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arxiv: 2404.14599 · v3 · pith:WXLY6RZAnew · submitted 2024-04-22 · 🧮 math.GR

Rank type conditions on commutators in finite groups

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

Pith reviewed 2026-05-24 02:27 UTC · model grok-4.3

classification 🧮 math.GR
keywords finite groupsp-soluble groupsSylow subgroupscommutatorsgroup rankderived subgroupcoprime automorphismsbounded generation
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The pith

In p-soluble finite groups, an r-generation condition on subsets of commutators with a Sylow p-subgroup implies that [G,P] has rank bounded by a function of r.

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

The paper shows that if G is p-soluble with Sylow p-subgroup P, and every subgroup generated by any collection of elements from the set I_G(P) of commutators is generated by at most r elements, then the subgroup [G,P] generated by all those commutators has rank at most some number depending only on r. This controls the minimal number of generators needed for the commutator subgroup using a local generation hypothesis. The p-solubility assumption is essential, since the authors construct counterexamples in groups that are not p-soluble. With an added hypothesis that the same r-generation property holds for I_G(x) whenever x lies in I_G(P), the bounded-rank conclusion is recovered without p-solubility. A global version states that if the r-generation condition holds for every Sylow subgroup across all primes, then the derived subgroup G' itself has r-bounded rank.

Core claim

If G is a p-soluble finite group with Sylow p-subgroup P such that any subgroup generated by a subset of I_G(P) is r-generated, then [G,P] has r-bounded rank. The same conclusion holds for arbitrary finite G provided the extra condition that every subgroup generated by a subset of I_G(x) is r-generated for each x in I_G(P). If the r-generated condition on I_G(P) holds for a Sylow p-subgroup P for every prime p dividing the order of G, then the derived subgroup G' has r-bounded rank. As an auxiliary result, if a finite group G admits a coprime automorphism group A such that any subgroup generated by a subset of I_G(A) is r-generated, then the rank of [G,A] is r-bounded.

What carries the argument

The set I_G(S) of all commutators [g,s] with g in G and s in S, under the hypothesis that every subgroup generated by an arbitrary subset of these commutators is r-generated.

If this is right

  • The rank of [G,P] is bounded by a function of r alone.
  • Under the additional hypothesis on I_G(x), the same r-bounded rank for [G,P] holds without assuming p-solubility.
  • When the r-generation condition holds for all Sylow subgroups for every prime, the derived subgroup G' has r-bounded rank.
  • The coprime-automorphism theorem supplies an independent tool for obtaining r-bounded rank of [G,A] whenever A acts coprimely and satisfies the generation condition.

Where Pith is reading between the lines

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

  • The global version for all primes may imply that soluble groups satisfying the condition everywhere have bounded rank for G itself or for its Fitting subgroup.
  • The coprime-automorphism result could be applied to fixed-point-free actions or to questions about the rank of fixed-point subgroups in representation theory.
  • Similar generation hypotheses might be used to bound ranks of other verbal subgroups generated by commutators or higher commutators.

Load-bearing premise

The group G must be p-soluble (or satisfy the extra condition on I_G(x) for each x in I_G(P) in the general case).

What would settle it

A p-soluble finite group G with Sylow p-subgroup P in which every subset of I_G(P) generates an r-generated subgroup yet the rank of [G,P] exceeds every bound depending only on r.

read the original abstract

For a subgroup $S$ of a group $G$, let $I_G(S)$ denote the set of commutators $[g,s]=g^{-1}g^s$, where $g\in G$ and $s\in S$, so that $[G,S]$ is the subgroup generated by $I_G(S)$. We prove that if $G$ is a $p$-soluble finite group with a Sylow $p$-subgroup $P$ such that any subgroup generated by a subset of $I_G(P)$ is $r$-generated, then $[G,P]$ has $r$-bounded rank. We produce examples showing that such a result does not hold without the assumption of $p$-solubility. Instead, we prove that if a finite group $G$ has a Sylow $p$-subgroup $P$ such that (a) any subgroup generated by a subset of $I_G(P)$ is $r$-generated, and (b) for any $x\in I_G(P)$, any subgroup generated by a subset of $I_G(x)$ is $r$-generated, then $[G,P]$ has $r$-bounded rank. We also prove that if $G$ is a finite group such that for every prime $p$ dividing $|G|$ for any Sylow $p$-subgroup $P$, any subgroup generated by a subset of $I_G(P)$ can be generated by $r$ elements, then the derived subgroup $G'$ has $r$-bounded rank. As an important tool in the proofs, we prove the following result, which is also of independent interest: if a finite group $G$ admits a group of coprime automorphisms $A$ such that any subgroup generated by a subset of $I_G(A)$ is $r$-generated, then the rank of $[G,A]$ is $r$-bounded.

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

0 major / 3 minor

Summary. The paper proves that if G is a p-soluble finite group with Sylow p-subgroup P such that every subgroup generated by a subset of I_G(P) is r-generated, then [G,P] has r-bounded rank. It gives counterexamples showing p-solubility is necessary, and proves a version for general finite groups under the extra hypothesis that for x in I_G(P), subgroups generated by subsets of I_G(x) are r-generated. It also shows that if the r-generation condition on I_G(P) holds for every Sylow p-subgroup P (all p dividing |G|), then G' has r-bounded rank. The key independent tool is: if G admits a coprime automorphism group A with the r-generation condition on subsets of I_G(A), then [G,A] has r-bounded rank.

Significance. If the derivations hold, the results supply new, explicitly conditioned criteria for r-bounded rank of commutator subgroups [G,P] and G' in finite groups, with counterexamples establishing necessity of the p-solubility (or extra I_G(x)) hypotheses. The coprime-automorphism theorem is stated as independently interesting and is used directly without apparent circularity or parameter-fitting. The work distinguishes the soluble and general cases cleanly and supplies falsifiable statements.

minor comments (3)
  1. The abstract and introduction should explicitly reference the section containing the coprime-automorphism theorem (the main tool) so readers can locate the independent result immediately.
  2. Notation for I_G(S) is defined clearly in the abstract, but the first numbered section should restate the definition of rank (as the minimal number of generators of a subgroup) to avoid any ambiguity for readers outside finite-group theory.
  3. The counterexample constructions (showing necessity of p-solubility) would benefit from a short table or explicit small-order examples with their I_G(P) subsets listed, to make the failure mode concrete.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report, so we have no points requiring response or revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states and proves a sequence of direct implications in finite group theory: given explicit hypotheses on r-generation of subgroups generated by subsets of I_G(P) (or I_G(A) for coprime automorphisms), it concludes r-bounded rank of [G,P] or [G,A]. The central tool theorem is presented as proved within the paper and of independent interest, with no equations or definitions that reduce the conclusion to the input by construction. p-solubility is an explicit hypothesis whose necessity is demonstrated by counterexamples supplied in the paper; no self-citation chain, fitted parameters renamed as predictions, or ansatz smuggling is indicated in the abstract or described structure. The derivation chain consists of standard group-theoretic arguments conditioned on the stated assumptions rather than any self-referential or load-bearing reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard axioms of finite group theory (existence of Sylow subgroups, definitions of commutators and p-solubility) with no free parameters, invented entities, or ad-hoc axioms introduced.

axioms (1)
  • standard math Finite groups possess Sylow p-subgroups for each prime p and the usual properties of commutators and derived subgroups hold.
    Invoked throughout the statements of the theorems.

pith-pipeline@v0.9.0 · 5891 in / 1195 out tokens · 22400 ms · 2026-05-24T02:27:01.555805+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    math.GR 2025-05 unverdicted novelty 5.0

    In finite π-soluble groups, the rank of [G,A] for a π-group A of automorphisms is bounded in terms of r whenever every subset of commutators generates an r-generated subgroup.

Reference graph

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