Subindices and subfactors of infinite groups and numbers

Abstract

The theory of subfactors of groups, together with the associated notions of subindices and index stability for groupsandtheirsubsets, hasrecentlybeenintroducedandsystematicallydeveloped. Theseconceptsexhibitdeepconnections with additive combinatorics and number theory, relating to important topics such as packing and covering numbers, syndetic sets, group diameters, special integer sequences (e.g., primes and Fibonacci numbers), and classical rational sequences (e.g., Bernoulli numbers). Following the initial paper presented in 2020, two subsequent works further investigated these ideas within the framework of finite groups. In the present paper, in addition to advancing several aspects of the topic, we focus on infinite groups, with particular emphasis on groups of numbers. In this context, we introduce the RSFA (Right Subfactor Algorithm) for infinite groups and resolve several previously open problems. One of the important results is that every infinite group is index-unstable. We also correct several earlier inaccuracies and establish a weak version of a conjecture concerning differences of prime numbers. Furthermore, we determine the exact subindices for several notable sequences of integers and provide a general criterion for index stability and non-index stability of subsets in countable groups. Finally, we investigate the index stability of infinite groups and present a collection of related projects, problems, questions, and conjectures.