Subsets of abelian groups closed under addition or subtraction

TL;DR

This paper characterizes specific subsets of integers and abelian groups that are closed under addition or subtraction, identifying all such small and infinite sets, and explores related problems in nonabelian groups.

Contribution

It provides a complete classification of certain subsets of integers and abelian groups with closure properties, extending the understanding to nonabelian groups.

Findings

Sets with at most two elements, including zero, satisfy the property.

Infinite sets of the form {rk} with r positive and k not divisible by 3 are characterized.

Partial results are discussed for nonabelian groups.

Abstract

In this article, we first describe all nonempty sets of integers S with the property that for all n and m in S, not necessarily distinct, the set {n-m,n+m} intersected with S consists of a single element. These are the sets with at most two elements, one of which is 0, and the infinite sets {rk}, where r is a fixed positive integer and k runs over all integers not divisible by 3. In the later sections, we solve the analogous problem for subsets of abelian groups. We also discuss, but do not completely solve, the analogous problem for nonabelian groups.