aut (D) Aut (D) of a symmetric design associated to a known Spence difference set and identifying a regular subgroup of Aut (D) Aut (D), uses (3, 3, 3, 1) -relative difference sets to describe the difference set.