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.

Spence [9] 构造 \(\left( \frac{3^{d+1}(3^{d+1}-1)}{2}, \frac{3^d(3^{d+1}) +1)}{2}, \frac{3^d(3^d+1)}{2}\right) \)-组中的差分集 \(K \times C_3^{d+1}\) 为d 为任意正整数，K 为任意阶 \(\frac{3^{d+1}-1}{2}\) 的群。 Smith 和 Webster [8] 详尽地研究了 \(d=1\) 情况，而不要求群具有上面列出的形式，并发现了许多构造。其中，一个有趣的例子通过使用与 \(C_3^2\) 同构的非正态子群中的 (3, 3, 3, 1) 相对差分集来构造 \(A_4 \times C_3\) 中的 Spence 差分集。 Drisko [3] 有一个注释暗示他的技术允许在具有同构于 \(C_3^{d+1}\) 的非中心正态子群的群中构造 Spence 差分集，只要 \(\frac{3^{d+ 1}-1}{2}\) 是素数幂。我们通过在相似的群族中构造 Spence 差分集来推广这个结果，但我们放弃了 \(\frac{3^{d+1}-1}{2}\) 是素数幂的要求。我们推测任何阶 \(\frac{3^{d+1}(3^{d+1}-1)}{2}\) 的群，其正规子群同构于 \(C_3^{d+1) }\) 将有一个 Spence 差分集（这类似于 Dillon 的 2 群猜想，并且该结果在 Drisko 的工作中得到了证明）。最后，我们提出了群中 Spence 差分集的第一个已知示例，其中 Sylow 3 子群是非阿贝尔群且指数大于 3。这种新构造是通过计算完全自同构群 \(\textrm{Aut}( \mathcal {D})\) 的对称设计与已知的 Spence 差分集相关并识别 \(\textrm{Aut}(\mathcal {D})\) 的正则子群，使用 (3, 3, 3, 1）-相对差异集，描述差异集。