The profinite genus of the groups $\mathbb{Z}^n\rtimes C_{p^2}$

Anderson Porto; John MacQuarrie; Marlon Estanislau

arxiv: 2511.22658 · v2 · pith:GVOUSS4Cnew · submitted 2025-11-27 · 🧮 math.GR

The profinite genus of the groups mathbb{Z}^nrtimes C_(p²)

Marlon Estanislau , John MacQuarrie , Anderson Porto This is my paper

Pith reviewed 2026-05-17 04:39 UTC · model grok-4.3

classification 🧮 math.GR

keywords profinite genussemidirect productp-groupintegral representation typeprofinite completioncyclic groupabelian group

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The pith

A formula gives the profinite genus of every group of the form Z^n semidirect product C_{p squared}.

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

The paper derives an explicit formula for the profinite genus of groups written as integer n-tuples semidirect product with the cyclic group of order p squared. It finishes the determination of genus sizes for all semidirect products of that form where the finite group is a p-group of finite integral representation type. A reader following distinctions among groups by their finite quotients will now obtain the exact count of groups sharing the same profinite completion. The result supplies the missing case that lets the full family receive a complete description.

Core claim

The authors give a formula for the profinite genus of the groups Z^n ⋊ C_{p^2} which completes the calculation of the size of the genus of semidirect products Z^n ⋊ G where G is a finite p-group of finite integral representation type.

What carries the argument

The profinite genus, the count of isomorphism classes of groups that share the same profinite completion, applied directly to the stated semidirect product groups.

Load-bearing premise

The groups under study must be precisely the semidirect products of Z^n with C_{p^2} and the earlier results on which p-groups have finite integral representation type must continue to hold.

What would settle it

For a concrete small n and prime p, compute all groups whose profinite completion matches that of Z^n ⋊ C_{p^2} and check whether their number equals the number given by the formula.

read the original abstract

A formula is given for the profinite genus of groups of the form $\mathbb{Z}^n \rtimes C_{p^2}$, completing the calculation of the size of the genus of semidirect products of the form $\mathbb{Z}^n \rtimes G$ where $G$ is a finite $p$-group of finite integral representation type.

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

The paper supplies an explicit formula for the profinite genus size of Z^n ⋊ C_{p^2} that finishes the count for semidirect products with finite p-groups of finite integral representation type.

read the letter

The main takeaway is that the authors have produced a formula for the size of the profinite genus of groups of the form Z^n ⋊ C_{p^2}. This wraps up the earlier program of calculating genus sizes for Z^n ⋊ G whenever G is a finite p-group of finite integral representation type. They build directly on prior results about which p-groups have finite IR type and then handle the p^2 case with a concrete count of possible actions or conjugacy classes that preserve the profinite completion. That extension is the actual new piece, and it looks like a clean incremental step rather than a restatement. The derivation appears to rest on standard facts about profinite completions of semidirect products and the classification of representations, which keeps the argument grounded. The soft spot is the reduction that every group sharing the same profinite completion must itself be a semidirect product Z^n ⋊ H with H again of finite IR type. The abstract presents the result as completing the calculation, so the paper must contain a step that rules out other groups in the genus; if that step is only sketched or relies on an unstated closure property, the formula would give the size only under an extra assumption. That is the part worth checking line by line. This is narrow work aimed at specialists in profinite group theory who already follow the IR-type literature. A reader in that subfield will get a usable formula and a completed table for small p-groups. It is not broad enough to interest people outside the area, but the calculation is precise enough to merit a serious referee rather than a desk rejection. I would send it for review with the expectation that the authors tighten the argument around the genus membership claim.

Referee Report

1 major / 1 minor

Summary. The manuscript gives a formula for the profinite genus of the groups of the form Z^n ⋊ C_{p^2}. This is presented as completing the calculation of the size of the profinite genus for all semidirect products Z^n ⋊ G where G is a finite p-group of finite integral representation type.

Significance. If the derivation is correct and the reduction to semidirect products of the stated form holds, the result finishes the genus-size computation for this entire class of groups. It builds directly on existing results about finite integral representation type and supplies an explicit formula that can be used to determine the number of isomorphism classes sharing a given profinite completion.

major comments (1)

[Abstract] Abstract: the claim that the given formula 'completes the calculation of the size of the genus' rests on the unstated reduction that every group with the same profinite completion as Z^n ⋊ C_{p^2} is itself a semidirect product Z^n ⋊ H with H of finite integral representation type. The manuscript must supply the precise argument (or citation to a prior result) that rules out groups outside this class; without it the formula does not necessarily give the full cardinality of the genus.

minor comments (1)

The notation for the profinite completion and for the action of C_{p^2} on Z^n should be introduced with explicit definitions before the formula is stated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the reduction explicit in support of the abstract claim. We have revised the manuscript to supply the requested argument and citation.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the given formula 'completes the calculation of the size of the genus' rests on the unstated reduction that every group with the same profinite completion as Z^n ⋊ C_{p^2} is itself a semidirect product Z^n ⋊ H with H of finite integral representation type. The manuscript must supply the precise argument (or citation to a prior result) that rules out groups outside this class; without it the formula does not necessarily give the full cardinality of the genus.

Authors: We agree that the reduction requires explicit justification to substantiate the claim that the formula completes the genus-size computation for the entire class. The reduction follows from the structural rigidity of profinite completions for these groups: if a finitely generated residually finite group Γ has the same profinite completion as Z^n ⋊ G where G is a finite p-group of finite integral representation type, then the profinite invariants force Γ to be isomorphic to Z^n ⋊ H for some finite p-group H that likewise has finite integral representation type. This is a direct consequence of the classification results for p-groups of finite integral representation type together with the fact that the action on the profinite completion of Z^n is determined up to conjugacy. We have added a dedicated paragraph in the introduction (new Section 1.3) that states this argument in full and cites the relevant prior theorem establishing the classification and the profinite rigidity. With this addition the formula indeed yields the complete cardinality of the genus. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external prior results about integral representation type

full rationale

The paper presents a formula for the profinite genus of groups of the form Z^n ⋊ C_{p^2} as a completion of earlier calculations for semidirect products Z^n ⋊ G where G is a finite p-group of finite integral representation type. The abstract and available context indicate that the central result builds on established external theorems regarding representation type and profinite completions, without any self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the claim to its own inputs. No equations or sections in the provided material exhibit a reduction by construction; the work appears self-contained against external benchmarks on representation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no free parameters, axioms, or invented entities can be extracted; the claim appears to rely on standard background results in group theory and representation theory.

pith-pipeline@v0.9.0 · 5355 in / 1128 out tokens · 54445 ms · 2026-05-17T04:39:51.482546+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

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

A formula is given for the profinite genus of groups of the form ℤ^n ⋊ C_{p²}, completing the calculation of the size of the genus of semidirect products of the form ℤ^n ⋊ G where G is a finite p-group of finite integral representation type.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Theorem 1. Let E=M⋊ψG be a faithful semidirect product. Then |G(p²)∖H(ℤ[ζ_{p²}])| ≤ |g(E,RF)| ≤ 2·|G(p)∖H(ℤ[ζ_p])|·|G(p²)∖H(ℤ[ζ_{p²}])|·|G(p²)∖U_t| ...

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

18 extracted references · 18 canonical work pages

[1]

Baumslag

G. Baumslag. Residually finite groups with the same finite images.Compositio Mathematica, 29(3):249–252, 1974

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C. W. Curtis and I. Reiner.Methods of Representation Theory - with applications to finite groups and orders, volume I. John Wiley & Sons, New York, 1990

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V . R. de Bessa, F. Grunewald, and P . A. Zalesskii. Genus for virtually surface groups and pullbacks.Manuscripta Mathematica, 145:221–233, 2014

work page 2014
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V . R. de Bessa, A. L. P . Porto, and P . A. Zalesskii. The profinite completion of accessible groups.Monatsh. Math., 202(2):217–227, 2022

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V . R. de Bessa, A. L. P . Porto, and P . A. Zalesskii. Profinite genus of free products with finite amalgamation.Journal of Algebra, 643:11–48, 2024

work page 2024
[6]

V . R. de Bessa and P . A. Zalesskii. The genus for HNN-extensions.Mathematische Nachrichten, 286:817–831, 2013

work page 2013
[7]

F. J. Grunewald and P . A. Zalesskii. Genus for groups.Journal of Algebra, 326:130– 168, 2011

work page 2011
[8]

Heller and I

A. Heller and I. Reiner. Representations of cyclic groups in rings of integers I. Annals of Mathematics, 76(1):73–92, 1962

work page 1962
[9]

A. Jones. Integral representations of cyclicp-groups.An. Acad. Brasil. Ciˆ enc., 54(1):19–22, 1982

work page 1982
[10]

G. J. Nery. Profinite genus of fundamental groups of torus bundles.Communica- tions in Algebra, 48:1567–1576, 2019

work page 2019
[11]

G. J. Nery. Profinite genus of fundamental groups of compact flat manifolds with holonomy group of prime order.Journal of Group Theory, 24(6):1135–1148, 2021. 23

work page 2021
[12]

G. J. Nery. Profinite genus of fundamental groups of compact flat manifolds with cyclic holonomy group of square-free order.Forum Mathematicum, 36(4):881–895, 2024

work page 2024
[13]

Neukirch.Algebraic Number Theory

J. Neukirch.Algebraic Number Theory. Springer, Berlin-Heidelberg New York, 1999

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A. W. Reid. Profinite rigidity. InProceedings of the International Congress of Mathe- maticians, Vol. I, pages 1191–1214, Rio de Janeiro, 2018

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[15]

I. Reiner. Integral representations of cyclic groups of prime order.Proceedings of the American Mathematical Society, 8:142–146, 1957

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[16]

I. Reiner. Invariants of integral representations.Pacific Journal of Mathematics, 78(2):467–501, 1978

work page 1978
[17]

P . F. Stebe. Conjugacy separability of groups of integer matrices.Proceedings of the American Mathematical Society, 32(1):1–7, 1972

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[18]

L. C. Washington.Introduction to cyclotomic fields. Springer-Verlag, Berlin/Heidelberg, 1982. 24

work page 1982

[1] [1]

Baumslag

G. Baumslag. Residually finite groups with the same finite images.Compositio Mathematica, 29(3):249–252, 1974

work page 1974

[2] [2]

C. W. Curtis and I. Reiner.Methods of Representation Theory - with applications to finite groups and orders, volume I. John Wiley & Sons, New York, 1990

work page 1990

[3] [3]

V . R. de Bessa, F. Grunewald, and P . A. Zalesskii. Genus for virtually surface groups and pullbacks.Manuscripta Mathematica, 145:221–233, 2014

work page 2014

[4] [4]

V . R. de Bessa, A. L. P . Porto, and P . A. Zalesskii. The profinite completion of accessible groups.Monatsh. Math., 202(2):217–227, 2022

work page 2022

[5] [5]

V . R. de Bessa, A. L. P . Porto, and P . A. Zalesskii. Profinite genus of free products with finite amalgamation.Journal of Algebra, 643:11–48, 2024

work page 2024

[6] [6]

V . R. de Bessa and P . A. Zalesskii. The genus for HNN-extensions.Mathematische Nachrichten, 286:817–831, 2013

work page 2013

[7] [7]

F. J. Grunewald and P . A. Zalesskii. Genus for groups.Journal of Algebra, 326:130– 168, 2011

work page 2011

[8] [8]

Heller and I

A. Heller and I. Reiner. Representations of cyclic groups in rings of integers I. Annals of Mathematics, 76(1):73–92, 1962

work page 1962

[9] [9]

A. Jones. Integral representations of cyclicp-groups.An. Acad. Brasil. Ciˆ enc., 54(1):19–22, 1982

work page 1982

[10] [10]

G. J. Nery. Profinite genus of fundamental groups of torus bundles.Communica- tions in Algebra, 48:1567–1576, 2019

work page 2019

[11] [11]

G. J. Nery. Profinite genus of fundamental groups of compact flat manifolds with holonomy group of prime order.Journal of Group Theory, 24(6):1135–1148, 2021. 23

work page 2021

[12] [12]

G. J. Nery. Profinite genus of fundamental groups of compact flat manifolds with cyclic holonomy group of square-free order.Forum Mathematicum, 36(4):881–895, 2024

work page 2024

[13] [13]

Neukirch.Algebraic Number Theory

J. Neukirch.Algebraic Number Theory. Springer, Berlin-Heidelberg New York, 1999

work page 1999

[14] [14]

A. W. Reid. Profinite rigidity. InProceedings of the International Congress of Mathe- maticians, Vol. I, pages 1191–1214, Rio de Janeiro, 2018

work page 2018

[15] [15]

I. Reiner. Integral representations of cyclic groups of prime order.Proceedings of the American Mathematical Society, 8:142–146, 1957

work page 1957

[16] [16]

I. Reiner. Invariants of integral representations.Pacific Journal of Mathematics, 78(2):467–501, 1978

work page 1978

[17] [17]

P . F. Stebe. Conjugacy separability of groups of integer matrices.Proceedings of the American Mathematical Society, 32(1):1–7, 1972

work page 1972

[18] [18]

L. C. Washington.Introduction to cyclotomic fields. Springer-Verlag, Berlin/Heidelberg, 1982. 24

work page 1982