New spence difference sets

Abstract

AbstractSpence [9] constructed $$\left( \frac{3^{d+1}(3^{d+1}-1)}{2}, \frac{3^d(3^{d+1}+1)}{2}, \frac{3^d(3^d+1)}{2}\right) $$ 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 $$K \times C_3^{d+1}$$ K × C 3 d + 1 for d any positive integer and K any group of order $$\frac{3^{d+1}-1}{2}$$ 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 $$A_4 \times C_3$$ A 4 × C 3 by using (3, 3, 3, 1)-relative difference sets in a non-normal subgroup isomorphic to $$C_3^2$$ 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 $$C_3^{d+1}$$ C 3 d + 1 as long as $$\frac{3^{d+1}-1}{2}$$ 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 $$\frac{3^{d+1}-1}{2}$$ 3 d + 1 - 1 2 is a prime power. We conjecture that any group of order $$\frac{3^{d+1}(3^{d+1}-1)}{2}$$ 3 d + 1 ( 3 d + 1 - 1 ) 2 with a normal subgroup isomorphic to $$C_3^{d+1}$$ 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 $$\textrm{Aut}(\mathcal {D})$$ Aut ( D ) of a symmetric design associated to a known Spence difference set and identifying a regular subgroup of $$\textrm{Aut}(\mathcal {D})$$ Aut ( D ) , uses (3, 3, 3, 1)-relative difference sets to describe the difference set.