New spence difference sets

Spence [9] constructed \(\left( \frac{3^{d+1}(3^{d+1}-1)}{2}, \frac{3^d(3^{d+1}+1)}{2}, \frac{3^d(3^d+1)}{2}\right) \)-difference sets in groups \(K \times C_3^{d+1}\) for d any positive integer and K any group of order \(\frac{3^{d+1}-1}{2}\). Smith and Webster [8] have exhaustively studied the \(d=1\) 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 \(A_4 \times C_3\) by using (3, 3, 3, 1)-relative difference sets in a non-normal subgroup isomorphic to \(C_3^2\). Drisko [3] has a note implying that his techniques allow constructions of Spence difference sets in groups with a noncentral normal subgroup isomorphic to \(C_3^{d+1}\) as long as \(\frac{3^{d+1}-1}{2}\) is a prime power. We generalize this result by constructing Spence difference sets in similar families of groups, but we drop the requirement that \(\frac{3^{d+1}-1}{2}\) is a prime power. We conjecture that any group of order \(\frac{3^{d+1}(3^{d+1}-1)}{2}\) with a normal subgroup isomorphic to \(C_3^{d+1}\) 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 \(\textrm{Aut}(\mathcal {D})\) of a symmetric design associated to a known Spence difference set and identifying a regular subgroup of \(\textrm{Aut}(\mathcal {D})\), uses (3, 3, 3, 1)-relative difference sets to describe the difference set.