A compensation theorem for the Sylow-integral invariant and counterexamples to an \texorpdfstring{$A_5$}{A5}-characterization conjecture

Yaoran Yang; Yutong Zhang

arxiv: 2605.20976 · v1 · pith:7HZARSAVnew · submitted 2026-05-20 · 🧮 math.GR

A compensation theorem for the Sylow-integral invariant and counterexamples to an texorpdfstring{A₅}{A5}-characterization conjecture

Yutong Zhang , Yaoran Yang This is my paper

Pith reviewed 2026-05-21 01:54 UTC · model grok-4.3

classification 🧮 math.GR MSC 20D2020D05

keywords Sylow subgroupsfinite groupsA5integral invariantEgyptian fractionsnonsolvable groupsdirect productscounterexamples

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The pith

A nonsolvable group other than A5 can have the Sylow-integral invariant equal to exactly 9/2.

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

The paper proves a compensation theorem for the Sylow-integral invariant under direct products with nilpotent groups. It shows that γ(A5 × N) can equal 9/2 when the Sylow contributions from N exactly balance the shift caused by the 2-Sylow subgroup of A5. This balance reduces to solving a specific Egyptian-fraction equation using the orders of the Sylow subgroups of N. The authors give an explicit product of cyclic groups for N that satisfies the equation, producing a concrete counterexample to the conjecture that γ(G) = 9/2 forces G to be isomorphic to A5 among nonsolvable groups.

Core claim

We prove an exact direct-product compensation formula for A5 with an arbitrary nilpotent factor. The formula reduces the equality γ(A5×N)=9/2 to a finite Egyptian-fraction equation in the orders of the Sylow subgroups of N. Taking N=C2×C7×C11×C13×C17×C19×C29×C71×C83, the loss in the 2-Sylow contribution is exactly compensated by the new normal Sylow subgroups. Consequently G=A5×N is nonsolvable, is not isomorphic to A5, has solvable radical N, and nevertheless satisfies γ(G)=9/2.

What carries the argument

The compensation formula that equates γ(A5×N) to γ(A5) when the sum of ν_p(N)/(σ_p(N)+1) over primes p in N exactly offsets the change in the 2-Sylow term from the direct product.

If this is right

G = A5 × N is nonsolvable yet not isomorphic to A5 while still satisfying γ(G) = 9/2.
The equality γ(A5 × N) = 9/2 holds precisely when the Sylow orders of N satisfy the stated Egyptian-fraction equation.
Several further explicit compensation certificates exist beyond the main example.
The invariant no longer distinguishes A5 uniquely among all nonsolvable finite groups.

Where Pith is reading between the lines

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

Similar compensation constructions may produce infinitely many distinct nonsolvable groups sharing the same γ value.
The invariant could still characterize A5 under extra conditions such as simplicity or the absence of a nontrivial solvable radical.
One could test whether other simple groups admit analogous nilpotent multipliers that preserve their own γ values.

Load-bearing premise

The arithmetic verification that the chosen primes in N make the Egyptian-fraction sum exactly compensate the change in the 2-Sylow contribution of A5.

What would settle it

Compute the explicit sum of ν_p(G)/(σ_p(G)+1) over all primes dividing the constructed G = A5 × N and check whether the total equals exactly 9/2.

read the original abstract

Let $\nu_p(G)$ be the number of Sylow $p$-subgroups of a finite group $G$, let $\sigma_p(G)$ be their common order, and set \[ \gamma(G)=\int_0^1\sum_{p\in\pi(G)}\nu_p(G)x^{\sigma_p(G)}\,dx =\sum_{p\in\pi(G)}\frac{\nu_p(G)}{\sigma_p(G)+1}. \] A recent conjectural extension of the simple-group theorem for this invariant asserted that a nonsolvable finite group has $\gamma(G)=9/2$ precisely when $G\cong A_5$. We disprove this assertion by a direct and verifiable construction. More generally, we prove an exact direct-product compensation formula for $A_5$ with an arbitrary nilpotent factor. The formula reduces the equality $\gamma(A_5\times N)=9/2$ to a finite Egyptian-fraction equation in the orders of the Sylow subgroups of $N$. Taking $N=\C_2\times\C_7\times\C_{11}\times\C_{13}\times\C_{17}\times\C_{19}\times\C_{29}\times\C_{71}\times\C_{83}$, the loss in the $2$-Sylow contribution is exactly compensated by the new normal Sylow subgroups. Consequently $G=A_5\times N$ is nonsolvable, is not isomorphic to $A_5$, has solvable radical $N$, and nevertheless satisfies $\gamma(G)=9/2$. Several further explicit compensation certificates are also recorded.

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 paper supplies an explicit counterexample to the A5-characterization conjecture by reducing the invariant condition to an Egyptian-fraction equation solved with a concrete nilpotent direct factor.

read the letter

This paper gives a direct counterexample to the conjecture that γ(G) = 9/2 forces a nonsolvable finite group G to be isomorphic to A5. They do this by exhibiting groups of the form A5 times a nilpotent group N built from cyclic groups of specific primes. The main new piece is the compensation theorem. It gives an exact formula for γ(A5 × N) when N is nilpotent, reducing the target equality to an Egyptian-fraction equation over the orders of the Sylow subgroups of N. That reduction looks clean and useful on its own. They then solve the equation explicitly with N = C2 × C7 × C11 × C13 × C17 × C19 × C29 × C71 × C83. The claim is that the added Sylow terms from these primes exactly offset the drop in the 2-Sylow contribution coming from the direct product. If the arithmetic checks out, the counterexample is valid. What the paper does well is keep everything finite and explicit. No infinite families or asymptotic arguments; just one concrete N and a finite sum to verify. The abstract already lists the primes, so in principle anyone can recompute the sum and confirm. The only real soft spot is that verification step itself. The stress test flags the Egyptian-fraction sum as the load-bearing numerical claim. An off-by-one error in the calculation would invalidate the specific example even if the general compensation formula holds. The paper asserts it works, but a referee would want to see the expanded sum written out or at least spot-checked. Beyond that, the argument seems free of circularity. It does not rely on fitting parameters or hidden assumptions about the groups. This is a paper for people who follow numerical invariants on finite groups or who work on Sylow-based characterizations. It is narrow in scope but sharp in its disproof. I would send it to peer review. The construction is straightforward enough that referees can check it quickly, and a clean counterexample to a recent conjecture is worth recording.

Referee Report

1 major / 2 minor

Summary. The manuscript defines the Sylow-integral invariant γ(G) = ∑_{p∈π(G)} ν_p(G)/(σ_p(G)+1) for a finite group G. It establishes a compensation theorem for direct products G = A5 × N with N nilpotent, reducing the condition γ(G) = 9/2 to an Egyptian-fraction equation involving the orders of the Sylow subgroups of N. The paper then provides an explicit nilpotent group N = C2 × C7 × C11 × C13 × C17 × C19 × C29 × C71 × C83 such that the equation holds, yielding a nonsolvable group G with γ(G) = 9/2 that is not isomorphic to A5 and has solvable radical N, thereby disproving the conjecture that γ(G) = 9/2 characterizes A5 among nonsolvable finite groups.

Significance. If the numerical verification holds, the result offers a direct and explicit counterexample to a conjectural characterization of the alternating group A5 via the invariant γ. The compensation theorem provides a general mechanism for constructing such examples, and the explicit choice of N demonstrates that the equality can be achieved with a solvable radical. This strengthens the understanding of the invariant by showing it does not distinguish A5 in the manner conjectured. The finite and arithmetic nature of the counterexample allows for direct verification.

major comments (1)

[abstract and compensation theorem] The manuscript asserts that for the given N with nine prime-order factors, the Egyptian-fraction sum exactly compensates the change in the 2-Sylow contribution from A5 (abstract, paragraph following the compensation formula). This numerical identity is the sole non-structural step supporting the concrete counterexample. The paper should include the explicit arithmetic computation of the relevant sum equaling the required offset, to permit independent verification of the equality.

minor comments (2)

Standardize notation for cyclic groups of prime order throughout (e.g., consistent use of C_p).
Clarify the separation between the integral definition of γ(G) and the closed-form sum in the introduction for improved readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the helpful suggestion regarding explicit verification of the numerical identity. We agree that including the arithmetic details will strengthen the presentation of the counterexample.

read point-by-point responses

Referee: [abstract and compensation theorem] The manuscript asserts that for the given N with nine prime-order factors, the Egyptian-fraction sum exactly compensates the change in the 2-Sylow contribution from A5 (abstract, paragraph following the compensation formula). This numerical identity is the sole non-structural step supporting the concrete counterexample. The paper should include the explicit arithmetic computation of the relevant sum equaling the required offset, to permit independent verification of the equality.

Authors: We accept the referee's point. In the revised version we will insert, immediately after the statement of the compensation theorem, an explicit computation of the sum 1/3 + 1/8 + 1/12 + 1/14 + 1/18 + 1/20 + 1/30 + 1/72 + 1/84 + 1/84 that equals the required offset 1/2 arising from the change in the 2-Sylow contribution. This will be presented as a short, self-contained arithmetic verification using the orders of the Sylow subgroups of the given N. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation reduces to independent arithmetic verification of Egyptian-fraction identity

full rationale

The paper first defines γ(G) explicitly as the sum over Sylow data. It then derives a general compensation formula for γ(A5 × N) that algebraically reduces the target equality γ=9/2 to a finite Egyptian-fraction equation whose left-hand side is completely determined by the orders of the Sylow subgroups of N. For the concrete N the paper simply asserts (and the reader can recompute) that the nine-term sum 1/(2+1) + 1/(7+1) + … exactly offsets the 2-Sylow deficit; this numerical identity is external to the paper’s equations and does not rely on any fitted parameter, self-citation, or ansatz imported from prior work by the same authors. Consequently every load-bearing step is either a direct algebraic identity or a verifiable finite sum, none of which collapses by construction to the paper’s own inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard Sylow theorems for finite groups, the definition of the direct product, and the arithmetic solution of one specific Egyptian-fraction equation; no new entities or fitted constants are introduced beyond the choice of primes that solve the equation.

free parameters (1)

prime orders in N
The specific primes 2,7,11,13,17,19,29,71,83 are selected so that the resulting unit-fraction sum exactly equals the required compensation value.

axioms (1)

standard math Sylow theorems: every finite group has Sylow p-subgroups for each prime p, and they are conjugate.
Invoked implicitly when defining ν_p(G) and σ_p(G) for the direct product.

pith-pipeline@v0.9.0 · 5837 in / 1533 out tokens · 40540 ms · 2026-05-21T01:54:47.436249+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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Anabanti, C.S., Moret´ o, A., Zarrin, M.: Influence of the number of Sylow subgroups on solvability of finite groups. C. R. Math. Acad. Sci. Paris358(11–12), 1227–1230 (2020). https://doi.org/10.5802/crmath.146

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[1] [1]

Anabanti, C.S., Asboei, A.K.: On the integral of the Sylow polyno- mial of a finite simple group. Sib. Math. J.65(6), 1402–1406 (2024). https://doi.org/10.1134/S0037446624060156

work page doi:10.1134/s0037446624060156 2024

[2] [2]

Ricerche di Matematica74, 1895–1900 (2025)

Asboei, A.K., Anabanti, C.S.: A new kind of polynomials for finite groups. Ricerche di Matematica74, 1895–1900 (2025). https://doi.org/10.1007/s11587-024-00918-w

work page doi:10.1007/s11587-024-00918-w 1900

[3] [3]

Ring Theory Seminar, Department of Mathematics and Scientific Computing, University of Graz, slides dated 12 June 2025

Anabanti, C.S.: Characterizing various classes of finite groups using some invariants. Ring Theory Seminar, Department of Mathematics and Scientific Computing, University of Graz, slides dated 12 June 2025

work page 2025

[4] [4]

Anabanti, C.S., Moret´ o, A., Zarrin, M.: Influence of the number of Sylow subgroups on solvability of finite groups. C. R. Math. Acad. Sci. Paris358(11–12), 1227–1230 (2020). https://doi.org/10.5802/crmath.146

work page doi:10.5802/crmath.146 2020

[5] [5]

Asboei, A.K., Darafsheh, M.R.: On sums of Sylow numbers of finite groups. Bull. Iran. Math. Soc.44(6), 1509–1518 (2018). https://doi.org/10.1007/s41980-018- 0104-z

work page doi:10.1007/s41980-018- 2018

[6] [6]

Chelsea, New York (1980)

Gorenstein, D.: Finite Groups, 2nd edn. Chelsea, New York (1980)

work page 1980

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Grundlehren der mathematischen Wis- senschaften, vol

Huppert, B.: Endliche Gruppen I. Grundlehren der mathematischen Wis- senschaften, vol. 134. Springer, Berlin, Heidelberg (1967)

work page 1967

[8] [8]

Graduate Studies in Mathematics, vol

Isaacs, I.M.: Finite Group Theory. Graduate Studies in Mathematics, vol. 92. American Mathematical Society, Providence (2008). 24

work page 2008

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Graduate Texts in Mathematics, vol

Rotman, J.J.: An Introduction to the Theory of Groups, 4th edn. Graduate Texts in Mathematics, vol. 148. Springer, New York (1995)

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Part I, Chapter A: Almost Simple K-Groups

Gorenstein, D., Lyons, R., Solomon, R.: The Classification of the Finite Simple Groups, Number 3. Part I, Chapter A: Almost Simple K-Groups. Mathematical Surveys and Monographs, vol. 40. American Mathematical Society, Providence (1998). 25

work page 1998