Some additive properties of $\mathbb{Z}_n$ and $k$-index stability of finite groups

Abstract

We study some additive combinatorial properties of the group of least nonnegative residues modulo $n$ that are related to index stability of groups and packing numbers. We prove several theorems that not only confirm a conjecture and resolve some open problems about index stability of such groups, but also provide basic tools for the characterization of finite $k$-index stable groups. As a consequence, we completely characterize all 2-element index-stable subsets of $\mathbb{Z}_n$, obtain an exact closed formula for their density, and determine all $n$ for which every 2-subset is index-unstable. Finally, we present a problem and a research project extending the study to 3-subsets and general $k$-subsets.