Simple groups with narrow prime spectrum: Extended list

arxiv: 2605.16450 · v1 · pith:27TEJSGPnew · submitted 2026-05-15 · 🧮 math.GR

Simple groups with narrow prime spectrum: Extended list

Andrei V. Zavarnitsine This is my paper

Pith reviewed 2026-05-19 21:52 UTC · model grok-4.3

classification 🧮 math.GR

keywords finite simple groupsprime spectrumgroup ordersclassification of finite simple groupscomputational group theorynarrow spectrum

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The pith

All non-abelian finite simple groups whose order has largest prime divisor at most 10000 have been determined.

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

The paper extends an earlier computation to identify every non-abelian finite simple group in which the biggest prime dividing the group order stays no larger than 10 to the fourth power. It relies on the known list of all such groups furnished by the classification theorem and checks their orders for the prime-divisor condition. A reader interested in the structure of finite groups would care because the result produces an explicit finite collection that can be examined for further properties such as generation, representations, or embeddings. The authors also release the code used for the check, allowing the same method to be rerun for changed bounds or updated tables.

Core claim

We determine all non-abelian finite simple groups whose order has largest prime divisor not exceeding 10^4. This generalises a previous result by raising the bound on the largest prime and is carried out by systematic inspection of the orders of all groups supplied by the classification of finite simple groups.

What carries the argument

Systematic computational check of the prime factors of |G| for every non-abelian finite simple group G supplied by the classification theorem.

If this is right

The groups satisfying the condition form a finite explicit list whose individual properties can now be studied in full.
The same verification procedure applies directly to any larger prime bound once sufficient computational resources are available.
Any future simple group whose order introduces a prime larger than 10000 automatically falls outside the narrow-spectrum class.
The released code supplies a reusable tool for repeating the spectrum check after any revision of the order tables.

Where Pith is reading between the lines

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

The method scales with improvements in group-order databases and could be rerun periodically to incorporate newly confirmed simple groups.
The resulting list offers a concrete test bed for questions about the distribution of primes in the orders of simple groups.
Similar spectrum restrictions might be examined for other algebraic objects once their classification or enumeration becomes available.

Load-bearing premise

The classification of finite simple groups is complete and the orders of all known simple groups are correctly tabulated in the literature or databases used by the program.

What would settle it

The appearance of a non-abelian finite simple group not on the current list whose order has no prime factor larger than 10000, or the discovery that the tabulated order of a known simple group has an incorrect largest prime factor.

read the original abstract

Generalising a previous result, we determine all non-abelian finite simple groups whose order has largest prime divisor not exceeding $10^4$. The computer code for this and similar calculations is made available.

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.

Desk Editor's Note private letter to a colleague

This is a direct computational update to the author's prior list of non-abelian simple groups whose orders have largest prime factor at most 10^4, with code supplied for the check.

read the letter

The main point is that Zavarnitsine has extended his own earlier bound and produced an explicit list of all non-abelian finite simple groups whose order is divisible only by primes up to 10^4. He also releases the code used for the enumeration. That is the concrete output here. The work checks the known simple groups against their order formulas, factors out the primes, and keeps only those satisfying the bound. The result is a longer but still finite list that generalizes the previous version. Code availability is the clearest strength; anyone can rerun the prime-factor checks and parameter bounds without starting from scratch. The approach stays within the standard toolkit of finite group theory: rely on the classification and tabulated orders, then filter. No new theoretical machinery appears, but the execution is transparent for what it sets out to do. The obvious limitation is dependence on the completeness of the classification of finite simple groups and the accuracy of the order tables in the literature or databases. The paper does not re-prove either; it applies them. That is standard for this style of result, but it means the claim inherits whatever gaps or errors exist in those external sources. The increase in bound simply means more cases to verify, yet the method itself does not change. This paper is for people already working inside finite group theory on questions that involve groups with restricted prime factors, such as certain classification problems or bounds on subgroup structure. A reader who needs an updated concrete list for such arguments will find it directly useful. Outside that niche it adds little. I would send it to peer review. The claim is modest and well-scoped, the computation is reproducible by design, and these kinds of explicit lists do get referenced in the subfield even when they are incremental.

Referee Report

0 major / 2 minor

Summary. The manuscript generalizes a prior result by determining all non-abelian finite simple groups whose order has largest prime divisor at most 10^4. The determination proceeds by exhaustive enumeration against the complete list of known simple groups furnished by the classification of finite simple groups together with their explicit order formulas; the authors supply the computer code used for the prime-factor checks and case-by-case parameter bounds.

Significance. If the enumeration is accurate, the paper supplies an extended, reproducible catalog of simple groups with narrow prime spectra. Such lists support work on the prime graph, spectrum recognition, and order-restricted properties of finite simple groups. The explicit release of the code is a clear strength, directly addressing reproducibility of the computational checks.

minor comments (2)

The main theorem or result statement should include a concise summary or categorization of the groups that satisfy the bound (e.g., which sporadic groups appear and the ranges of parameters for groups of Lie type), rather than leaving the reader to consult only the code output.
A brief comparison table or paragraph contrasting the new bound (10^4) with the previous result being generalized would help readers assess the extension achieved.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of the manuscript, the recognition of its potential utility for work on prime graphs and spectrum recognition, and the recommendation of minor revision. We are pleased that the explicit release of the computer code is highlighted as a strength for reproducibility.

Circularity Check

0 steps flagged

Minor self-citation to prior enumeration; central computation independent of fitted inputs or self-definition

full rationale

The paper performs an exhaustive computational enumeration of non-abelian finite simple groups whose orders have largest prime factor ≤ 10^4, generalizing a previous result while relying on the external Classification of Finite Simple Groups together with tabulated order formulas and explicit parameter bounds from the literature. The author supplies the code for the prime-factor checks and case analysis, making the enumeration reproducible against external benchmarks without any quantity defined in terms of a fitted parameter from the present work or any reduction of the central claim to a self-citation chain. This is a standard honest use of prior results and does not meet the criteria for load-bearing circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on the completeness of the classification of finite simple groups and on the accuracy of tabulated group orders; no new free parameters or invented entities are introduced.

axioms (1)

domain assumption The classification of finite simple groups is complete and all orders are correctly known.
Invoked implicitly when the program enumerates over the known simple groups.

pith-pipeline@v0.9.0 · 5537 in / 1073 out tokens · 24351 ms · 2026-05-19T21:52:51.046430+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1. There are 15072 isomorphism types of finite non-abelian simple groups whose order has all prime divisors less than 10000.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wil- son, Atlas of finite groups: maximal subgroups and ordinary characters for simple groups. Oxford. Clarendon Press (1985), xxxiii + 252 pp

work page 1985
[2]

URL: http://www.gap-system.org

The GAP Group, GAP — Groups, Algorithms, and Program- ming, Version 4.15.1 (2025). URL: http://www.gap-system.org

work page 2025
[3]

V. D. Mazurov, On the set of orders of elements of a finite group, Algebra and Logic,33, N 1 (1994), 49–55

work page 1994
[4]

A. V. Zavarnitsine, Finite simple groups with narrow prime spectrum,Sib. Elect. Math. Reports,6(2009), 1–12. URL: http://semr.math.nsc.ru/v6/p1-12.pdf

work page 2009
[5]

A. V. Zavarnitsine, GAP code accompanying this paper (2026). URL:https://github.com/zavandr/pi-simple The tables T able 1: Primes p∈ { 1000, . . . ,10000} with generic Sp 1009, 1013, 1019, 1033, 1039, 1097, 1103, 1151, 1163, 1187, 1193, 1213, 1217, 1249, 1259, 1279, 1307, 1319, 1361, 1381, 1409, 1439, 1453, 1481, 1523, 1559, 1579, 1627, 1667, 1669, 1721, ...

work page 2026

[1] [1]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wil- son, Atlas of finite groups: maximal subgroups and ordinary characters for simple groups. Oxford. Clarendon Press (1985), xxxiii + 252 pp

work page 1985

[2] [2]

URL: http://www.gap-system.org

The GAP Group, GAP — Groups, Algorithms, and Program- ming, Version 4.15.1 (2025). URL: http://www.gap-system.org

work page 2025

[3] [3]

V. D. Mazurov, On the set of orders of elements of a finite group, Algebra and Logic,33, N 1 (1994), 49–55

work page 1994

[4] [4]

A. V. Zavarnitsine, Finite simple groups with narrow prime spectrum,Sib. Elect. Math. Reports,6(2009), 1–12. URL: http://semr.math.nsc.ru/v6/p1-12.pdf

work page 2009

[5] [5]

A. V. Zavarnitsine, GAP code accompanying this paper (2026). URL:https://github.com/zavandr/pi-simple The tables T able 1: Primes p∈ { 1000, . . . ,10000} with generic Sp 1009, 1013, 1019, 1033, 1039, 1097, 1103, 1151, 1163, 1187, 1193, 1213, 1217, 1249, 1259, 1279, 1307, 1319, 1361, 1381, 1409, 1439, 1453, 1481, 1523, 1559, 1579, 1627, 1667, 1669, 1721, ...

work page 2026