A categorical proof of the nonexistence of (120, 35, 10)-difference sets

TL;DR

This work resolves the long-standing open question of the existence of a nontrivial $(v,k,\lambda)=(120,35,10)$-difference set in nonabelian groups by developing a categorical framework based on relation partitions and equi-distributed functions. The pushout theorem shows that equidistribution is preserved under quotients, enabling an inductive, quotient-chain search that reduces the problem to a linear programming formulation with quadratic constraints. Applying this method to all six order-$120$ groups that survive previous eliminations, the authors demonstrate that no such difference set can exist, thus proving nonexistence in this challenging case. The results showcase how category-theoretic generalizations, notably relation partitions and Schurian schemes, can drive decisive conclusions in combinatorial design problems and related computational searches.

Abstract

A difference set with parameters $(v, k, λ)$ is a subset $D$ of cardinality $k$ in a finite group $G$ of order $v$, such that the number $λ$ of occurrences of $g \in G$ as the ratio $d^{-1}d'$ in distinct pairs $(d, d')\in D\times D$ is independent of $g$. We prove the nonexistence of $(120, 35, 10)$-difference sets, which has been an open problem for 70 years since Bruck introduced the notion of nonabelian difference sets. Our main tools are 1. a generalization of the category of finite groups to that of association schemes (actually, to that of relation partitions), 2. a generalization of difference sets to equi-distributed functions and its preservation by pushouts along quotients, 3. reduction to a linear programming in the nonnegative integer lattice with quadratic constraints.

Theorems & Definitions (29)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Proposition 2.5
• Definition 2.6
• Definition 2.7
• Remark 2.8
• Definition 2.9
• Definition 2.10