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arxiv: 2409.18410 · v2 · submitted 2024-09-27 · 🧮 math.GR · math.QA· math.RA

On Gr\"{u}n's lemma for perfect skew braces

Cindy Tsang This is my paper

Pith reviewed 2026-05-23 20:32 UTC · model grok-4.3

classification 🧮 math.GR math.QAmath.RA
keywords skew bracesGrün's lemmaannihilatorsocleperfect skew bracestwo-sided skew braces
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The pith

The analog of Grün's lemma holds for perfect two-sided skew braces when the annihilator replaces the socle, but fails in general.

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

Earlier work established that Grün's lemma fails for perfect skew left braces if the socle is used as the stand-in for a group's center. This paper switches to the annihilator and proves the lemma holds exactly for the two-sided case. A sympathetic reader cares because the result isolates the structural feature that lets the classical group statement transfer to skew braces.

Core claim

For perfect two-sided skew braces the annihilator functions as the correct analog of the center, so the analog of Grün's lemma holds; the same statement is false for skew braces that are not two-sided.

What carries the argument

The annihilator of a skew brace, serving as the structural analog of the center in the adapted form of Grün's lemma.

If this is right

  • Grün's lemma transfers to all perfect two-sided skew braces under the annihilator definition.
  • The lemma does not transfer to perfect skew braces that lack the two-sided property.
  • The choice between annihilator and socle determines whether the statement holds.
  • Two-sidedness is required for this particular group-theoretic analog to survive.

Where Pith is reading between the lines

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

  • Two-sidedness may be the minimal extra condition needed for other center-like properties to carry over from groups to skew braces.
  • The result suggests testing whether the annihilator also recovers other classical lemmas for two-sided braces.

Load-bearing premise

The annihilator, rather than the socle, is the right analog of the center so that the classical proof adapts precisely when the skew brace is two-sided.

What would settle it

A concrete perfect two-sided skew brace in which the annihilator version of Grün's lemma fails would disprove the claim.

read the original abstract

By previous work of Ced\'{o}, Smoktunowicz, and Vendramin, one already knows that the analog of Gr\"{u}n's lemma fails to hold for perfect skew left braces when the socle is used as an analog of the center of a group. In this paper, we use the annihilator instead of the socle. We shall show that the analog of Gr\"{u}n's lemma holds for perfect two-sided skew braces but not in general.

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 / 2 minor

Summary. The paper studies an analog of Grün's lemma in the setting of perfect skew braces. Prior work showed that the lemma fails when the socle is used as the analog of the group center. The authors replace the socle by the annihilator and prove that the resulting statement holds for perfect two-sided skew braces while failing for general skew left braces.

Significance. If the proofs are correct, the result supplies a precise structural distinction: the annihilator functions as the appropriate center-like object precisely when the skew brace is two-sided and perfect. This refines earlier negative results and isolates a positive case in which an adaptation of the classical argument succeeds, which may be useful for further classification or cohomology questions in skew-brace theory.

minor comments (2)
  1. [Abstract] The abstract states the main claim but does not indicate the section or theorem number where the positive result for two-sided braces is proved; adding a forward reference would improve readability.
  2. Notation for the annihilator (e.g., Ann(A) or a similar symbol) should be introduced explicitly in the first section where it appears, together with a short comparison to the socle definition used in the cited prior work.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of our manuscript and for recommending minor revision. The report correctly captures the main result: the analog of Grün's lemma holds for perfect two-sided skew braces when the annihilator is used, but fails in general for skew left braces. No specific major comments are raised in the report.

Circularity Check

0 steps flagged

No significant circularity; standard proof adaptation on external foundation

full rationale

The paper explicitly distinguishes the socle (used in prior external work by Cedó, Smoktunowicz, and Vendramin) from the annihilator, cites that external result for the failure case in general skew left braces, and then proves a positive result specifically for perfect two-sided skew braces. No load-bearing step reduces by definition or self-citation to the target claim; the two-sided restriction and choice of annihilator are presented as the technical adjustment enabling the adaptation. The derivation is self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no information on free parameters, background axioms, or new entities introduced by the paper.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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