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arxiv: 2510.03754 · v2 · submitted 2025-10-04 · 🧮 math.GR

Coprime commutators in profinite groups

Cristina Acciarri , Pavel Shumyatsky This is my paper

Pith reviewed 2026-05-18 10:57 UTC · model grok-4.3

classification 🧮 math.GR
keywords profinite groupscoprime commutatorspronilpotent residualprocyclic subgroupsgroup structurecommutator subgroups
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The pith

If the coprime commutators of a profinite group are covered by countably many procyclic subgroups, then its pronilpotent residual is finite-by-procyclic.

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

The paper shows that a countable covering of coprime commutators by procyclic subgroups in a profinite group G forces the pronilpotent residual to be finite-by-procyclic. This yields the further conclusion that G itself must be finite-by-pronilpotent-by-abelian. Readers care because conditions that bound the commutators in this way sharply restrict the possible structure of infinite profinite groups and their derived series. The argument rests on the known fact that the coprime commutators generate exactly the pronilpotent residual, allowing the covering hypothesis to control that residual directly.

Core claim

If the set of coprime commutators of a profinite group G is covered by countably many procyclic subgroups, then γ_∞(G) is finite-by-procyclic. In particular, it follows that G is finite-by-pronilpotent-by-abelian. The proof uses the identification that the subgroup generated by all coprime commutators equals the pronilpotent residual γ_∞(G).

What carries the argument

The covering of the set of coprime commutators by countably many procyclic subgroups, which is used to deduce the finite-by-procyclic structure of the pronilpotent residual γ_∞(G).

If this is right

  • γ_∞(G) is finite-by-procyclic.
  • G is finite-by-pronilpotent-by-abelian.
  • Finiteness-type conditions on coprime commutators continue to impose strong global restrictions on the structure of profinite groups.

Where Pith is reading between the lines

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

  • The same covering hypothesis might be tested on other residual subgroups or on pro-p groups specifically.
  • One could ask whether a finite covering (instead of countable) would force an even stronger conclusion such as finiteness of γ_∞(G).
  • The result suggests looking for analogous statements when commutators are replaced by other words or when the group is required to be finitely generated.

Load-bearing premise

That the subgroup generated by all coprime commutators equals the pronilpotent residual γ_∞(G).

What would settle it

A profinite group in which the coprime commutators lie in a countable union of procyclic subgroups yet the pronilpotent residual fails to be finite-by-procyclic.

read the original abstract

By a coprime commutator in a profinite group $G$ we mean any element of the form $[x, y]$, where $x,y\in G$ and $(|x|,|y|)=1$. It is well-known that the subgroup generated by the coprime commutators of $G$ is precisely the pronilpotent residual $\gamma_\infty(G)$. There are several recent works showing that finiteness conditions on the set of coprime commutators have strong impact on the properties of $\gamma_\infty(G)$ and, more generally, on the structure of $G$. In this paper we show that if the set of coprime commutators of a profinite group $G$ is covered by countably many procyclic subgroups, then $\gamma_\infty(G)$ is finite-by-procyclic. In particular, it follows that $G$ is finite-by-pronilpotent-by-abelian.

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.

Referee Report

1 major / 0 minor

Summary. The paper proves that if the set of coprime commutators in a profinite group G is covered by countably many procyclic subgroups, then the pronilpotent residual γ_∞(G) is finite-by-procyclic. It further concludes that G itself is finite-by-pronilpotent-by-abelian, building on the well-known fact that the subgroup generated by coprime commutators equals γ_∞(G).

Significance. If the result holds, it adds a new countable covering condition to the recent literature on how restrictions on coprime commutators control the structure of profinite groups and their residuals. The theorem supplies a concrete hypothesis that forces γ_∞(G) to have a finite normal subgroup with procyclic quotient, yielding a structural decomposition for G.

major comments (1)
  1. [Abstract] Abstract and the final paragraph of the introduction: the deduction that γ_∞(G) finite-by-procyclic implies G is finite-by-pronilpotent-by-abelian requires an explicit argument that a finite normal subgroup N of γ_∞(G) can be taken G-normal (or that the image of γ_∞(G) in G/N remains the pronilpotent residual of G/N). Without this, the 'in particular' claim does not follow directly from the main theorem on γ_∞(G).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying this point concerning the deduction in the abstract and introduction. We agree that an explicit argument is required to justify passing from the structure of γ_∞(G) to the claimed structure of G, and we will revise the paper to supply it.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the final paragraph of the introduction: the deduction that γ_∞(G) finite-by-procyclic implies G is finite-by-pronilpotent-by-abelian requires an explicit argument that a finite normal subgroup N of γ_∞(G) can be taken G-normal (or that the image of γ_∞(G) in G/N remains the pronilpotent residual of G/N). Without this, the 'in particular' claim does not follow directly from the main theorem on γ_∞(G).

    Authors: We accept the referee's observation. In the revised version we will insert a short paragraph (or a brief lemma) immediately after the statement of the main theorem. Let N be a finite subgroup of γ_∞(G) that is normal in γ_∞(G) with γ_∞(G)/N procyclic. Because G is profinite and N is finite, the conjugation action yields a continuous homomorphism G → Aut(N). The group Aut(N) is finite, so the kernel C_G(N) is open of finite index in G. Consequently N possesses only finitely many distinct G-conjugates. The normal closure N^G is therefore generated by finitely many finite subgroups and is itself finite. Moreover N^G ⊴ G and N^G ≤ γ_∞(G). The quotient γ_∞(G)/N^G is a quotient of the procyclic group γ_∞(G)/N and hence procyclic. We then verify that the image of γ_∞(G) in G/N^G coincides with the pronilpotent residual of G/N^G: the quotient (G/N^G)/(γ_∞(G)/N^G) ≅ G/γ_∞(G) is pronilpotent by definition of the residual, and any smaller normal subgroup of G/N^G whose quotient is pronilpotent would lift to a normal subgroup of G properly contained in γ_∞(G) whose quotient is pronilpotent, contradicting the minimality of γ_∞(G). With this justification in place the 'in particular' statement follows at once. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard facts

full rationale

The paper states as well-known that the subgroup generated by coprime commutators equals γ_∞(G), then proves the covering hypothesis implies γ_∞(G) is finite-by-procyclic. No quoted step reduces a prediction or central claim to a fitted input, self-definition, or self-citation chain by construction. The argument uses external properties of profinite groups and is self-contained against benchmarks; the 'in particular' structural claim for G is a separate deduction whose validity is a correctness issue, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard facts from profinite group theory and commutator calculus; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption The subgroup generated by the coprime commutators of G is precisely the pronilpotent residual γ_∞(G)
    Explicitly stated as well-known in the abstract; this identification is load-bearing for the conclusion about γ_∞(G).

pith-pipeline@v0.9.0 · 5684 in / 1294 out tokens · 34769 ms · 2026-05-18T10:57:06.818516+00:00 · methodology

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Reference graph

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