Combinatorial transfer: a new method for constructing infinite families of nonabelian difference sets, partial difference sets, and relative difference sets

TL;DR

This paper introduces the combinatorial transfer method, a novel approach for constructing infinite families of nonabelian difference sets and related combinatorial objects, expanding the scope beyond traditional abelian group techniques.

Contribution

The paper presents the first infinite families of various nonabelian difference sets and partial difference sets, demonstrating the effectiveness of the new combinatorial transfer method.

Findings

First infinite families of nonabelian Denniston partial difference sets.

First infinite family of Spence difference sets with non-normal Sylow 3-subgroup.

New infinite families of partial difference sets in nonabelian p-groups.

Abstract

For nearly a century, mathematicians have been developing techniques for constructing abelian automorphism groups of combinatorial objects, and, conversely, constructing combinatorial objects from abelian groups. While abelian groups are a natural place to start, recent computational evidence strongly indicates that the vast majority of transitive automorphism groups of combinatorial objects are nonabelian. This observation is the guiding motivation for this paper. We propose a new method for constructing nonabelian automorphism groups of combinatorial objects, which could be called the \textit{combinatorial transfer method}, and we demonstrate its power by finding (1) the first infinite families of nonabelian Denniston partial difference sets (including nonabelian Denniston PDSs of odd order), (2) the first infinite family of Spence difference sets in groups with a Sylow 3-subgroup that is non-normal and not elementary abelian, (3) the first infinite families of McFarland difference sets in groups with a Sylow $p$-subgroup that is non-normal and is not elementary abelian, (4) new infinite families of partial difference sets in nonabelian $p$-groups with large exponent, (5) an infinite family of semiregular relative difference sets whose forbidden subgroup is nonabelian, and (6) a converse to Dillon's Dihedral Trick in the PDS setting. We hope this paper will lead to more techniques to explore this largely unexplored topic.