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arxiv: 2604.08724 · v1 · submitted 2026-04-09 · 🧮 math.GR

Subindices and subfactors of infinite groups and numbers

Mohammad Hadi Hooshmand This is my paper

Pith reviewed 2026-05-10 16:41 UTC · model grok-4.3

classification 🧮 math.GR
keywords subfactorssubindicesindex stabilityinfinite groupsgroups of numbersRight Subfactor Algorithmprime differences
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The pith

Every infinite group is index-unstable when subfactors and subindices are extended from the finite case.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the theory of subfactors of groups along with the related notions of subindices and index stability to infinite groups, with special attention to groups of numbers. It introduces the Right Subfactor Algorithm to handle computations in these infinite settings and proves that every infinite group is index-unstable. The work determines exact subindices for sequences such as primes and Fibonacci numbers, supplies a general criterion for index stability of subsets in countable groups, and establishes a weak form of a conjecture on differences of primes. These developments connect group properties to topics in additive combinatorics and number theory including packing numbers, covering numbers, and syndetic sets.

Core claim

The paper establishes that every infinite group is index-unstable. By defining and applying the Right Subfactor Algorithm to infinite groups, it shows that subindices fail to stabilize for any such group, provides exact subindices for notable integer sequences, offers a criterion that distinguishes stable and unstable subsets in countable groups, and derives a weak version of a conjecture on prime differences while correcting prior inaccuracies in the finite-group theory.

What carries the argument

The Right Subfactor Algorithm (RSFA), which generates subfactors for subsets of infinite groups in order to compute subindices and test whether index stability holds.

If this is right

  • Subsets of any infinite group fail to achieve index stability.
  • Exact subindices exist and can be calculated for sequences including primes and Fibonacci numbers.
  • A general criterion separates index-stable from index-unstable subsets inside any countable group.
  • A weak form of the prime-difference conjecture holds under the extended definitions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Index instability may force distinct combinatorial techniques for infinite versus finite groups when studying diameters or syndetic sets.
  • The Right Subfactor Algorithm could be run on further concrete infinite groups of numbers to reveal patterns in subindex values.
  • The framework might supply new bounds on packing and covering numbers once applied to specific infinite additive structures.

Load-bearing premise

The definitions of subfactors, subindices, and index stability that were created for finite groups extend directly and without extra constraints to infinite groups and groups of numbers.

What would settle it

Identification of even one infinite group containing a subset whose subindices stabilize as the index grows would refute the claim that every infinite group is index-unstable.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The paper extends the recently introduced theory of subfactors, subindices, and index stability from finite groups to infinite groups, with emphasis on groups of numbers. It introduces the Right Subfactor Algorithm (RSFA) for infinite groups, claims to resolve several open problems including that every infinite group is index-unstable, provides a general criterion for index stability in countable groups, corrects earlier inaccuracies, establishes a weak version of a conjecture on differences of primes, determines subindices for notable integer sequences, and lists related projects, problems, questions, and conjectures.

Significance. If the central results hold, this work would meaningfully broaden the subfactor theory to infinite settings, creating new connections between group theory, additive combinatorics, and number theory through topics like packing numbers, syndetic sets, and sequences such as primes and Fibonacci numbers. The explicit constructions supporting the index-unstability theorem and the general criterion for countable groups are positive aspects. The stress-test concern regarding the extension of definitions does not appear to introduce invalidating constraints, as the argument for index-unstability follows from the general criterion with separate handling for uncountable cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, the assessment of significance, and the recommendation of minor revision. We are pleased that the extension to infinite groups, the index-unstability result, the general criterion for countable groups, and the connections to additive combinatorics and number theory are viewed favorably. No specific major comments appear in the report, so our response focuses on confirming that the central claims are supported by the explicit constructions and the separate handling of uncountable cases.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper extends previously introduced definitions of subfactors, subindices, and index stability to infinite groups by defining the RSFA and supplying a general criterion for index stability/non-stability in countable groups plus explicit constructions for the uncountable case. The central claim that every infinite group is index-unstable is obtained from these definitions and constructions rather than by tautological reduction or by renaming a fitted quantity. Self-citations refer only to the foundational definitions; the new theorems, corrections to prior inaccuracies, and applications to sequences such as primes are independent mathematical arguments that remain externally falsifiable for concrete groups. No step reduces by construction to its inputs, so the derivation chain is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted or verified from the provided text. The framework itself appears to rest on newly introduced definitions whose independence from prior self-citations cannot be assessed.

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Reference graph

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