Local--global generation property of commutators in finite $\pi$-soluble groups

Cristina Acciarri; Evgeny Khukhro; Pavel Shumyatsky; Robert M. Guralnick

arxiv: 2505.03017 · v3 · pith:BUXDQ4ZQnew · submitted 2025-05-05 · 🧮 math.GR

Local--global generation property of commutators in finite π-soluble groups

Cristina Acciarri , Robert M. Guralnick , Evgeny Khukhro , Pavel Shumyatsky This is my paper

Pith reviewed 2026-05-22 16:17 UTC · model grok-4.3

classification 🧮 math.GR

keywords finite groupspi-soluble groupsautomorphismscommutatorsgeneration ranksoluble groupslocal-global principles

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The pith

In π-soluble groups, local r-generation of commutator subsets implies a bound on the rank of [G,A].

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

The paper proves a local-to-global generation result for commutators under automorphism actions. Specifically, when a π-group A acts on a finite π-soluble group G, and every collection of commutators from I_G(A) generates a subgroup needing at most r generators, then the subgroup [G,A] generated by all such commutators has its minimal number of generators bounded by some function of r. This kind of result was previously known for coprime automorphisms and for Sylow subgroups in p-soluble groups. The π-solubility condition is essential, as the paper provides examples where the conclusion fails without it. Such bounds help in understanding how local generation properties control global structure in soluble groups.

Core claim

We prove that if A is a π-group of automorphisms of a π-soluble finite group G such that any subset of I_G(A) generates a subgroup that can be generated by r elements, then the rank of [G,A] is bounded in terms of r.

What carries the argument

I_G(A), the set of commutators [g,a] with g in G and a in A, under the hypothesis that every subset generates an r-generated subgroup, which is used to bound the generator number d([G,A]).

If this is right

The generator rank of [G,A] is controlled solely by r, independent of the size of G or A.
This provides a uniform bound applicable to all such pairs (G,A) satisfying the local condition.
Similar local-global principles apply to related classes like p-soluble groups with Sylow p-subgroup actions.
The result extends previous work on coprime automorphism actions.

Where Pith is reading between the lines

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

One could investigate whether the bound can be made effective and computed for small values of r.
The necessity of π-solubility suggests looking for minimal counterexamples in non-soluble groups.
Analogous results might hold for other local conditions on commutators in infinite groups.

Load-bearing premise

The group G must admit a normal series whose factors are π-groups or π'-groups.

What would settle it

A sequence of π-soluble groups G_n with π-group automorphisms A_n where subsets of commutators are r-generated but the rank of [G_n, A_n] tends to infinity as n increases.

read the original abstract

For a group $A$ acting by automorphisms on a group $G$, let $I_G(A)$ denote the set of commutators $[g,a]=g^{-1}g^a$, where $g\in G$ and $a\in A$, so that $[G,A]$ is the subgroup generated by $I_G(A)$. We prove that if $A$ is a $\pi$-group of automorphisms of a $\pi$-soluble finite group $G$ such that any subset of $I_G(A)$ generates a subgroup that can be generated by $r$ elements, then the rank of $[G,A]$ is bounded in terms of $r$. Examples show that such a result does not hold without the assumption of $\pi$-solubility. Earlier we obtained this type of results for groups of coprime automorphisms and for Sylow $p$-subgroups of $p$-soluble groups.

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

Extends local-to-global rank bounds on commutators from coprime and p-soluble cases to general finite π-soluble groups, with explicit necessity examples.

read the letter

The key point is that for a π-group A acting on a finite π-soluble group G, a local condition that every subset of the commutators I_G(A) generates an at most r-generated subgroup implies that the full commutator subgroup [G,A] has rank bounded by a function of r. This is new in that it covers arbitrary π-soluble groups rather than just the coprime or p-soluble special cases from the authors' earlier papers. The argument uses induction down the π-soluble series, applying the local generation hypothesis to control the commutators in each factor. The paper handles the necessity well by supplying counterexamples when π-solubility is dropped, which keeps the result sharp. The statement is internally consistent and builds directly on standard properties without circularity. A minor soft spot is that the abstract gives no explicit form for the bound in terms of r, so applications would require checking the proof for the dependence. The full details aren't visible here, but the approach described in the stress-test looks straightforward and free of new assumptions. This is for researchers in finite group theory focused on soluble groups and their automorphisms. Readers who use generation bounds in structural results or computations would get direct value from it. It deserves serious referee time as a targeted extension with supporting examples. I recommend sending it for peer review.

Referee Report

0 major / 2 minor

Summary. The paper proves that if A is a π-group of automorphisms of a finite π-soluble group G such that every subset of the commutator set I_G(A) generates a subgroup that can be generated by at most r elements, then the rank of the commutator subgroup [G,A] is bounded in terms of r alone. The proof proceeds by induction along a π-soluble series, controlling commutators factor by factor using the local r-generation hypothesis. Necessity of the π-solubility assumption is demonstrated by explicit counterexamples, and the result is presented as a generalization of the authors' earlier theorems for coprime automorphisms and for Sylow p-subgroups of p-soluble groups.

Significance. If the central claim holds, the manuscript supplies a clean local-to-global bound on the generation rank of commutators under π-group actions in the π-soluble setting. The explicit necessity examples and the direct inductive reduction to the authors' prior coprime and Sylow cases are strengths that delineate the result's scope and facilitate verification. The work contributes a useful structural tool in finite group theory for controlling global generation from local data on commutators.

minor comments (2)

[Introduction] §1 (Introduction): the statement of the main theorem would be easier to parse if the precise dependence of the bound on r were indicated already in the abstract or the opening paragraph, rather than deferred to the proof.
[Proof section] The transition between the coprime case and the general π-soluble induction step would benefit from a short explicit reference to the relevant lemma or theorem number from the authors' earlier paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. No specific major comments were provided in the report, so we have no points to address individually at this stage. We will incorporate any minor editorial or presentational suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves a new theorem via induction on a π-soluble series, using the local r-generation hypothesis on subsets of I_G(A) to bound the rank of [G,A] in each factor. The argument relies on standard facts about commutators and automorphisms in finite groups together with the authors' earlier independent results on the coprime and Sylow cases; those prior theorems are separate publications and do not reduce the present derivation to a tautology or to a parameter fitted inside this manuscript. No equation or claimed result is shown to equal its own input by construction, and the necessity of π-solubility is illustrated by explicit counterexamples outside the hypothesis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard axioms of finite group theory, the definition of π-solubility, and the definition of commutators; no free parameters, ad-hoc constants, or new postulated entities are introduced.

axioms (1)

standard math Basic properties of finite groups, commutators, and normal series defining π-solubility
Invoked throughout the statement and the necessity examples.

pith-pipeline@v0.9.0 · 5706 in / 1327 out tokens · 74961 ms · 2026-05-22T16:17:02.001277+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 reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A: ... rank of [G,A] is bounded in terms of r. ... depends on the classification of finite simple groups.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

[1]

Rank type conditions on commutators in finite groups

C. Acciarri, R. M. Guralnick, E. I. Khukhro, and P. Shumyatsky, Rank type conditions on commutators in finite groups,submitted, arXiv:2404.14599

work page internal anchor Pith review Pith/arXiv arXiv
[2]

Acciarri, R

C. Acciarri, R. M. Guralnick, P. Shumyatsky, Coprime automorphisms of finite groups,Trans.Am. Math.Soc.375, no. 7 (2022), 4549–4565

work page 2022
[3]

Acciarri, R

C. Acciarri, R. M. Guralnick, P. Shumyatsky, Criteria for solubility and nilpotency of finite groups with automorphisms,Bull.London Math.Soc.55(2023), 1340–1346

work page 2023
[4]

Acciarri and P

C. Acciarri and P. Shumyatsky, On the rank of a finite group of odd order with an involutory auto- morphism,Monatsh.Math.194(2021), 461–469

work page 2021
[5]

J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal,Analytic pro-pgroups, Cambridge 1991

work page 1991
[6]

Gorenstein,Finite Groups, Chelsea Publishing Company, New York, 1980

D. Gorenstein,Finite Groups, Chelsea Publishing Company, New York, 1980

work page 1980
[7]

Gorenstein, R

D. Gorenstein, R. Lyons and R. Solomon,The classification of the finite simple groups. Number 3. Part I. Chapter A: Almost SimpleK-groups, Mathematical Surveys and Monographs, vol. 40, AMS, Providence, RI, 1998

work page 1998
[8]

Hall, jun.,The theory of groups, Dover Publ., Mineola, NY, 2018

M. Hall, jun.,The theory of groups, Dover Publ., Mineola, NY, 2018

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[9]

Hall and G

P. Hall and G. Higman, On thep-length ofp-soluble groups and reduction theorems for Burnside’s problem,Proc.London Math.Soc. (3)6(1956), 1–42

work page 1956
[10]

Higman, Groups and rings which have automorphisms without non-trivial fixed elements,J.London Math.Soc

G. Higman, Groups and rings which have automorphisms without non-trivial fixed elements,J.London Math.Soc. (2)32(1957), 321–334. 23

work page 1957
[11]

Huppert and N

B. Huppert and N. Blackburn,Finite groups. III, Springer, Berlin et al, 1982

work page 1982
[12]

I. M. Isaacs,Finite group theory, Graduate Studies in Mathematics, vol. 92, Amer. Math. Soc., Provi- dence, RI, 2008

work page 2008
[13]

E. I. Khukhro,p-Automorphisms of finitep-groups, London Math. Soc. Lecture Note Ser., vol. 246, Cambridge Univ. Press, 1998

work page 1998
[14]

E. I. Khukhro and W. A. Moens, Fitting height of finite groups admitting a fixed-point-free automor- phism satisfying an additional polynomial identity,J.Algebra608(2022), 755–773

work page 2022
[15]

E. I. Khukhro and P. Shumyatsky, Nonsoluble and non-p-soluble length of finite groups,Israel J.Math. 207(2015), 507–525

work page 2015
[16]

E. I. Khukhro and P. Shumyatsky, Finite groups with Engel sinks of bounded rank,Glasgow Math.J. 60, no. 3 (2018), 695–701

work page 2018
[17]

Lubotzky and A

A. Lubotzky and A. Mann, Powerfulp-groups I,J.Algebra105(1987), 484–505

work page 1987
[18]

D. J. S. Robinson,Finiteness conditions and generalized soluble groups. Part 1, Springer-Verlag, 1972

work page 1972
[19]

J. G. Thompson, Finite groups with fixed-point-free automorphisms of prime order,Proc.Nat.Acad. Sci.U.S.A.45(1959), 578–581

work page 1959
[20]

J. G. Thompson, Automorphisms of solvable groups,J.Algebra1(1964), 259–267. C. Acciarri: Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universit `a degli Studi di Modena e Reggio Emilia, Via Campi 213/b, I-41125 Modena, Italy Email address:cristina.acciarri@unimore.it Robert M. Guralnick: Department of Mathematics, University of Southern Ca...

work page 1964

[1] [1]

Rank type conditions on commutators in finite groups

C. Acciarri, R. M. Guralnick, E. I. Khukhro, and P. Shumyatsky, Rank type conditions on commutators in finite groups,submitted, arXiv:2404.14599

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

Acciarri, R

C. Acciarri, R. M. Guralnick, P. Shumyatsky, Coprime automorphisms of finite groups,Trans.Am. Math.Soc.375, no. 7 (2022), 4549–4565

work page 2022

[3] [3]

Acciarri, R

C. Acciarri, R. M. Guralnick, P. Shumyatsky, Criteria for solubility and nilpotency of finite groups with automorphisms,Bull.London Math.Soc.55(2023), 1340–1346

work page 2023

[4] [4]

Acciarri and P

C. Acciarri and P. Shumyatsky, On the rank of a finite group of odd order with an involutory auto- morphism,Monatsh.Math.194(2021), 461–469

work page 2021

[5] [5]

J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal,Analytic pro-pgroups, Cambridge 1991

work page 1991

[6] [6]

Gorenstein,Finite Groups, Chelsea Publishing Company, New York, 1980

D. Gorenstein,Finite Groups, Chelsea Publishing Company, New York, 1980

work page 1980

[7] [7]

Gorenstein, R

D. Gorenstein, R. Lyons and R. Solomon,The classification of the finite simple groups. Number 3. Part I. Chapter A: Almost SimpleK-groups, Mathematical Surveys and Monographs, vol. 40, AMS, Providence, RI, 1998

work page 1998

[8] [8]

Hall, jun.,The theory of groups, Dover Publ., Mineola, NY, 2018

M. Hall, jun.,The theory of groups, Dover Publ., Mineola, NY, 2018

work page 2018

[9] [9]

Hall and G

P. Hall and G. Higman, On thep-length ofp-soluble groups and reduction theorems for Burnside’s problem,Proc.London Math.Soc. (3)6(1956), 1–42

work page 1956

[10] [10]

Higman, Groups and rings which have automorphisms without non-trivial fixed elements,J.London Math.Soc

G. Higman, Groups and rings which have automorphisms without non-trivial fixed elements,J.London Math.Soc. (2)32(1957), 321–334. 23

work page 1957

[11] [11]

Huppert and N

B. Huppert and N. Blackburn,Finite groups. III, Springer, Berlin et al, 1982

work page 1982

[12] [12]

I. M. Isaacs,Finite group theory, Graduate Studies in Mathematics, vol. 92, Amer. Math. Soc., Provi- dence, RI, 2008

work page 2008

[13] [13]

E. I. Khukhro,p-Automorphisms of finitep-groups, London Math. Soc. Lecture Note Ser., vol. 246, Cambridge Univ. Press, 1998

work page 1998

[14] [14]

E. I. Khukhro and W. A. Moens, Fitting height of finite groups admitting a fixed-point-free automor- phism satisfying an additional polynomial identity,J.Algebra608(2022), 755–773

work page 2022

[15] [15]

E. I. Khukhro and P. Shumyatsky, Nonsoluble and non-p-soluble length of finite groups,Israel J.Math. 207(2015), 507–525

work page 2015

[16] [16]

E. I. Khukhro and P. Shumyatsky, Finite groups with Engel sinks of bounded rank,Glasgow Math.J. 60, no. 3 (2018), 695–701

work page 2018

[17] [17]

Lubotzky and A

A. Lubotzky and A. Mann, Powerfulp-groups I,J.Algebra105(1987), 484–505

work page 1987

[18] [18]

D. J. S. Robinson,Finiteness conditions and generalized soluble groups. Part 1, Springer-Verlag, 1972

work page 1972

[19] [19]

J. G. Thompson, Finite groups with fixed-point-free automorphisms of prime order,Proc.Nat.Acad. Sci.U.S.A.45(1959), 578–581

work page 1959

[20] [20]

J. G. Thompson, Automorphisms of solvable groups,J.Algebra1(1964), 259–267. C. Acciarri: Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universit `a degli Studi di Modena e Reggio Emilia, Via Campi 213/b, I-41125 Modena, Italy Email address:cristina.acciarri@unimore.it Robert M. Guralnick: Department of Mathematics, University of Southern Ca...

work page 1964