Subindices and subfactors of ${\mathbb{Z}}_n$ and $k$-index stability of finite groups

Abstract

We study subindices, subfactors, and index stability in the cyclic group ${\mathbb{Z}}_n$. 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 some problems and a research project extending the study to 3-subsets and general $k$-subsets.