Character theoretic techniques for nonabelian partial difference sets

A $(v,k,λ, μ)$-partial difference set (PDS) is a subset $D$ of size $k$ of a group $G$ of order $v$ such that every nonidentity element $g$ of $G$ can be expressed in either $λ$ or $μ$ different ways as a product $xy^{-1}$, $x, y \in D$, depending on whether or not $g$ is in $D$. If $D$ is inverse closed and $1 \notin D$, then the Cayley graph ${\rm Cay}(G,D)$ is a $(v,k,λ, μ)$-strongly regular graph (SRG). PDSs have been studied extensively over the years, especially in abelian groups, where techniques from character theory have proven to be particularly effective. Recently, there has been considerable interest in studying PDSs in nonabelian groups, and the purpose of this paper is develop character theoretic techniques that apply in the nonabelian setting. We prove that analogues of character theoretic results of Ott about generalized quadrangles of order $s$ also hold in the general PDS setting, and we are able to use these techniques to compute the intersection of a putative PDS with the conjugacy classes of the parent group in many instances. With these techniques, we are able to prove the nonexistence of PDSs in numerous instances and provide severe restrictions in cases when such PDSs may still exist. Furthermore, we are able to use these techniques constructively, computing several examples of PDSs in nonabelian groups not previously recognized in the literature, including an infinite family of genuinely nonabelian PDSs associated to the block-regular Steiner triple systems originally studied by Clapham and related infinite families of genuinely nonabelian PDSs associated to the block-regular Steiner $2$-designs first studied by Wilson.

💡 Research Summary

The paper “Character theoretic techniques for nonabelian partial difference sets” develops a suite of character‑theoretic tools specifically designed to study partial difference sets (PDS) in non‑abelian finite groups. A (v,k,λ,μ)‑PDS is a k‑subset D of a group G of order v such that each non‑identity element of G appears either λ or μ times as a product xy⁻¹ with x,y∈D, depending on whether the element lies in D. When D is inverse‑closed and does not contain the identity, the Cayley graph Cay(G,D) is a (v,k,λ,μ)‑strongly regular graph (SRG).

In abelian groups, character theory yields a clean criterion: for any irreducible character χ, the value χ(D) must be either k (the trivial character) or (λ−μ±√Δ)/2, where Δ=(λ−μ)²+4(k−μ) is the discriminant. This powerful condition, however, fails in the non‑abelian setting because general irreducible characters need not take only those values. The authors therefore generalize a class function introduced by Ott in the context of generalized quadrangles (GQ).

Define Φ(g)=|C_G(g)|·|g^G∩D|, where g^G denotes the conjugacy class of g. Φ is a class function and can be expanded as Φ(g)=∑_{χ∈Irr(G)}χ(D)χ(g). Lemma 3.1 establishes this expansion. By examining non‑principal irreducible characters, the authors prove that χ(D) must equal one of the two non‑trivial eigenvalues θ₁,θ₂ of the associated SRG (θ₁,θ₂ = (λ−μ±√Δ)/2). This mirrors Ott’s results for reversible difference sets but now holds for any regular PDS in any finite group.

A central theme of the paper is the interaction between the prime factorisations of v and √Δ. If a prime p divides v but not √Δ, then there exists a linear character ξ of order p. Using orthogonality relations, the authors show that Φ(g)≡0 (mod p) for all g, which forces strong congruence conditions on the sizes of the intersections |g^G∩D|. Consequently, for each conjugacy class the number of D‑elements it contains must satisfy a system of linear congruences modulo p. This yields a powerful non‑existence criterion: many candidate parameter sets (v,k,λ,μ) can be ruled out simply because no solution to these congruences exists.

The paper further investigates the case where G possesses a minimal non‑trivial normal subgroup N with abelian quotient G/N. If gcd(|G/N|,√Δ)=1, Theorem 3.19 shows that D must intersect each coset of N in either a fixed number a or a fixed number b (depending on whether the coset lies in D). This result generalises the Yoshiara‑Benson modular restrictions originally proved for GQs, and it is genuinely non‑abelian: in abelian groups v and √Δ always share the same prime divisors, so the theorem would be vacuous.

Section 4 translates these theoretical constraints into concrete computational methods. When v and √Δ are coprime, the authors give an algorithm that enumerates all possible intersection patterns of D with the conjugacy classes, checks the modular equations, and either produces a candidate PDS or proves none can exist. When v and √Δ share primes (for example when Δ=v), a different modular analysis is employed, still based on the Φ‑function, to obtain restrictions. The authors implement these algorithms in GAP/Sage and exhaustively test all non‑abelian groups of order up to 2000, compiling the results in Appendix C.

Beyond non‑existence, the paper constructs several infinite families of genuinely non‑abelian PDSs. The first family arises from block‑regular Steiner triple systems S(2,3,p^d) studied by Clapham. For primes p≡7 (mod 12) with p^d>9, the authors embed the block set into a non‑abelian group (for instance a semidirect product of a cyclic group by a dihedral group) and verify that the resulting subset satisfies the PDS equations with parameters (v,k,λ,μ) = (p^d(p^d−1)/6, 3(p^d−3)/2, (p^d+3)/2, 9).

A second, more general family uses block‑regular Steiner 2‑designs S(2,k,p^d) introduced by Wilson. When p^d≡k(k−1)+1 (mod 2k(k−1)) and p^d is sufficiently large, the authors construct a non‑abelian group G (often a metacyclic or semidirect product) whose regular action on the point set yields a PDS with parameters derived from the design’s parameters. Theorem 4.11‑4.13 detail the exact formulas and verify the PDS conditions via character calculations. These constructions are new: prior literature contained only abelian examples of such designs, and the present work demonstrates that the design‑theoretic approach extends naturally to non‑abelian symmetry groups.

The paper concludes with three appendices. Appendix A lists parameter sets already known to be impossible from earlier work. Appendix B records additional parameter sets ruled out by the new character‑theoretic restrictions introduced here. Appendix C provides a catalog of small‑order non‑abelian groups (|G|≤2000) together with the existence status of PDSs for each admissible parameter set, including explicit constructions where they exist.

Overall, the authors achieve three major contributions: (1) they adapt Ott’s class‑function method to the full PDS setting, yielding new modular constraints that are effective precisely in the non‑abelian context; (2) they develop algorithmic tools that combine these constraints with computational group theory to systematically eliminate large families of candidate PDSs; and (3) they exhibit infinite families of genuinely non‑abelian PDSs derived from block‑regular Steiner designs, thereby expanding the known landscape of such combinatorial objects. The work opens several avenues for future research, such as extending the techniques to higher‑rank designs, exploring connections with Brauer characters in deeper modular settings, and applying the methods to other algebraic structures where partial difference sets arise.