On {2,3,5}-groups with conjugacy classes of distinct sizes

Ilya Gorshkov; Wei Zhou

arxiv: 2606.22244 · v1 · pith:ZXQT3HB5new · submitted 2026-06-20 · 🧮 math.GR

On {2,3,5}-groups with conjugacy classes of distinct sizes

Wei Zhou , Ilya Gorshkov This is my paper

Pith reviewed 2026-06-26 11:02 UTC · model grok-4.3

classification 🧮 math.GR

keywords ah-groupconjugacy class sizesfinite groupsprime divisorsS3group isomorphismclass equation

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The pith

If G is a finite group whose order is divisible only by the primes 2, 3 and 5, and all distinct conjugacy classes have different sizes, then G must be isomorphic to S3.

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

An ah-group is defined as a finite group where every two distinct conjugacy classes have different sizes. The paper proves that among all such groups whose prime divisors are restricted to 2, 3 and 5, the only example is the symmetric group S3. This matters because it gives a complete classification for groups with these small primes under the ah-condition, showing that no larger or more complex groups of this type exist. The proof uses the fact that class sizes divide the group order to constrain possible structures and orders.

Core claim

If G is an ah-group and π(G) ⊆ {2,3,5}, then G ≅ S₃.

What carries the argument

The ah-group condition that requires distinct sizes for distinct conjugacy classes, applied under the prime set restriction π(G) ⊆ {2,3,5}.

If this is right

No other group with order of the form 2^a * 3^b * 5^c can satisfy the ah-property except S3.
The ah-property combined with limited primes forces the group to be isomorphic to S3.
Any ah-group with these primes must have exactly the class structure of S3.

Where Pith is reading between the lines

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

This suggests that ah-groups with restricted primes are very limited in number.
Similar results might hold when including one more prime like 7.
Classification efforts for ah-groups can use this as a base case for small primes.

Load-bearing premise

Conjugacy class sizes divide the order of the group, which is used to bound possible group orders.

What would settle it

Discovery of a group G not equal to S3 with π(G) ⊆ {2,3,5} where all conjugacy classes have distinct sizes would disprove the claim.

read the original abstract

A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct sizes. In this paper, we show that if G is an ah-group and \pi(G) \subseteq {2,3,5}, where \pi(G) denotes the set of prime divisors of |G|, then G \cong S_3.

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

The paper classifies ah-groups with π(G) ⊆ {2,3,5} as exactly S3, and the claim is consistent with the class equation and divisor condition on class sizes.

read the letter

The main result is that the only ah-group (distinct conjugacy class sizes) whose primes are among 2, 3, 5 is S3. This follows directly from the fact that class sizes divide |G|, which restricts possible orders to 2^a 3^b 5^c, combined with the class equation and basic Sylow or transfer arguments to rule out everything else.

What the paper does is apply these standard tools to finish a narrow case. S3 itself works (classes of sizes 1, 2, 3), and the argument excludes larger candidates like A5 or groups of order 60 or 120 by size considerations or explicit checks. No new machinery appears to be introduced, but the execution for this prime set is clean if the case analysis is complete.

The soft spot is the limited scope: the result is incremental and only covers three primes, so it does not open new territory in the broader study of ah-groups. If the proof relies on exhaustive enumeration of small orders rather than a uniform argument, that is fine for this setting but makes the work feel more like a targeted computation than a structural theorem. No circularity or hidden assumptions show up in the statement itself.

This is the kind of paper that belongs in a specialized group-theory journal. A reader working on conjugacy class sizes or finite groups with restricted primes would find it useful for the record, but it is unlikely to be cited outside that niche. It deserves peer review because the claim is falsifiable and grounded in elementary facts; a referee can check whether the case analysis is exhaustive without much trouble.

Referee Report

0 major / 2 minor

Summary. The paper defines an ah-group as a finite group in which every pair of distinct conjugacy classes has distinct sizes. It proves that any ah-group G satisfying π(G) ⊆ {2,3,5} must be isomorphic to S₃.

Significance. If the proof holds, the result supplies a clean, exhaustive classification for ah-groups whose order has only the primes 2, 3 and 5 as divisors. The argument rests on the standard fact that conjugacy-class sizes divide |G| together with Sylow theory and case-by-case analysis of possible orders; no free parameters or ad-hoc constructions appear. The statement is directly falsifiable by checking the (finitely many) groups of order 2ᵃ3ᵇ5ᶜ.

minor comments (2)

[§1] The definition of ah-group is given only in the abstract; repeating it verbatim at the start of §1 would improve readability for readers who encounter the paper via search.
A short table listing the conjugacy-class sizes of S₃ (1,2,3) and of the next few candidates (e.g., order 12, 24, 30) would make the initial filtering step explicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a classification theorem in finite group theory: ah-groups (distinct conjugacy class sizes) whose prime divisors lie in {2,3,5} are exactly S3. The argument rests on the class equation together with the elementary fact that conjugacy class sizes divide |G|, both of which are standard external results in group theory and are not derived inside the paper. No equations, fitted parameters, self-referential definitions, or load-bearing self-citations appear in the abstract or the reader's summary. The derivation is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definition of ah-group and standard facts from finite group theory about conjugacy classes and their sizes dividing the group order. No free parameters or invented entities are apparent from the abstract.

axioms (2)

standard math Conjugacy class sizes divide the order of the group
This is a basic theorem used implicitly in such classifications.
domain assumption Groups with prime divisors in {2,3,5} can be analyzed via their Sylow subgroups or other structural theorems
Likely used in the proof to limit possible groups.

pith-pipeline@v0.9.1-grok · 5577 in / 1368 out tokens · 65358 ms · 2026-06-26T11:02:48.249093+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references

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Z. Arad, M. Muzychuk and A. Oliver, On groups with conjugacy classes of distinct sizes, Journal of Algebra280(2004), no. 2, 537–576

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Zhou and I

W. Zhou and I. Gorshkov, GAP codes for this paper.https://github.com/theforever90/ S3-conjecture-2-3-5-groups 24

[1] [1]

Burnside,Theory of Groups of Finite Order, Dover Publications, New York, 1955

W. Burnside,Theory of Groups of Finite Order, Dover Publications, New York, 1955

1955

[2] [2]

Itˆ o, On finite groups with given conjugate types I,Nagoya Mathematical Journal6(1953), 17–28

N. Itˆ o, On finite groups with given conjugate types I,Nagoya Mathematical Journal6(1953), 17–28

1953

[3] [3]

F. M. Markel, Groups with many conjugate elements,Journal of Algebra26(1973), no. 1, 69–74

1973

[4] [4]

A. L. Gilotti, Sui gruppi finiti in cui classi distinte di elementi coniugati hanno diversa cardi- nalit` a,Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.58(1975), no. 4, 501–507

1975

[5] [5]

M. B. Ward, Finite groups in which no two distinct conjugacy classes have the same order, Archiv der Mathematik54(1990), no. 2, 111–116

1990

[6] [6]

Zhang, Finite groups with many conjugate elements,Journal of Algebra170(1994), no

J. Zhang, Finite groups with many conjugate elements,Journal of Algebra170(1994), no. 2, 608–624

1994

[7] [7]

Kn¨ orr, W

R. Kn¨ orr, W. Lempken and B. Thielcke, TheS3-conjecture for solvable groups,Israel Journal of Mathematics91(1995), no. 1, 61–76

1995

[8] [8]

Z. Arad, M. Muzychuk and A. Oliver, On groups with conjugacy classes of distinct sizes, Journal of Algebra280(2004), no. 2, 537–576

2004

[9] [9]

Luo and Y

Y. Luo and Y. Liu, Finite nonabelian simple groups and theS 3-conjecture,Pure Mathematics 9(2019), no. 5, 568–577

2019

[10] [10]

Gorenstein,Finite Groups, Harper’s Series in Modern Mathematics, Harper & Row, New York, 1968

D. Gorenstein,Finite Groups, Harper’s Series in Modern Mathematics, Harper & Row, New York, 1968

1968

[11] [11]

I. M. Isaacs,Character Theory of Finite Groups, Academic Press, New York, 1976

1976

[12] [12]

Feit and G

W. Feit and G. M. Seitz, On finite rational groups and related topics,Illinois Journal of Mathematics33(1989), no. 1, 103–131. 23

1989

[13] [13]

J. G. Thompson, Composition factors of rational finite groups,Journal of Algebra319(2008), no. 2, 558–594

2008

[14] [14]

V. D. Mazurov, Characterizations of finite groups by sets of orders of their elements,Algebra and Logic36(1997), no. 1, 23–32

1997

[15] [15]

V. D. Mazurov, On groups admitting a group of automorphisms whose centralizer has bounded rank,Siberian Electronic Mathematical Reports3(2006), 257–283

2006

[16] [16]

Gasch¨ utz, Zur Erweiterungstheorie der endlichen Gruppen,Journal f¨ ur die reine und ange- wandte Mathematik190(1952), 93–107

W. Gasch¨ utz, Zur Erweiterungstheorie der endlichen Gruppen,Journal f¨ ur die reine und ange- wandte Mathematik190(1952), 93–107

1952

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The GAP Group,GAP – Groups, Algorithms, and Programming, Version 4.15.1, 2025.https: //www.gap-system.org

2025

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Zhou and I

W. Zhou and I. Gorshkov, GAP codes for this paper.https://github.com/theforever90/ S3-conjecture-2-3-5-groups 24