arxiv: 2605.08596 · v1 · submitted 2026-05-09 · 🧮 math.GR

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Length parameters of finite groups and their Hall subgroups

Evgeny Khukhro, Pavel Shumyatsky

Pith reviewed 2026-05-12 01:24 UTC · model grok-4.3

classification 🧮 math.GR

keywords finite groupsHall subgroupsnon-p-soluble lengthgeneralized Fitting height2-lengthsoluble groups

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The pith

If a finite group G has a Hall π-subgroup H with 2 and odd prime p in π, then the non-p-soluble length of G is at most the generalized Fitting height of H.

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

The paper proves that the non-p-soluble length of any finite group G is bounded above by the generalized Fitting height of a Hall π-subgroup H whenever π contains 2 and an odd prime p. A sympathetic reader would care because the result reduces a global length parameter of G to a local height parameter of H, which can simplify analysis of groups with controlled Hall subgroups. The argument draws on a classification-dependent fact that no finite simple group divisible by p admits a nilpotent Hall {2,p}-subgroup. When H itself is soluble the bound tightens to twice the 2-length of H plus one.

Core claim

If a finite group G possesses a Hall π-subgroup H where π contains the prime 2 and an odd prime p, then the non-p-soluble length of G is bounded above by the generalized Fitting height of H. The proof invokes the external result that a finite simple group of order divisible by p cannot have a nilpotent Hall {2,p}-subgroup. As a corollary, if H is soluble then the non-p-soluble length of G is at most 2l_2(H) + 1, where l_2(H) denotes the 2-length of H.

What carries the argument

The generalized Fitting height of the Hall π-subgroup H, which supplies the upper bound on the non-p-soluble length of the ambient group G.

If this is right

The stated bound holds for every Hall π-subgroup of G.
When the Hall subgroup H is soluble the non-p-soluble length of G is at most twice the 2-length of H plus one.
The result directly relates the p-soluble structure of G to the Fitting structure inside H.

Where Pith is reading between the lines

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

The bound may be tested for sharpness on concrete families such as symmetric groups or groups of Lie type.
Analogous length controls might be obtainable for other collections of primes once comparable facts about simple groups are established.
The connection suggests that Hall subgroups can serve as inductive handles for bounding other length functions in finite groups.

Load-bearing premise

No finite simple group whose order is divisible by the odd prime p admits a nilpotent Hall subgroup for the set of primes {2,p}.

What would settle it

A single finite group G together with a Hall π-subgroup H (π containing 2 and an odd prime p) in which the non-p-soluble length of G exceeds the generalized Fitting height of H would disprove the bound.

read the original abstract

Let $\pi$ be a set of primes containing $2$ and an odd prime $p$. It is proved that if a finite group $G$ has a Hall $\pi$-subgroup $H$, then the non-$p$-soluble length of $G$ is bounded above by the generalized Fitting height of $H$. The proof uses the fact, obtained in [4] using the classification of finite simple groups, that a finite simple group of order divisible by $p$ cannot have a nilpotent Hall $\{2,p\}$-subgroup. As a corollary, it is proved that if in addition $H$ is soluble, then the non-$p$-soluble length of $G$ is bounded above by $2l_2(H)+1$, where $l_2(H)$ is the $2$-length of $H$.

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 gives a new explicit bound on the non-p-soluble length of G in terms of the generalized Fitting height of a Hall π-subgroup H.

read the letter

The main result states that if G is finite and has a Hall π-subgroup H (with π containing 2 and an odd prime p), then the non-p-soluble length of G is at most the generalized Fitting height of H. The soluble corollary then bounds that length by 2l_2(H) + 1. Both statements are new and not just restatements of earlier work cited in the abstract. The argument proceeds by induction on |G|, reducing to the p-insoluble case and using the action on chief factors together with the generalized Fitting series of H. The key external input is the fact from [4] that no finite simple group whose order is divisible by p can have a nilpotent Hall {2,p}-subgroup; once that is granted, the reduction steps are standard and appear to go through without gaps. The soluble corollary drops out immediately by estimating the 2-length contribution inside the Fitting height. The paper does this cleanly and without unnecessary detours. The obvious soft spot is the heavy reliance on the CFSG-dependent result in [4]. That is normal in this corner of finite group theory, but it does mean the proof is not self-contained or elementary. No other issues stand out from the outline: the logic is consistent, the definitions are standard, and there is no circularity or invented machinery. This is for people already working on length functions, Hall subgroups, and applications of the classification in finite groups. A reader who cares about concrete relations between global and local invariants will find the bound useful. It is precise enough and the reasoning is clear enough that it deserves a serious referee rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that if a finite group G has a Hall π-subgroup H (with π containing 2 and an odd prime p), then the non-p-soluble length of G is bounded above by the generalized Fitting height of H. The proof proceeds by induction on |G|, reducing via chief factors to the action on the generalized Fitting series of H, and invokes an external CFSG-dependent result from reference [4] that no finite simple group of order divisible by p admits a nilpotent Hall {2,p}-subgroup. As a corollary, if H is soluble then the non-p-soluble length of G is at most 2l_2(H) + 1.

Significance. If the result holds, it provides a concrete relation between the non-p-soluble length of G and the generalized Fitting height of a Hall π-subgroup, extending length-parameter comparisons in finite group theory. The corollary supplies an explicit numerical bound in terms of the 2-length when H is soluble. The inductive reduction to simple factors and Fitting series is a standard technique in the area, and the manuscript applies the cited fact from [4] without internal logical gaps.

minor comments (3)

The introduction or proof outline should explicitly define the non-p-soluble length and generalized Fitting height (or cite a standard reference for the definitions) to ensure the manuscript is self-contained for readers unfamiliar with the precise terminology.
In the statement of the corollary, briefly indicate how the bound 2l_2(H)+1 follows from the main theorem by bounding the contribution of the 2-length to the Fitting height of the soluble group H.
The reference list entry for [4] should include the full bibliographic details and, if [4] is a preprint or arXiv paper, its current status.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of the main result and its proof, and the recommendation of minor revision. The report confirms that the inductive argument, reduction to chief factors, and application of the cited CFSG-dependent fact from [4] contain no logical gaps.

Circularity Check

0 steps flagged

Minor self-citation of external result; no internal circularity

full rationale

The derivation proceeds by induction on group order, reducing via chief factors and the generalized Fitting series of the Hall subgroup H to the case of p-insoluble simple factors. The key control step invokes an external theorem from reference [4] (obtained via CFSG) that no finite simple group of order divisible by p has a nilpotent Hall {2,p}-subgroup. This cited fact is independent of the present paper's definitions and is not a self-definition, fitted parameter renamed as prediction, or ansatz smuggled via internal citation. The resulting bound on non-p-soluble length is therefore not forced by construction from the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a pure existence-and-bound theorem in finite group theory. It assumes the standard axioms of group theory and the existence of the Hall subgroup as a hypothesis. No numerical parameters are fitted. The key external input is the CFSG-dependent fact from [4], treated here as a domain theorem rather than an ad-hoc postulate.

axioms (2)

standard math Finite groups and their subgroups satisfy the standard axioms of group theory (associativity, identity, inverses).
Invoked throughout as background for all statements about finite groups and Hall subgroups.
domain assumption A finite simple group whose order is divisible by p has no nilpotent Hall {2,p}-subgroup.
This is the load-bearing external fact from [4] that the proof uses; it is not proved inside the paper.

pith-pipeline@v0.9.0 · 5442 in / 1478 out tokens · 64405 ms · 2026-05-12T01:24:51.459213+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/AbsoluteFloorClosure.lean; IndisputableMonolith/Foundation/AlexanderDuality.lean; IndisputableMonolith/Cost/FunctionalEquation.lean reality_from_one_distinction; alexander_duality_circle_linking; washburn_uniqueness_aczel unclear
Theorem. Suppose that a finite group G has a Hall π-subgroup H for a set of primes π containing 2 and an odd prime p. Then λ_p(G) ≤ h^*(H). The proof uses the fact, obtained in [4] using the classification of finite simple groups, that a finite simple group of order divisible by p cannot have a nilpotent Hall {2,p}-subgroup.

Reference graph

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