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The Order of the Monster Finite Simple Group

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Counting axes in the Griess algebra determines the order of the Monster group from first principles; it also yields the order of the Baby Monster and the full automorphism group of the Moonshine module.

desk verdict The Monster's order is finally computed without assuming the Baby Monster, via heavy but well-documented computation; the main caveat is an explicitly unproven axis-reduction routine that the certificate does not fully cover. read the letter →

arxiv 2508.01037 v1 pith:FUSQ766G submitted 2025-08-01 math.GR hep-thmath.QAmath.RT

Gerald Höhn , Martin Seysen This is my paper · ORCID
classification math.GRhep-thmath.QAmath.RT MSC 20D0817B69
keywords MonstergroupGriessalgebraaxesLeechlatticeBabyMoonshinemodulevertexoperatorcomputationaltheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the order of the Monster, the largest sporadic simple group, can be determined from first principles by counting special idempotents, called axes, inside the 196884-dimensional Griess algebra, using only the Leech lattice and its Conway group symmetry together with an additional triality automorphism. The authors show that the Monster has exactly twelve orbits on these axes under the subgroup $G_{x_0} \cong 2^{1+24}_+.Co_1$, totalling $97{,}239{,}461{,}142{,}009{,}186{,}000$ axes, and that a complementary count of feasible axes equals $11{,}707{,}448{,}673{,}375$; multiplying by the stabilizer order $|2^{1+23}.Co_2|$ yields the classical order $808{,}017{,}424{,}794{,}512{,}875{,}886{,}459{,}904{,}961{,}710{,}757{,}005{,}754{,}368{,}000{,}000{,}000$. If correct, this computation requires no prior construction or character table of the Monster or of other sporadic groups, and it also yields the order of the Baby Monster, the simplicity of the Monster, and the statement that the Monster is the full automorphism group of both the Griess algebra and the Moonshine module.

What carries the argument

The central objects are axes: idempotent vectors $v$ in the Griess algebra with $v * v = 16v$ and $(v, v) = 8$, each in bijection with a 2A involution of the Monster. The enumeration decomposes the set of axes into twelve $G_{x_0}$-orbits, refines them into 251 orbits under $N_{x_0}$ and 123 orbits under $N_0$, and builds a column-stochastic triality transition matrix whose unique Perron–Frobenius eigenvector gives the orbit sizes once the single orbit size $196560$ is known; an analogous transition matrix for the stabilizer $M_{v_+}$ acting on feasible axes gives the second count. The implementation is carried by an axis-reduction routine in the paper's computational package, and the heavy transition-matrix data are distilled into a small certificate that a reader can verify directly.

What would settle it

Run the supplied certificate and independently re-enumerate the $N_{x_0}$-orbits: if any axis cannot be reduced to one of the twelve listed $G_{x_0}$-orbit representatives, or if any column of the triality transition matrix sums to a number other than $16{,}584{,}750$, then the claimed order is not established.

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Extended reading notes

Core claim

The central claim is Theorem 5.1: $|M| = 2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71 = 808{,}017{,}424{,}794{,}512{,}875{,}886{,}459{,}904{,}961{,}710{,}757{,}005{,}754{,}368{,}000{,}000{,}000$. The proof counts $|X_+| = 97{,}239{,}461{,}142{,}009{,}186{,}000$ axes in the Griess algebra, counts the feasible axes $|X_-| = 11{,}707{,}448{,}673{,}375$, and observes that the common stabilizer of the two axes $v_+$ and $v_-$ has structure $2^{1+23}.Co_2$, so $|M| = |X_+| \cdot |X_-| \cdot |2^{1+23}.Co_2|$. From the same data the paper derives the order of the Baby Monster, and with bounds from the literature on the order of $\mathrm{Aut}(V^\natural)$ together with the Monstrous Moonshine proof, it concludes $M = \mathrm{Aut}(B) = \mathrm{Aut}(V^\natural)$ and that the Monster has exactly two conjugacy classes of involutions.

Load-bearing premise

The axis-reduction routine, which assigns every axis encountered to one of the twelve $G_{x_0}$-orbits, is not proven to work on all possible axes; the paper states that it suffices that the routine succeeded on all axes occurring in the practical computations.

Editorial extensions

If this is right

  • The order of the Monster is obtained without assuming the existence or order of any other sporadic simple group; the only imported group orders are those of $Co_1$, $Co_2$, and $M_{24}$.
  • The order of the Baby Monster, $2^{41} \cdot 3^{13} \cdot 5^6 \cdot 7^2 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 31 \cdot 41 \cdot 47$, follows from the same two axis counts.
  • The Monster is the full automorphism group of the Griess algebra and of the Moonshine module: $M = \mathrm{Aut}(B) = \mathrm{Aut}(V^\natural)$.
  • The Monster has exactly two conjugacy classes of involutions, with centralizers a double cover of the Baby Monster and $2^{1+24}_+.Co_1$, respectively.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the axis-reduction routine can be turned into a proved algorithm for all axes, the order computation becomes a fully human-checkable proof; the certificate makes the output auditable, but not the reduction routine itself.
  • The same template — a visible subgroup plus a triality automorphism, then counting idempotent axes through a Perron–Frobenius transition matrix — could be applied to other sporadic groups whose definitions come from an algebra rather than from a character table.
  • A reader could test the method's portability by re-running the certificate with an independent implementation of the linear algebra only, isolating any residual dependency on the unproved reduction routine.
  • Because the argument uses only Leech lattice and Conway group input, it makes the Monster's order a direct reflection of Leech lattice geometry, a connection previously visible mainly through the character table.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents a computation of the order of the Monster group M as a subgroup of the automorphism group of the 196884-dimensional Griess algebra, generated by a group G_{x0} of type 2^{1+24}_+.Co_1 and a triality automorphism τ. The strategy is to count the M-orbit X_+ of an axis v_+ and the M_{v_+}-orbit X_- of an orthogonal axis v_-; the stabilizer of the pair is identified with 2^{1+23}.Co_2, so |M| = |X_+| · |X_-| · |2^{1+23}.Co_2|. The counts are obtained by decomposing axes into twelve G_{x0}-orbits, computing a column-stochastic transition matrix for the action of τ, and applying Perron–Frobenius plus a known base orbit size. An analogous procedure for feasible axes yields ten H-orbits. The paper also derives the order of the Baby Monster, a new proof that Aut(B) = Aut(V^♮) = M, and the statement that M has exactly two involution classes. Computations are performed with the mmgroup package, and certificates are provided in Appendix D for the transition Tables 2 and 4.

Significance. If the computational completeness premises are accepted, this is a landmark result: it gives the first computation of |M| that does not assume the existence of the Monster or its character table, using only the Leech lattice, Co_1, Co_2, and M_24 as inputs. The paper is also notable for shipping machine-checkable certificates, explicit orbit representatives, and code that lets a reader re-verify the tabulated data. The derivation is genuinely parameter-free: the orbit sizes are computed, not fitted, and the order of Co_2 is an established input from the Leech lattice, so there is no circularity of the kind that would arise from using the Monster character table. The main caveat, discussed below, is that the completeness of the twelve-orbit (and ten-orbit) classifications is not proven, and the certificate does not close that gap.

major comments (3)
  1. [Appendix A.1; Lemma 3.4; Appendix D] The totality premise of the axis-reduction routine is load-bearing and unproven. Appendix A.1 states: 'We will not prove that every axis can be reduced as described here. It suffices that we are able to reduce all axes occurring during our practical computations.' Lemma 3.4 concludes 'There are precisely twelve orbits on the axes under G_{x0}' from the success of this routine on the axes encountered in the enumeration of §3.2. A hypothetical additional G_{x0}-orbit whose axes are not reducible by the routine would be invisible to the computation, and would change |X_+| and hence the order in Theorem 5.1. The certificate in Appendix D re-verifies the stored transition data and the consistency of the supplied transformations, but it does not certify that every axis in the relevant transversal reduces to one of the twelve representatives; it only checks the representatives that the reduction routine produced. This is not an internal inconsistency, but it is a genuine completeness gap in the central numerical claim.
  2. [Appendix A.2; Lemma 4.4; Proposition 4.1] Exactly the same issue affects the feasible-axis count. Appendix A.2 says: 'As in Appendix A.1, we will not prove that every feasible axis can be reduced. It suffices that we are able to reduce all axes occurring during the calculation.' Lemma 4.4 ('There are precisely ten orbits on the feasible axes under H') and therefore Proposition 4.1 (|X_-| = 11,707,448,673,375) inherit the same unproven totality premise. Since |X_-| enters Theorem 5.1 multiplicatively, this is equally load-bearing.
  3. [§3.1, §3.2; Table 2] The orbit identification of transformed axes relies on a 'watermarking' technique whose discriminative power is asserted but not proven to be complete. The paper says that the eigenvalues and dimensions of the projection to 300x 'are sufficient to distinguish between the putative twelve G_{x0}-orbits' and refers to [Sey24] for a more sophisticated disambiguation. The final classification of every axis in the 200-million-axis enumeration as belonging to one of the twelve listed orbits depends on the reduction routine's success. Consequently, Table 2 and the Perron–Frobenius step in §3.1 are conditional on the same unverified completeness assumption as Lemma 3.4.
minor comments (5)
  1. [Abstract and §1] The phrase 'self-contained' should be read as 'self-contained relative to established Leech-lattice and Conway-group facts', as the paper itself carefully states at the end of §1; the abstract could make this qualification slightly more prominent to avoid overstatement.
  2. [§2.1, notation] The notation such as 2^{1+24}_+.Co_1 and 2^{2+11+22}.(M_24×2) is standard in the field, but a beginner would benefit from a one-sentence reminder of the meaning of the +/– signs and the dot notation; currently this is only implicit.
  3. [Table 2 caption] The caption says the column totals are scaled to 16,584,750 = |G_{x0}:N_{xyz}|; it would be clearer to state explicitly that the matrix M/16584750 is column-stochastic and that the Perron–Frobenius eigenvector is computed for that matrix, not for the raw integer table.
  4. [Appendix E] The GAP script in Appendix E uses the character table of the Monster, which is only presented as an independent cross-check; the main proof does not use it. The text could say this explicitly in the appendix to prevent a misunderstanding, since the paper elsewhere emphasizes that Monster character information is not assumed.
  5. [§5.7] The sentence 'there is only one completely replicable function of order 11 and none of order 112' should give the precise classification reference (it cites [ACMS92] in the bibliography, but the statement is a specific classification fact worth a pointer to the relevant table).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Monster order is derived from independently anchored orbit counts; the unproven axis-reduction completeness in Appendix A.1 is a computational gap, not a circular step.

full rationale

The central derivation is not circular. Theorem 5.1 computes |M| = |X+|·|X−|·|S|, where |S| = |2^{1+23}.Co2| is read from Table 3 and uses only the independently known order of Co2 from Leech-lattice data. The factor |X+| is obtained in Proposition 3.1 as the Perron–Frobenius eigenvector of the transition matrix Table 2, normalized by the 2A orbit size 196560 = 2·98280, where 98280 is the number of short vectors in Λ/2Λ (Lemma 3.3). The factor |X−| is obtained in Proposition 4.1 from the analogous matrix Table 4, normalized by the single-axis orbit 2A1 (Lemma 4.3). No fitted parameter is fed into the order; the orbit sizes are outputs of the enumeration, not inputs tuned to reproduce the target integer. The computations rely on the mmgroup package developed by the second author, and Section 3.2 states that the reduction routine is used to identify the orbit of every axis encountered. Appendix A.1 explicitly concedes: 'We will not prove that every axis can be reduced as described here. It suffices that we are able to reduce all axes occurring during our practical computations.' This is a genuine computational completeness caveat: a hypothetical non-reducible 13th orbit would be invisible to the computation. However, that is a reliability/completeness gap in the computational verification, not a circularity of the kind where the conclusion is presupposed by definition, by fitted inputs, or by a load-bearing self-citation chain. The certificate in Appendix D re-verifies the stored transition data and transformations, and the accompanying program code allows independent re-checking of the reductions that occurred, so the mmgroup self-citation is code-backed evidence rather than an unverified assertion of the target result. Peripheral self-citations, such as [DGH98] and [HM23], support the separate Aut(B) = Aut(V♮) and involution-classification results, not the order computation. Thus the derivation of the Monster order is self-contained against the stated Leech-lattice and Conway-group inputs; the explicit limitation in Appendix A.1 is flagged here and should be treated as a verification risk, but it does not make the argument circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted: the orbit sizes and matrix entries are computed rather than tuned. The central claim rests on standard and prior results (Conway's construction, Leech lattice, Co1/Co2, Borcherds and Carnahan) and on the unproven-for-all-axes reduction routine in mmgroup. No new particles, forces, or mathematical entities are introduced.

assumptions (5)
  • domain assumption All theorems of Conway 1985, including the construction of the Griess algebra B, the action of G_x0 and the triality automorphism tau, and the properties of the Parker loop, are accepted as input.
    Invoked in Section 2.1: 'We take for granted all theorems proved in [Con85], §§ 1-12, and Appendices 1-7.' The paper builds its computation on this construction rather than reproving it.
  • domain assumption The Conway group Co1 and its automorphism group on the Leech lattice, along with the order of Co2 = 42,305,421,312,000, are assumed known.
    Used in Lemma 2.5, Theorem 5.1, and throughout; the paper states 'the order ... of Co2 is assumed to be known'.
  • ad hoc to paper The axis reduction routine in mmgroup correctly reduces all axes that occur in the enumeration.
    Appendix A.1: 'We will not prove that every axis can be reduced ... It suffices that we are able to reduce all axes occurring during our practical computations.' This is an unproven computational premise of the orbit counts.
  • standard math The transition matrix M in Table 2 is regular, so the Perron-Frobenius argument recovers orbit sizes uniquely.
    The paper says 'one checks that it is regular' in Section 3.1; the check is not shown and is an input to the counting.
  • domain assumption Carnahan's bound on |Aut(V#)|/|M| being a power of 11, Borcherds' proof of Monstrous Moonshine, and the classification of completely replicable functions are used for Theorem 5.7.
    Invoked in the proof of Theorem 5.7 to exclude an 11^3 Sylow subgroup and to identify the full automorphism group; these are external results, not proven in this paper.

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Pith. "Pith review of The Order of the Monster Finite Simple Group." pith.science (2026). https://pith.science/paper/FUSQ766G

@misc{pith2026250801037,
  author       = {Pith},
  title        = {Pith review of: The Order of the Monster Finite Simple Group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FUSQ766G}},
  note         = {Machine review of arXiv:2508.01037}
}
abstract

We determine the order of the largest of the twenty-six sporadic simple groups known as the Monster, using a straightforward computational approach. The Monster is here defined as a subgroup of the symmetry group of the 196884-dimensional Griess algebra generated by a group of type $2^{1+24}_+.Co_1$ and an additional triality automorphism. Our approach is based on counting arguments for certain idempotents of the Griess algebra called axes. Our proof is self-contained, requiring only established properties of the Conway group as the automorphism group of the Leech lattice, and some of its subgroups. Although our approach is conceptually simple, it requires extensive calculation inside a 196884-dimensional matrix group that current computer algebra systems cannot easily handle directly. Instead, we use the software package mmgroup, developed by the second author, which supports fast calculations inside the Monster. To our knowledge, this paper contains the first self-contained computation of the order of the Monster. The Monster also acts on the Moonshine module V^#, which is a vertex operator algebra of central charge c=24. We provide a new proof that the Monster is the full automorphism group of the Griess algebra and of the Moonshine module using Borcherds' proof of the Monstrous Moonshine conjectures. In addition, we show that the Monster has exactly two conjugacy classes of involutions. The order of the Baby Monster, the second largest of the sporadic simple groups, is also determined.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Finding 59:29 in the Monster

    math.GR 2026-07 conditional novelty 6.0 of 10

    Explicit mmgroup generators for the Monster maximal subgroup 59:29 are given, completing the explicit classification and yielding a short proof that PSL2(59) is not a subgroup.

Reference graph

Works this paper leans on

7 extracted references · 5 canonical work pages · cited by 1 Pith paper

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