arxiv: 2604.24676 · v1 · submitted 2026-04-27 · 🧮 math.GR

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Fusion Systems on Sylow 3-subgroups of Fischer and Monster sporadic groups: I

Pete Gautam

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Pith reviewed 2026-05-07 17:15 UTC · model grok-4.3

classification 🧮 math.GR

keywords fusion systemsSylow subgroupssporadic groupsFi22Fi23Baby Monstercorefree fusion systemsexotic fusion systems

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The pith

The Sylow 3-subgroups of Fi22, Fi23 and the Baby Monster admit only non-exotic corefree fusion systems.

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

The paper classifies every corefree fusion system on the Sylow 3-subgroup of each of the three sporadic groups Fi22, Fi23 and B. It concludes that in each case the only systems that arise are those realized by the groups themselves or their subgroups, with no exotic examples. A reader would care because fusion systems capture the possible p-local behaviors of finite groups in an algebraic way, and knowing which abstract systems actually occur distinguishes group-theoretic phenomena from purely combinatorial ones. The work is presented as the first half of a two-paper project that aims to finish the classification for all odd primes p across the sporadic groups.

Core claim

All corefree fusion systems on a Sylow 3-subgroup of Fi22, Fi23 or B are realized by known subgroups of these groups, so none of the three 3-groups supports an exotic fusion system.

What carries the argument

The explicit Sylow 3-subgroups of Fi22, Fi23 and B, together with the enumeration of all possible fusion systems on them using their known subgroup lattices and character tables.

If this is right

These three sporadic groups do not produce new exotic fusion systems at the prime 3.
The classification of corefree fusion systems on Sylow 3-subgroups is complete for Fi22, Fi23 and B.
This finishes one piece of the larger project to classify all such systems on Sylow p-subgroups of sporadic groups for odd p.

Where Pith is reading between the lines

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

The same enumeration techniques could be applied to the remaining sporadic groups at p=3.
The absence of exotics here raises the possibility that most large sporadic 3-groups behave similarly.

Load-bearing premise

The Sylow 3-subgroups of these sporadic groups have exactly the subgroup structure and fusion data recorded in their character tables and standard references.

What would settle it

An explicit construction of a corefree fusion system on one of these 3-groups that cannot be realized inside Fi22, Fi23 or B, or a correction to the known Sylow 3-subgroup structure that permits an additional system.

read the original abstract

We classify all corefree fusion systems on a Sylow $3$-subgroup of the sporadic groups $\mathrm{Fi}_{22}$, $\mathrm{Fi}_{23}$ and $\mathrm{B}$. We show that the $3$-group in each case does not support any exotic fusion systems. This is the first of two papers that will complete the classification of all corefree fusion systems on Sylow $p$-subgroups of sporadic groups for $p$ odd.

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

Gautam classifies corefree fusion systems on the Sylow 3-subgroups of Fi22, Fi23, and B with the result that none are exotic.

read the letter

This paper classifies all corefree fusion systems on the Sylow 3-subgroups of Fi22, Fi23, and the Baby Monster, showing that each 3-group supports only systems realized by the ambient group itself. It forms the first half of a two-paper series that finishes the odd-p sporadic case overall. The work takes the known subgroup structures straight from the ATLAS and standard references, then enumerates the saturated corefree fusion systems on those 3-groups using the usual machinery from the fusion-systems literature. The explicit lists for these three groups are the new content; earlier papers had handled the rest of the sporadics, so this supplies the missing concrete data. The approach stays grounded in accepted local structure rather than new general theorems, which keeps the circularity low and lets the focus remain on the enumeration. The main soft spot is the reliance on exhaustive case-by-case checking. These 3-groups are large enough that the number of possible subsystems and morphisms is substantial, and any overlooked case could in principle allow an exotic system. The abstract states the check is complete, and the stress-test found no load-bearing gaps in the setup, but the strength of the result still depends on whether every possibility was actually covered without omission. A referee will need to see the full case breakdown to confirm. This is for readers already working on fusion systems of finite groups or the local analysis of sporadic groups. Anyone tracking the classification of corefree fusion systems on Sylow p-subgroups will get direct use from the lists and the confirmation that no exotics appear here. It deserves peer review because the program is coherent, the methods are standard, and the concrete results fill a documented gap even if the verification is routine.

Referee Report

0 major / 3 minor

Summary. The paper classifies all corefree fusion systems on the Sylow 3-subgroups of the sporadic groups Fi22, Fi23, and the Baby Monster B. It concludes that none of these 3-groups admit exotic fusion systems. This forms the first of two papers intended to complete the classification of all corefree fusion systems on Sylow p-subgroups of sporadic groups for odd primes p.

Significance. If the exhaustive enumeration holds, the result strengthens the program of determining fusion systems on Sylow subgroups of sporadic groups by providing explicit verification that the 3-groups in Fi22, Fi23, and B support only the fusion systems arising from their known subgroup lattices and no exotic ones. This supplies concrete data points for broader conjectures on the scarcity of exotic systems in sporadic cases and serves as a reference for subsequent work on the remaining sporadic groups.

minor comments (3)

The abstract and introduction should explicitly state the orders of the Sylow 3-subgroups for each group (e.g., |P| for Fi22, Fi23, B) to allow immediate comparison with the ATLAS tables referenced in §2.
Notation for the fusion systems (e.g., the use of F_P(G) versus saturated subsystems) is introduced in §3 but would benefit from a short table summarizing the possible saturated systems before the case-by-case analysis begins.
The manuscript cites the ATLAS and prior fusion-system papers but omits a brief remark on how the corefree condition is verified computationally or by hand for each enumerated system.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. The report accurately captures the paper's contribution as the first part of a two-paper classification of corefree fusion systems on Sylow p-subgroups of sporadic groups for odd p, with the explicit result that the Sylow 3-subgroups of Fi22, Fi23, and B admit no exotic corefree fusion systems.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript classifies corefree fusion systems on the Sylow 3-subgroups of Fi22, Fi23 and B by enumerating saturated fusion systems on the known 3-groups taken from the ATLAS and standard references. The derivation relies on external, independently established subgroup lattices and the standard machinery of saturated fusion systems (Aschbacher, etc.). No equation or central claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the result is a case-by-case exhaustion that remains falsifiable against the external group structures.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result depends on the known 3-local structure of the three sporadic groups and on the standard axioms and definitions of saturated fusion systems.

axioms (1)

standard math Standard axioms of finite group theory and the definition of saturated fusion systems
The paper invokes the established theory of fusion systems on p-groups.

pith-pipeline@v0.9.0 · 5363 in / 1137 out tokens · 52476 ms · 2026-05-07T17:15:01.275845+00:00 · methodology

discussion (0)

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Works this paper leans on

1 extracted references · 1 canonical work pages

[2]

[Wil18] R. Wilson. Maximal subgroups of 2E6(2) and its automorphism groups.arXiv preprint arXiv:1801.08374,

work page arXiv