Subsets of abelian groups closed under addition or subtraction
Artūras Dubickas, Chris Smyth
TL;DR
The paper introduces and classifies one-subsets, i.e., nonempty subsets $S$ of a group where for all $g,h\in S$, exactly one of $g+h$ or $g-h$ lies in $S$, with a complete classification for $S\subseteq\mathbb Z$ and a general structure for abelian groups. It develops a two-part framework: if $0\in S$ the structure reduces to an involution subgroup $I$ or a union $I\cup(g+I)$ with $2g\notin I$, and if $0\notin S$ the set forms two opposite cosets $(a+H)\cup(-a+H)$ modulo an index-3 subgroup $H$ of the generated subgroup. The paper then provides explicit constructions and lists for finite abelian groups, $\mathbb Z^n$, and $\mathbb Z_3$, and discusses nonabelian analogues via index-3 subgroups, along with observations about groups not generated by a one-subset. It also proves that divisible groups cannot be generated by a one-subset, illustrating the limitations of the framework and highlighting connections to root-system structures and subgroup generation properties.
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.