The article delves into the construction of Spence difference sets within finite groups, particularly focusing on the properties of noncentral Sylow 3-subgroups and the application of relative difference sets. It introduces new methods for finding difference sets in groups of varying orders, emphasizing the use of relative difference sets analogous to hyperplanes. The paper also highlights the potential for further constructions and research directions in the study of difference sets, encouraging exploration into nonabelian techniques and computational approaches.
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Abstract
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.
Notes
The authors express our deep sadness at the untimely death of our dear friend and colleague Kai-Uwe Schmidt. We dedicate this article to his memory.
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1 Introduction
If G is a finite group of order v and D is a k-subset with the property that every nonidentity element \(g \in G\) has exactly \(\lambda \) representations of the form \(g = x_1 x_2^{-1}, x_1, x_2 \in D\), then we say that D is a \((v,k,\lambda )\)-difference set in G. We note that we are writing our groups multiplicatively. A useful characterization of difference sets is the following equation in the group ring \({\mathbb {Z}}[G]\), where in this equation we use the conventions that
and \(1_G\) is the identity element of G as a group ring element:
$$\begin{aligned} D D^{(-1)} = (k-\lambda ) 1_G + \lambda G. \end{aligned}$$
A variant of a difference set is known as a relative difference set: if G is a group of order mn with a (typically although not necessarily normal) subgroup N of order n and a k-subset R that satisfies the group ring equation \(RR^{(-1)} = k 1_G + \lambda (G-N)\), then we say that R is a \((m,n,k,\lambda )\)-relative difference set relative to N. In our final section, we will use (3, 3, 3, 1)-relative difference sets in \(C_3 \times C_3\) to construct difference sets in groups of two different orders.
The central problem in the study of difference sets is to determine which groups of order v contain a \((v,k,\lambda )\)-difference set. Difference sets are often classified into families based on their parameters. When \(\gcd (v,k-\lambda )>1\), the known examples include Hadamard, McFarland, Spence, Davis-Jedwab, and Chen (see [1, Chapter VI, Sect. 9, pp. 363–369] for a description of these families). In this paper, we focus on the Spence family, the members of which have parameters \(\left( \frac{3^{d+1}(3^{d+1}-1)}{2}, \frac{3^{d}(3^{d+1}+1)}{2}, \frac{3^{d}(3^{d}+1)}{2}\right) \). The original Spence construction [9] found examples in all groups \(G \cong K \times C_3^{d+1}\) for K any group (including nonabelian) of order \(\frac{3^{d+1}-1}{2}\). That construction parallels the McFarland construction [6] while taking advantage of the trivial (but critical) property that \(3-1=2\). The main ingredient in the original construction is a collection of hyperplanes (i.e. subgroups of order \(3^d\)) \(H_0, H_1, \ldots , H_{\frac{3^{d+1}-1}{2}-1}\) in the subgroup \(H \le G, H \cong C_3^{d+1}\). The complement of one of these hyperplanes within H, say \(H^c_0 = H \backslash H_0\), is a set with \(3^{d+1} - 3^d = 2 \cdot 3^d\) elements (this is where the \(3-1=2\) factor comes into play). If \(K = \{ k_0, k_1, \ldots , k_{\frac{3^{d+1}-1}{2}-1} \}\), then the following theorem gives the original Spence construction.
is a \(\left( \frac{3^{d+1}(3^{d+1}-1)}{2}, \frac{3^d(3^{d+1}+1)}{2}, \frac{3^d(3^d+1)}{2}\right) \)-difference set in \(K \times H\).
Although somewhat technical, the group ring calculations used in the proof of Theorem 1.1 (see [9, Proof of Theorem 1]) are enlightening and show why subsets that behave nicely with respect to multiplication, such as subgroups (or “hyperplanes,” when viewing an elementary abelian group as a vector space) and relative difference sets, are useful for constructions. We remark that different choices for \(k_i\)’s can lead to nonisomorphic difference sets; see, for example [8, Sect. 3], where four inequivalent Spence (36, 15, 6)-difference sets are constructed in the group with K isomorphic to a cyclic group of order 4 and three inequivalent Spence (36, 15, 6)-difference sets are constructed in the group with K isomorphic to a Klein 4-group using a hyperplane construction (referred to there as a “spread construction”).
As noted by Drisko [3, Corollary 8], when \(\frac{3^{d+1} - 1}{2}\) is a prime power, any group G of order \(\frac{3^{d+1}(3^{d+1}-1)}{2}\) with a (not necessarily central) normal elementary abelian subgroup of order \(3^{d+1}\) contains a difference set with the Spence parameters. This begs the question of exactly how large the Spence family actually is. For example, Jia [4] proved that abelian (351, 126, 45)-difference sets only exist in groups with exponent 39; a natural question is whether difference sets in the Spence family exist in groups with a non-elementary abelian Sylow 3-subgroup.
The purpose of this paper is to address precisely these questions and to provide evidence that the Spence family likely extends far beyond the known examples. This paper is organized as follows. In Sect. 2, we prove that there exists at least one member of the Spence family with a noncentral Sylow 3-subgroup for every \(d \ge 1\). In Sect. 3, we summarize the known results regarding the (36, 15, 6)-difference sets, highlighting a particular group whose Spence difference sets arise not from a “hyperplane construction” but from a similar construction involving (3, 3, 3, 1)-relative difference sets. In Sect. 4, we present an example of a (351, 126, 45)-difference set with a non-normal, nonabelian Sylow 3-subgroup with exponent 9 that is related to (3, 3, 3, 1)-relative difference sets, illustrating the promise in using relative difference sets analogously to hyperplanes in constructions. Finally, in Sect. 5, we highlight a number of potential directions and open questions.
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2 New hyperplane constructions
Dillon gave a generalization of McFarland’s construction that also applies to Spence difference sets; we present that result here in a form specific to the Spence family. (Note the additive notation here, which represents an element of the group ring \({\mathbb {Z}}[G]\) and is used interchangeably with set theoretic notation.)
Theorem 2.1
([2, Theorem]) Let \(H \cong C_3^{d+1}\) be a normal subgroup of G of index \(\frac{3^{d+1}-1}{2}\) and let \(\{k_0, \dots , k_{\frac{3^{d+1}-1}{2}-1} \}\) be a transversal for the cosets of H in G so that
In Spence’s original paper, any assignment of elements of the transversal to the hyperplanes will trivially be a permutation of the hyperplanes since the \(k_i\)’s commute with all elements of H. Drisko’s paper [3] deals precisely with the problem of finding such a transversal, and it is proved that such a transveral always exists when \(\frac{3^{d+1}-1}{2}\) is a prime power.
Corollary 2.2
([3, Corollary 8]) Suppose \(r = \frac{3^{d+1}-1}{2}\) is a prime power. Then any group G of order \(r 3^{d+1}\) which has a normal elementary abelian subgroup E of order \(3^{d+1}\) has a difference set with Spence parameters.
As examples of cases where r is a prime power, we have at least the following: \(d = 2\), where \((3^3 - 1)/2 = 13\); \(d=4\), where \((3^5-1)/2 = 121 = 11^2\); \(d=6\), where \((3^7-1)/2 = 1093\); \(d=10\), where \((3^{11}-1)/2 = 88573\); and \(d=12\), where \((3^{13}-1)/2 = 797161\). In each case we could use Corollary 2.2 to prove the existence of a Spence difference set in a nonabelian semidirect product group isomorphic to \(C^{d+1} \rtimes K\).
We now modify the construction to find many more groups with Spence difference sets in addition to those given by Corollary 2.2.
Theorem 2.3
Given an integer \(d \ge 1\) and an integer m so that \(m | \frac{3^{d+1}-1}{2}\) and \(\gcd (m+1,\frac{3^{d+1}-1}{2}) = 1\), then the group
contains a \((\frac{3^{d+1}(3^{d+1}-1)}{2}\), \(\frac{3^d(3^{d+1}+1)}{2}, \frac{3^d(3^d+1)}{2})\)-difference set in the Spence family.
Proof
Consider \(\textrm{GF}(3^{d+1})^+\), the additive group of the field with \(3^{d+1}\) elements. This field has a Singer cycle: that is, an automorphism \(\phi \) of the field that multiplies each element x of the field by a primitive element \(\alpha \), so \(\phi (x) = \alpha x\). If we define \(\{ 1, \alpha , \alpha ^2, \ldots , \alpha ^{d-1} \}\) to be a basis for the hyperplane \(H_0\) of the \(d+1\)-dimensional vector space \(\textrm{GF}(3^{d+1})\), then we have that
$$\begin{aligned}&\psi : \textrm{GF}(3^{d+1})^+ \rightarrow \left\langle x_i, 1 \le i \le d+1 | x_i^3 = [x_i,x_j] = 1, 1 \le i \ne j \le d+1 \right\rangle \\&\psi : \alpha ^k \mapsto x_{k+1}, 0 \le k \le d \end{aligned}$$
be an isomorphism from the additive group of \(\textrm{GF}(3^{d+1})\) to the Sylow 3-subgroup of \({{\mathcal {G}}}_m\).
If we also have that \(\gcd (m+1,\frac{3^{d+1}-1}{2}) = 1\), then we claim that we can construct a Spence difference set in \({{\mathcal {G}}}_m\). To see this, first observe that \(w^j \psi (H_i) w^{-j} = \psi (H_{i+mj})\) in \({{\mathcal {G}}}_m\). Since we read the subscripts of the hyperplanes modulo \(\frac{3^{d+1}-1}{2}\) and \(\gcd (m+1, \frac{3^{d+1}-1}{2}) = 1\), the map
is a permutation of the hyperplanes. The result now follows from Theorem 2.1 by assigning the coset representative \(w^i\) to the hyperplane \(\psi (H_i)\). \(\square \)
As specific examples that are, as far as we know, new constructions of Spence difference sets, we present the following two corollaries.
Corollary 2.4
If d is an even positive integer, then the group \({{\mathcal {G}}}_1\) contains a Spence difference set with parameters \(\left( \frac{3^{d+1}(3^{d+1}-1)}{2}, \frac{3^d(3^{d+1}+1)}{2}, \frac{3^d(3^d+1)}{2}\right) \).
Proof
Observe that \(\frac{3^{d+1}-1}{2}\) is an odd number when d is even and we can apply Corollary 2.3 since \(m+1 = 2\) is relatively prime to \(\frac{3^{d+1}-1}{2}\). \(\square \)
Corollary 2.5
If d is an odd positive integer, then the group \({{\mathcal {G}}}_2\) contains a Spence difference set with parameters \(\left( \frac{3^{d+1}(3^{d+1}-1)}{2}, \frac{3^d(3^{d+1}+1)}{2}, \frac{3^d(3^d+1)}{2}\right) \).
Proof
Observe that \(m+1=3\) is relatively prime to \(\frac{3^{d+1}-1}{2}\) when d is an odd positive integer. \(\square \)
More generally, we could consider groups of order \(\frac{3^{d+1}(3^{d+1}-1)}{2}\) with a normal elementary abelian subgroup of order \(3^{d+1}\) and make selections for the coset representatives so that mapping \(H_j \mapsto g_j H_j g_j^{-1}\) is a permutation of the hyperplanes. We are optimistic this selection process can always be done, so we conjecture the following:
Conjecture 2.6
Suppose G is a group of order \(\frac{3^{d+1}(3^{d+1}-1)}{2}\) with an elementary abelian normal subgroup of order \(3^{d+1}\). Then G will contain a Spence difference set.
A difficult case seems to be the group \({{\mathcal {G}}}_1\) when d is an odd integer. We cannot apply Corollary 2.5, since \(m+1=2\) is not relatively prime to \(\frac{3^{d+1}-1}{2}\). We think this would be a good test case for anyone wanting to make progress on Conjecture 2.6.
3 The \(d=1\) case
Smith and Webster [8] did an exhaustive computer search for Spence difference sets with the parameters (36, 15, 6) (this is the \(d=1\) case) and found there are 35 inequivalent examples. We list the 32 inequivalent difference sets in eight groups that can be constructed via Theorem 1.1 directly or through a modification. Note that \(D_i\) represents the \(i^{\text {th}}\) difference set as given in [5] and [7]. In each case, the Spence difference set indicated by a row of the table has the form
We note that in each case the conjugation of the hyperplanes by the selected coset representative gives a permutation of the hyperplanes as required in Theorem 2.1.
An intriguing example that is not constructed via Theorem 2.1 is the following. The group \(G_9 \cong A_4 \times C_3\), where \(A_4\) is the alternating group on four letters, contains three inequivalent difference sets. We observe that \(G_9\) has four distinct subgroups isomorphic to \(C_3 \times C_3\), but none of these subgroups are normal and hence we cannot use hyperplanes to build a difference set. Since this construction is different than the others, we explicitly give a formula for getting the difference sets. Let
In the following, we use \(H_0 = \langle b \rangle , R_0 = \{ 1, a, a^2b \}\), and \(R_1 = \{ 1, a, a^2b^2 \}\) and note that both \(R_0\) and \(R_1\) are (3, 3, 3, 1)-relative difference sets relative to \(H_0\). Then,
$$\begin{aligned} D = H_0^c + R_0 c R_1 \end{aligned}$$
is a (36, 15, 6)-difference set in the Spence family. In fact, all difference sets in \(G_9\) may be described by fixing \(H_0\) and \(R_0\) and letting \(R_1\) vary across other relative difference sets in H:
In this section, we exhibit an interesting example of a Spence difference set in a group of order 351 whose Sylow 3-subgroup is nonabelian and has exponent 9. To the best of our knowledge, this is the first example of a Spence difference set in a group whose Sylow 3-subgroup that is not elementary abelian, and a proper understanding of this example will likely lead to further constructions.
this is SmallGroup(351,7), and \(G \cong C_{13} \rtimes (C_9 \rtimes C_3)\), where \(C_9 \rtimes C_3\) is the extraspecial group of order 27 with exponent 9. To find a difference set in G, we constructed the symmetric design \({\mathcal {D}}\) associated to several of the original Spence difference set in \(C_3^3 \times C_{13}\) (we have choices about how to assign the coset representatives \(k_i\) to the hyperplanes \(H_i\), and these choices can lead to nonisomorphic designs) and then computed the full automorphism group \(\textrm{Aut}(\mathcal {D})\) of each of those designs. We used GAP [10] to find subgroups \(G_{{\mathcal {D}}}\) of \(\textrm{Aut}(\mathcal {D})\) of order 351, and then we checked to see whether \(G_{{\mathcal {D}}}\) was regular on the design. In one of the designs, we found an example of a Spence difference set in G, and we write the difference set as follows.
It took GAP [10] nearly 5000 random assignments of coset representatives to find a design \(\mathcal {D}\) such that a Sylow 3-subgroup of \(\textrm{Aut}(\mathcal {D})\) had order at least 81.
If \(H = \langle b,c \rangle \), then H has a unique subgroup isomorphic to \(C_3 \times C_3\); namely \(H_0 = \langle b^3, c \rangle \). The set \((\langle b \rangle - \langle b^3 \rangle ) \langle c \rangle \) is in fact the complement to \(H_0\) in H and consists of all eighteen elements of order 9 in H.
When we contract by \(Z(G) = \langle b^3 \rangle \), we get
We note that we are using bar notation to indicate that we have moved to the factor group.
After contracting by Z(G), each of the group ring elements \((\overline{1+b+b^2c^2})\), \((\overline{1+bc+b^2c})\), and \((\overline{1+b^2+bc^2})\) is a (3, 3, 3, 1)-relative difference set in \(\langle \overline{b}, \overline{c} \rangle \) relative to the subgroup \(\langle \overline{c} \rangle \).
Finally, it seems relevant that the elements of order 9, namely \(b, b^2, b^4, b^5, b^7,\) and \(b^8\) are used in rows 2 through 4 in the definition of D, and they pair up in interesting ways.
Example 4.1 could lead to interesting new constructions of Spence difference sets. Much like the Hadamard difference set investigations (and to a lesser extent McFarland, Davis-Jedwab, and Chen), we hope this will lead to many new constructions of Spence difference sets.
5 Future directions
This paper demonstrates that the Spence family will likely have many more examples of difference sets. Most of the effort in finding difference sets over the past few decades has targeted the Hadamard family, and we have made significant progress. We hope the new results in this paper will motivate people to take a look at the Spence family (and possibly the others) to determine whether nonabelian techniques will produce new and deeper understanding.
Some specific directions that we suggest are worth pursuing include the following.
Conjecture 2.6:
This natural analogue to Dillon’s conjecture (proved by Drisko) would be a good start to understanding the nonabelian groups that can support a Spence difference set.
Groups of order 351:
Smith and Webster exhaustively examined the \(d=1\) case. The \(d=2\) case, groups of order \(\frac{3^{2+1}(3^{2+1}-1)}{2} = 351\), is the next largest case. These groups are currently too large for an exhaustive search, but simply determining which of these groups have Spence difference sets would be very interesting.
Nonisomorphic designs:
The original Spence difference sets were constructed by pairing hyperplanes \(H_i\) with coset representatives \(k_i\). To the surprise of the authors, different choices of assignment of coset representatives can lead to nonisomorphic symmetric designs. This is likely due to a transversal that is naturally compatible with outer automorphisms, but it would be interesting to understand precisely when and why this happens. Moreover, it is likely that examining these designs will lead to further examples in nonabelian groups.
Computational issues:
When doing computer searches for Spence difference sets, we have a variety of choices, some of which were mentioned in this paper. If we are to use computer searches for larger values of d, it is imperative that we have smart ways of doing those searches; otherwise, they will be out of reach.
Factor group:
Example 4.1 has a description of a contracted Spence difference set. In working with Hadamard difference sets, various authors have used contracted difference sets to “transfer” from one group to another. It would be very interesting to investigate whether this can also occur in Spence difference sets.
The authors feel as though this paper could potentially open whole new understandings of how difference sets work. We hope others will push these ideas forward.
Declarations
Conflict of interest
The authors declare no conflict of interest.
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