Subsets of finite groups exhibiting additive regularity

TL;DR

This paper develops a foundational theory of sum sets and partial sum sets in finite groups, establishing their properties, non-existence conditions, and construction methods, especially in groups with specific subgroup structures.

Contribution

It introduces the first principles theory of sum sets, characterizes their regularity, and provides new construction techniques using properties of dihedral and Frobenius groups.

Findings

Sum sets must exhibit higher-order regularity.

Abelian sum sets are necessarily reversible difference sets.

Several infinite classes of sum sets are constructed using dihedral and Frobenius groups.

Abstract

In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In particular, we show that any sum set must exhibit higher-order regularity and that an abelian sum set is necessarily a reversible difference set. We next develop several general construction techniques under the hypothesis that the over-riding group contains a normal subgroup of order 2. Finally, by exploiting properties of dihedral groups and Frobenius groups, several infinite classes of sum sets and partial sum sets are introduced.