Profinite groups with restricted centralizers of powers

Cristina Acciarri; Pavel Shumyatsky

arxiv: 2602.18839 · v2 · submitted 2026-02-21 · 🧮 math.GR

Profinite groups with restricted centralizers of powers

Cristina Acciarri , Pavel Shumyatsky This is my paper

Pith reviewed 2026-05-15 21:10 UTC · model grok-4.3

classification 🧮 math.GR MSC 20E18

keywords profinite groupscentralizersfinite exponentopen subgroupsvirtual propertiespower mapsgroup theory

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The pith

If a profinite group has an n such that every centralizer of an nth power is finite or open, then it admits an open normal subgroup T with G/Z(T) of finite exponent.

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

The paper considers profinite groups G equipped with an integer n where the centralizer of x to the n is either finite or open in G for every element x. This condition weakens the full restricted-centralizer property studied by Shalev, which forces virtual abelianness. The authors prove that G must contain an open normal subgroup T such that the quotient G divided by the center of T has finite exponent. A reader would care because the result gives a precise structural description: after passing to an open normal piece and then modding out its center, the group satisfies a uniform power law. This places the groups between virtually abelian and fully of finite exponent in the profinite setting.

Core claim

Let G be a profinite group. Suppose there exists an integer n such that, for every element x of G, the centralizer C_G(x^n) is either finite or open in G. Then G possesses an open normal subgroup T with the property that the quotient group G/Z(T) has finite exponent.

What carries the argument

The open normal subgroup T whose central quotient G/Z(T) carries finite exponent, obtained by exploiting the finiteness-or-openness restriction on centralizers of nth powers.

If this is right

The group is virtually of finite exponent once its center is quotiented out in an open normal piece.
The result recovers Shalev's virtual abelianness when the power is 1, since exponent 1 forces the quotient to be trivial.
Power-centralizer restrictions are strictly weaker than full centralizer restrictions yet still force strong finiteness in the quotient.
The exponent of G/Z(T) is bounded in terms of the given n and the structure of the finite centralizers.

Where Pith is reading between the lines

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

The same argument may adapt to pro-p groups or other compact groups where openness replaces finite index.
One could test the conclusion on concrete examples such as the profinite completion of a free group or p-adic analytic groups.
If the exponent of G/Z(T) can be made explicit in n, the result would yield uniform bounds useful for computational checks in finite quotients.

Load-bearing premise

The group must be profinite and satisfy the global condition that centralizers of nth powers are finite or open.

What would settle it

A profinite group G together with an integer n such that C_G(x^n) is finite or open for all x, yet no open normal subgroup T exists making G/Z(T) of finite exponent.

read the original abstract

A group $G$ is said to have restricted centralizers if for every $x\in G$ the centralizer $C_G(x)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Here we take interest in profinite groups $G$ for which there is an integer $n$ such that $C_G(x^n)$ is either finite or open whenever $x\in G$. It is shown that such a group $G$ has an open normal subgroup $T$ with the property that $G/Z(T)$ has finite exponent.

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 extends Shalev by replacing centralizers of elements with centralizers of nth powers and gets an open normal T where G/Z(T) has finite exponent instead of virtual abelianness.

read the letter

The main takeaway is that a profinite group G with an integer n such that C_G(x^n) is finite or open for every x has an open normal subgroup T with G/Z(T) of finite exponent. This is a straightforward weakening of Shalev's result on restricted centralizers, and the proof sketch in the abstract follows the usual route of intersecting open centralizers and their cores to produce T, then reading off the exponent bound directly from the hypothesis. No circularity or extra assumptions on generation show up in the outline. The argument stays within standard profinite topology and centralizer properties, which is the right level for this kind of incremental statement. The soft spot is simply that the conclusion is weaker than virtual abelianness, exactly as one would expect from relaxing the condition to powers; that is not a flaw but a feature of the setup. The paper does not claim more than it delivers. This is aimed at people who already work with profinite groups and centralizer restrictions. A reader who follows Shalev-type results will find the extension useful and easy to place. It is solid enough on its own terms to deserve a serious referee who can check the details of the intersection argument and confirm there are no hidden finiteness issues.

Referee Report

0 major / 2 minor

Summary. The paper studies profinite groups G that satisfy a restricted-centralizer condition on powers: there exists a fixed integer n such that for every x in G the centralizer C_G(x^n) is either finite or open in G. Building on Shalev's theorem that profinite groups with restricted centralizers (i.e., C_G(x) finite or open for all x) are virtually abelian, the authors prove that any such G possesses an open normal subgroup T for which the quotient G/Z(T) has finite exponent.

Significance. If the result holds, it supplies a clean structural description of profinite groups under this power-centralizer hypothesis, refining Shalev's virtual-abelian conclusion to a finite-exponent quotient by the center of an open normal subgroup. The argument relies only on the profinite topology, the openness/finiteness dichotomy, and standard facts about centralizers and normal cores, with no free parameters or ad-hoc constructions.

minor comments (2)

The abstract and introduction could explicitly recall the precise statement of Shalev's theorem (including the reference) to make the extension clearer for readers unfamiliar with the 1990s literature on restricted centralizers.
Notation for the fixed integer n is introduced in the abstract but not restated at the beginning of the main theorem; adding a sentence in §1 that fixes n once and for all would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and for recommending acceptance. The referee's summary correctly identifies the main theorem: that a profinite group G satisfying the restricted-centralizer condition on nth powers possesses an open normal subgroup T such that G/Z(T) has finite exponent. We are grateful for the recognition that the argument relies only on standard profinite topology and centralizer properties.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on the standard profinite topology and the openness/finiteness dichotomy for centralizers of powers. The open normal subgroup T is constructed by intersecting sufficiently many open centralizers C_G(x^n) (or their cores), directly forcing every element of G to have a power in Z(T) and yielding finite exponent for G/Z(T). No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the argument uses only established properties of profinite groups and the given hypothesis without renaming known results or smuggling ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard definitions and properties of profinite groups and centralizers with no free parameters, new entities, or ad hoc axioms introduced.

axioms (2)

standard math Profinite groups are compact totally disconnected topological groups that are inverse limits of finite groups.
Invoked implicitly as the setting for the theorem.
standard math Centralizers C_G(x) are always subgroups of G.
Basic fact from group theory used throughout.

pith-pipeline@v0.9.0 · 5394 in / 1245 out tokens · 32498 ms · 2026-05-15T21:10:19.551824+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Acciarri,C.; Shumyatsky, P.:Profinite groups with restricted centralizers ofπ-elements,Math.Z.301 (2022), 1039–1045

work page 2022
[2]

Azevedo,J.; Shumyatsky, P.: Compact groups with high commuting probability of monothetic sub- groups,J.Algebra623(2023), 34–41

work page 2023
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J.Math.255(2023), 955–973 3

Azevedo, J.; Shumyatsky, P.: Compact groups with probabilistically central monothetic subgroups,Isr. J.Math.255(2023), 955–973 3

work page 2023
[4]

Detomi, E.; Morigi, M.; Shumyatsky, P.: Profinite groups with restricted centralizers of commutators, Proc.Roy.Soc.Edin.A150(2020), 2301–2321

work page 2020
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Detomi E.; Morigi M.; Shumyatsky P.: Commutators, centralizers, and strong conciseness in profinite groups,Math.Nachr.296(2023), 4948–4960

work page 2023
[6]

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Shalev, A.: Profinite groups with restricted centralizers.Proc.Amer.Math.Soc.122(1994), 1279–1284

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Wilson, J.S.: On the structure of compact torsion groups,Monatsh.Math.96(1983), 57–66

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[15]

Zelmanov, E.I.: On periodic compact groups,Israel J.Math.77(1992), 83–95 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 P. Shumyatsky: Department of Mathematics, University of Brasilia, DF 70910-9...

work page 1992

[1] [1]

Acciarri,C.; Shumyatsky, P.:Profinite groups with restricted centralizers ofπ-elements,Math.Z.301 (2022), 1039–1045

work page 2022

[2] [2]

Azevedo,J.; Shumyatsky, P.: Compact groups with high commuting probability of monothetic sub- groups,J.Algebra623(2023), 34–41

work page 2023

[3] [3]

J.Math.255(2023), 955–973 3

Azevedo, J.; Shumyatsky, P.: Compact groups with probabilistically central monothetic subgroups,Isr. J.Math.255(2023), 955–973 3

work page 2023

[4] [4]

Detomi, E.; Morigi, M.; Shumyatsky, P.: Profinite groups with restricted centralizers of commutators, Proc.Roy.Soc.Edin.A150(2020), 2301–2321

work page 2020

[5] [5]

Detomi E.; Morigi M.; Shumyatsky P.: Commutators, centralizers, and strong conciseness in profinite groups,Math.Nachr.296(2023), 4948–4960

work page 2023

[6] [6]

I, Die Grundlehren der mathematischen Wissenschaften, Vol

Hewitt, E.; Ross, K.A.:Abstract Harmonic Analysis, vol. I, Die Grundlehren der mathematischen Wissenschaften, Vol. 115, Springer, Berlin–G¨ ottingen–Heidelberg (1963)

work page 1963

[7] [7]

II, Springer-Verlag, Grund

Hewitt, E.; Ross, K.A.:Abstract Harmonic Analysis, vol. II, Springer-Verlag, Grund. der Math. Wiss., Band 152, Heidelberg (1970)

work page 1970

[8] [8]

I.: Groups and Lie rings admitting almost regular automorphisms of prime order,Rend

Khukhro, E. I.: Groups and Lie rings admitting almost regular automorphisms of prime order,Rend. Circ.Mat.Palermo(2) Suppl.23(1990), 183–191

work page 1990

[9] [9]

de Gruyter-Verlag, Berlin (1993)

Khukhro, E.I.:Nilpotent Groups and Their Automorphisms. de Gruyter-Verlag, Berlin (1993)

work page 1993

[10] [10]

Mann, A.: The exponents of central factor and commutator groups,J.Group Theory(4)10(2007), 435–436

work page 2007

[11] [11]

L.:General topology, Grad

Kelley, J. L.:General topology, Grad. Texts in Math., vol. 27, Springer, New York (1975)

work page 1975

[12] [12]

Nachbin, L.:The Haar Integral, Van Nostrand, Princeton, NJ–Toronto, ON–London (1965)

work page 1965

[13] [13]

Shalev, A.: Profinite groups with restricted centralizers.Proc.Amer.Math.Soc.122(1994), 1279–1284

work page 1994

[14] [14]

Wilson, J.S.: On the structure of compact torsion groups,Monatsh.Math.96(1983), 57–66

work page 1983

[15] [15]

Zelmanov, E.I.: On periodic compact groups,Israel J.Math.77(1992), 83–95 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 P. Shumyatsky: Department of Mathematics, University of Brasilia, DF 70910-9...

work page 1992