On finite groups with bounded conjugacy classes of commutators

D\'ebora Senise; Pavel Shumyatsky

arxiv: 2502.04124 · v1 · submitted 2025-02-06 · 🧮 math.GR

On finite groups with bounded conjugacy classes of commutators

D\'ebora Senise , Pavel Shumyatsky This is my paper

Pith reviewed 2026-05-23 04:24 UTC · model grok-4.3

classification 🧮 math.GR

keywords finite groupsconjugacy classescommutatorsderived subgroupsprime power orderbounded order

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The pith

Finite groups where prime-power-order commutators have conjugacy classes of size at most m have second derived group of m-bounded order.

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

The paper establishes that if G is finite and every commutator of prime power order has conjugacy class size at most m, then the second derived subgroup G'' has order bounded by a number depending only on m. This relaxes the hypothesis of an earlier theorem that imposed the same bound on all commutators. A sympathetic reader would care because the result isolates a narrower collection of elements whose conjugacy classes suffice to force the derived series to stabilize at small size. The argument applies the condition selectively to prime-power-order commutators while still recovering the finiteness conclusion for G''.

Core claim

If G is a finite group such that |x^G| ≤ m whenever x is a commutator of prime power order, then G'' is finite and |G''| is bounded by a function of m alone.

What carries the argument

The hypothesis that conjugacy classes of prime-power-order commutators are bounded by m, used to control the size of the second derived subgroup.

If this is right

G'' is finite with order depending only on m.
The conclusion holds without requiring the class-size bound for commutators that are not of prime power order.
The result recovers the 2018 conclusion under a strictly weaker hypothesis on the commutators.

Where Pith is reading between the lines

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

The same conclusion might hold for certain infinite groups if additional finiteness conditions on commutators are imposed.
The method could be tested on whether it bounds higher terms of the derived series under analogous restrictions.
Explicit constructions of groups where |G''| grows with m would clarify whether the bound is sharp.

Load-bearing premise

The group G must be finite.

What would settle it

A finite group in which every prime-power-order commutator has conjugacy class size at most m, yet |G''| exceeds every fixed function of m.

read the original abstract

In 1954 B. H. Neumann discovered that if $G$ is a group in which all conjugacy classes have finite cardinality at most $m$, then the derived group $G'$ is finite of $m$-bounded order. In 2018 G. Dierings and P. Shumyatsky showed that if $|x^G| \le m$ for any commutator $x\in G$, then the second derived group $G''$ is finite and has $m$-bounded order. This paper deals with finite groups in which $|x^G|\le m$ whenever $x\in G$ is a commutator of prime power order. The main result is that $G''$ has $m$-bounded order.

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 weakens the 2018 Dierings-Shumyatsky hypothesis to prime-power-order commutators only and still concludes that G'' has m-bounded order in finite groups.

read the letter

The core claim is that if G is finite and every prime-power-order commutator has conjugacy class size at most m, then |G''| is bounded by a function of m. This is a direct relaxation of the earlier result that required the bound for every commutator. The abstract states the theorem cleanly and positions the work as an extension of the 1954 Neumann theorem and the 2018 paper, which matches what the authors cite as their starting point. The finiteness assumption is explicit in the title and abstract, so there is no hidden condition there. The result looks like a straightforward application of the same circle of ideas used in the predecessor papers, without introducing new parameters or fitting steps. That keeps the argument self-contained on its own terms. The main limitation is that the advance is incremental; it does not remove the finiteness hypothesis or produce a bound that works for infinite groups, and it does not address whether the same conclusion holds under even weaker conditions on the commutators. The proof itself is not visible in the abstract, so any referee would need to verify the derivation steps against the cited techniques. This is the kind of targeted note that specialists in finite-group commutator problems would want to see in print. It is not a major open-problem solver, but the claim is precise and the hypothesis change is verifiable. I would send it to peer review so the details can be checked.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that if G is a finite group such that |x^G| ≤ m for every commutator x of prime-power order, then the second derived subgroup G'' has order bounded by a function of m alone. This is positioned as a weakening of the Dierings–Shumyatsky theorem (2018), which required the conjugacy-class bound for all commutators rather than only those of prime-power order, while retaining the conclusion on G''.

Significance. If the result holds, it shows that the prime-power commutators already suffice to control the second derived subgroup in the finite case, thereby sharpening the 2018 theorem and extending the line of work begun by Neumann (1954). The paper uses standard techniques from finite group theory and the cited literature; the explicit restriction to finite groups removes any hidden infinitary assumptions.

minor comments (2)

The introduction would benefit from a brief comparison table or sentence contrasting the new hypothesis (prime-power commutators only) with the full commutator hypothesis of Dierings–Shumyatsky, to make the improvement immediately visible to readers.
Notation for the bounding function (e.g., whether it is denoted f(m) or left implicit) should be fixed consistently between the statement of the main theorem and the proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. The report correctly identifies the result as a sharpening of the Dierings–Shumyatsky theorem by restricting the conjugacy-class hypothesis to commutators of prime-power order while retaining the conclusion on the boundedness of G''. No major comments were raised.

Circularity Check

0 steps flagged

Minor self-citation of prior result; central claim remains independent

full rationale

The paper explicitly builds on the 1954 Neumann theorem and the 2018 Dierings-Shumyatsky theorem (the latter sharing an author) as background for a weakening to prime-power-order commutators in finite groups, but presents its own main result (G'' has m-bounded order) as a distinct theorem without any derivation that reduces the bound to a fitted input, self-definition, or load-bearing self-citation chain. The finiteness hypothesis is stated outright in the abstract and title. No equations or reductions in the provided text equate the claimed conclusion to its inputs by construction. This is the normal case of incremental progress on an externally established result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard axioms of group theory and the finiteness assumption; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

standard math Standard axioms of group theory (associativity, identity, inverses)
The entire paper operates inside the category of groups.
domain assumption The group G is finite
Explicitly stated in the abstract as the setting for the main result.

pith-pipeline@v0.9.0 · 5656 in / 1126 out tokens · 26047 ms · 2026-05-23T04:24:57.334029+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/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 1.3. Let m be a positive integer and G a finite group in which |x^G| ≤ m whenever x is a commutator of prime power order. Then G'' has m-bounded order.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Proof of Proposition 1.2 ... Neumann’s theorem

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

14 extracted references · 14 canonical work pages

[1]

Acciarri, P

C. Acciarri, P. Shumyatsky, A stronger form of Neumann’s BFC- theorem, Israel J. Math., 242 (2021), 269–278

work page 2021
[2]

Detomi, M

E. Detomi, M. Morigi, P. Shumyatsky, BFC-theorems for higher c ommutator subgroups, Quart. J. Math., 70 (2019), 849–858

work page 2019
[3]

Dierings, P

G. Dierings, P. Shumyatsky, Groups with boundedly ﬁnite conjug acy classes of commutators, Quart. J. Math., 69 (2018), 1047–1051

work page 2018
[4]

Doerk, T

K. Doerk, T. Hawkes, Finite Soluble Groups, de Gruyter, Berlin (1 992)

work page
[5]

Eberhard, P

S. Eberhard, P. Shumyatsky, Probabilistically nilpotent groups o f class two. Math. Ann. 388 (2024), 1879–1902

work page 2024
[6]

W. Feit, J. G. Thompson, Solvability of groups of odd order, Paciﬁ c J. Math., 13 (1963), 773–1029

work page 1963
[7]

Gorenstein, Finite Groups, Chelsea, New York (1980)

D. Gorenstein, Finite Groups, Chelsea, New York (1980)

work page 1980
[8]

R. M. Guralnick, A. Maroti, Average dimension of ﬁxed point space s with applications, J. Algebra 226 (2011), 298–308

work page 2011
[9]

B. H. Neumann, Groups covered by permutable subsets, J. Lon don Math. Soc., 29 (1954), 236–248

work page 1954
[10]

P. M. Neumann, M. R. Vaughan-Lee, An essay on BFC groups, P roc. Lond. Math. Soc., 35 (1977), 213–237

work page 1977
[11]

D. J. S. Robinson, A course in the theory of groups, 2nd edn. G raduate Texts in Mathematics, 80. Springer-Verlag, New York, 1996

work page 1996
[12]

Segal and A

D. Segal and A. Shalev, On groups with bounded conjugacy clas ses, Quart. J. Math. 50 (1999), 505–516

work page 1999
[13]

Shumyatsky, Bounded Conjugacy Classes, Commutators a nd Approximate Subgroups, Quart

P. Shumyatsky, Bounded Conjugacy Classes, Commutators a nd Approximate Subgroups, Quart. J. Math. 73 (2022), 679–684

work page 2022
[14]

Wiegold, Groups with boundedly ﬁnite classes of conjugate ele ments, Proc

J. Wiegold, Groups with boundedly ﬁnite classes of conjugate ele ments, Proc. Roy. Soc. London Ser. A, 238 (1957), 389–401. BOUNDED CONJUGACY CLASSES 9 D´ ebora Senise: Department of Mathematics, University of Bra silia, Brasilia, DF, Brazil Email address : deborasenise2502@gmail.com Pavel Shumyatsky: Department of Mathematics, University of Brasilia, Bra...

work page 1957

[1] [1]

Acciarri, P

C. Acciarri, P. Shumyatsky, A stronger form of Neumann’s BFC- theorem, Israel J. Math., 242 (2021), 269–278

work page 2021

[2] [2]

Detomi, M

E. Detomi, M. Morigi, P. Shumyatsky, BFC-theorems for higher c ommutator subgroups, Quart. J. Math., 70 (2019), 849–858

work page 2019

[3] [3]

Dierings, P

G. Dierings, P. Shumyatsky, Groups with boundedly ﬁnite conjug acy classes of commutators, Quart. J. Math., 69 (2018), 1047–1051

work page 2018

[4] [4]

Doerk, T

K. Doerk, T. Hawkes, Finite Soluble Groups, de Gruyter, Berlin (1 992)

work page

[5] [5]

Eberhard, P

S. Eberhard, P. Shumyatsky, Probabilistically nilpotent groups o f class two. Math. Ann. 388 (2024), 1879–1902

work page 2024

[6] [6]

W. Feit, J. G. Thompson, Solvability of groups of odd order, Paciﬁ c J. Math., 13 (1963), 773–1029

work page 1963

[7] [7]

Gorenstein, Finite Groups, Chelsea, New York (1980)

D. Gorenstein, Finite Groups, Chelsea, New York (1980)

work page 1980

[8] [8]

R. M. Guralnick, A. Maroti, Average dimension of ﬁxed point space s with applications, J. Algebra 226 (2011), 298–308

work page 2011

[9] [9]

B. H. Neumann, Groups covered by permutable subsets, J. Lon don Math. Soc., 29 (1954), 236–248

work page 1954

[10] [10]

P. M. Neumann, M. R. Vaughan-Lee, An essay on BFC groups, P roc. Lond. Math. Soc., 35 (1977), 213–237

work page 1977

[11] [11]

D. J. S. Robinson, A course in the theory of groups, 2nd edn. G raduate Texts in Mathematics, 80. Springer-Verlag, New York, 1996

work page 1996

[12] [12]

Segal and A

D. Segal and A. Shalev, On groups with bounded conjugacy clas ses, Quart. J. Math. 50 (1999), 505–516

work page 1999

[13] [13]

Shumyatsky, Bounded Conjugacy Classes, Commutators a nd Approximate Subgroups, Quart

P. Shumyatsky, Bounded Conjugacy Classes, Commutators a nd Approximate Subgroups, Quart. J. Math. 73 (2022), 679–684

work page 2022

[14] [14]

Wiegold, Groups with boundedly ﬁnite classes of conjugate ele ments, Proc

J. Wiegold, Groups with boundedly ﬁnite classes of conjugate ele ments, Proc. Roy. Soc. London Ser. A, 238 (1957), 389–401. BOUNDED CONJUGACY CLASSES 9 D´ ebora Senise: Department of Mathematics, University of Bra silia, Brasilia, DF, Brazil Email address : deborasenise2502@gmail.com Pavel Shumyatsky: Department of Mathematics, University of Brasilia, Bra...

work page 1957