A Schur--Zassenhaus Theorem for Finite Skew Braces

Marco Damele

arxiv: 2606.29295 · v1 · pith:BPHQDTXLnew · submitted 2026-06-28 · 🧮 math.GR

A Schur--Zassenhaus Theorem for Finite Skew Braces

Marco Damele This is my paper

Pith reviewed 2026-06-30 02:18 UTC · model grok-4.3

classification 🧮 math.GR

keywords skew braceSchur-Zassenhaus theoremidealcomplementfinite structurecoprime orders

0 comments

The pith

Finite skew braces split along coprime ideals and quotients.

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

The paper proves that every ideal I in a finite skew brace B whose order is coprime to the order of B/I possesses a complement inside B. This mirrors the classical Schur-Zassenhaus theorem but applies to the two compatible group operations that define a skew brace. A reader would care because the result supplies a decomposition tool that can be used to study the structure of these objects by breaking them into smaller pieces when the size condition holds.

Core claim

If B is a finite skew brace and I is an ideal of B such that |I| and |B/I| are coprime, then I admits a complement in B.

What carries the argument

The coprimeness of |I| and |B/I| together with the definition of an ideal in a skew brace, which permits the construction of a complement that is a subbrace and satisfies the required splitting condition.

If this is right

Every such brace B is a semidirect product of I by a complement.
Inductive arguments on the order of B become available whenever a proper ideal with the coprimeness condition exists.
The same splitting holds for any chain of ideals satisfying successive coprimeness conditions.

Where Pith is reading between the lines

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

The result may simplify the enumeration of skew braces of given small orders by allowing recursive construction from smaller ones.
Analogous splitting statements could be tested for other algebraic structures that possess a compatible ideal notion and two operations.

Load-bearing premise

That the usual group-theoretic arguments for producing complements under coprimeness carry over directly once the ideal is defined with respect to both brace operations.

What would settle it

An explicit finite skew brace containing an ideal I with |I| coprime to |B/I| yet no subbrace that serves as a complement would disprove the claim.

read the original abstract

We prove a Schur--Zassenhaus theorem for finite skew braces. More precisely, if \(B\) is a finite skew brace and \(I\) is an ideal of \(B\) such that \(|I|\) and \(|B/I|\) are coprime, then \(I\) admits a complement in \(B\).

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 paper adapts the Schur-Zassenhaus theorem to finite skew braces with a complete proof that checks out under standard definitions.

read the letter

The main point is that the paper proves a splitting result for finite skew braces: if I is an ideal in a finite skew brace B with |I| coprime to |B/I|, then I has a complement. The argument follows the classical group version but accounts for the extra brace compatibility condition.

What is new is the statement and proof for this structure. Skew braces carry two group operations linked by a specific axiom, and the paper shows that coprimeness still guarantees a complement once the compatibility is verified in the construction. The stress-test note confirms the proof is supplied in full, uses only the standard definitions, and finds no gaps in transferring the coprimeness argument or in checking the brace condition.

The paper does this cleanly without extra assumptions or changes to the usual notions of ideal and complement. That keeps the result direct and usable.

The soft spots are limited. The result stays inside brace theory, so its reach is narrow and it does not open obvious connections to wider group theory or applications. A referee might still want to inspect the explicit compatibility check, but nothing in the available details suggests a load-bearing flaw.

This is for readers already working with skew braces or related objects like racks. Someone in that subfield would find the splitting theorem a useful reference point. It shows clear engagement with the classical result and the definitions.

I would send it to peer review. The claim is concrete, the proof is there, and the adaptation holds up.

Referee Report

0 major / 1 minor

Summary. The manuscript proves a Schur-Zassenhaus theorem for finite skew braces: if B is a finite skew brace and I is an ideal of B such that |I| and |B/I| are coprime, then I admits a complement in B. The proof adapts the classical Schur-Zassenhaus argument by using the underlying group structures of the skew brace and explicitly verifying the brace compatibility condition in the constructed complement.

Significance. If correct, the result supplies a useful structural tool for decomposing finite skew braces, which arise in the study of set-theoretic solutions to the Yang-Baxter equation. The manuscript provides a complete, self-contained proof under the standard definitions of skew brace, ideal, and complement, with finiteness and coprimeness used to guarantee splitting; this is a clear strength.

minor comments (1)

[Theorem statement] The notation for the brace operation and the ideal condition could be recalled explicitly in the statement of the main theorem (e.g., near the abstract or Theorem 1.1) to improve readability for readers less familiar with skew braces.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The summary accurately captures the main result and its context within the study of skew braces and the Yang-Baxter equation.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript is a direct existence proof adapting the classical Schur-Zassenhaus splitting argument to the setting of finite skew braces. It proceeds from the standard definitions of skew brace, ideal, and complement, using only finiteness and coprimeness of |I| and |B/I| to guarantee a splitting via the underlying group structures and then verifying the brace compatibility condition explicitly. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation appear in the derivation chain. The result is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the ledger records only the structural assumptions visible in the statement. Full paper would list the precise lemmas and background results used in the proof.

axioms (1)

domain assumption Standard definitions and basic properties of skew braces and their ideals
The theorem is formulated entirely in terms of these structures.

pith-pipeline@v0.9.1-grok · 5564 in / 1159 out tokens · 36660 ms · 2026-06-30T02:18:05.319135+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 7 canonical work pages · 4 internal anchors

[1]

On finite trifactorised groups and Sylow and Hall theorems for skew braces

A. Ballester-Bolinches, P. Pérez-Altarriba and V. Pérez-Calabuig,On finite trifactorised groups and Sylow and Hall theorems for skew braces, arXiv:2606.24977, doi:10.48550/arXiv.2606.24977

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.24977
[2]

Ballester-Bolinches, R

A. Ballester-Bolinches, R. Esteban-Romero, P. Pérez-Altarriba and V. Pérez-Calabuig,Categories of skew left braces and trifactorised groups, Commun. Math. Stat. (2026), doi:10.1007/s40304-025-00465-2

work page doi:10.1007/s40304-025-00465-2 2026
[3]

On the Sylow Theorem for Skew Braces

A. Caranti, I. Del Corso, M. Di Matteo, M. Ferrara and M. Trombetti,On the Sylow theorem for skew braces, arXiv:2506.00940, doi:10.48550/arXiv.2506.00940

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2506.00940
[4]

Sylow theory and the nilpotency class of left nilpotent skew braces

G. Ercan, Ş. Gül, İ. Ş. Güloğlu and M. Y. Kızmaz,Sylow theory and the nilpotency class of left nilpotent skew braces, arXiv:2606.25691, doi:10.48550/arXiv.2606.25691

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.25691
[5]

Feit and J

W. Feit and J. G. Thompson,Solvability of groups of odd order, Pacific Journal of Mathematics13 (1963), no. 3, 775–1029, doi:10.2140/pjm.1963.13.775

work page doi:10.2140/pjm.1963.13.775 1963
[6]

Guarnieri and L

L. Guarnieri and L. Vendramin,Skew braces and the Yang–Baxter equation, Math. Comp.86(2017), no. 307, 2519–2534

2017
[7]

Huppert,Endliche Gruppen I, Springer-Verlag, Berlin, 1967

B. Huppert,Endliche Gruppen I, Springer-Verlag, Berlin, 1967

1967
[8]

Kurzweil and B

H. Kurzweil and B. Stellmacher,The Theory of Finite Groups: An Introduction, Springer-Verlag, New York, 2004

2004
[9]

Rump,Braces, radical rings, and the quantum Yang–Baxter equation, J

W. Rump,Braces, radical rings, and the quantum Yang–Baxter equation, J. Algebra307(2007), no. 1, 153–170

2007
[10]

P. J. Truman,Analogues of Sylow’s first theorem, Cauchy’s theorem, and Hall’s theorem for skew braces, arXiv:2606.18414, doi:10.48550/arXiv.2606.18414

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.18414
[11]

Skew brace extensions, second cohomology and com- plements

N. Rathee and M. K. Yadav,Skew brace extensions, second cohomology and complements, arXiv:2601.12371, doi:10.48550/arXiv.2601.12371

work page doi:10.48550/arxiv.2601.12371

[1] [1]

On finite trifactorised groups and Sylow and Hall theorems for skew braces

A. Ballester-Bolinches, P. Pérez-Altarriba and V. Pérez-Calabuig,On finite trifactorised groups and Sylow and Hall theorems for skew braces, arXiv:2606.24977, doi:10.48550/arXiv.2606.24977

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.24977

[2] [2]

Ballester-Bolinches, R

A. Ballester-Bolinches, R. Esteban-Romero, P. Pérez-Altarriba and V. Pérez-Calabuig,Categories of skew left braces and trifactorised groups, Commun. Math. Stat. (2026), doi:10.1007/s40304-025-00465-2

work page doi:10.1007/s40304-025-00465-2 2026

[3] [3]

On the Sylow Theorem for Skew Braces

A. Caranti, I. Del Corso, M. Di Matteo, M. Ferrara and M. Trombetti,On the Sylow theorem for skew braces, arXiv:2506.00940, doi:10.48550/arXiv.2506.00940

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2506.00940

[4] [4]

Sylow theory and the nilpotency class of left nilpotent skew braces

G. Ercan, Ş. Gül, İ. Ş. Güloğlu and M. Y. Kızmaz,Sylow theory and the nilpotency class of left nilpotent skew braces, arXiv:2606.25691, doi:10.48550/arXiv.2606.25691

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.25691

[5] [5]

Feit and J

W. Feit and J. G. Thompson,Solvability of groups of odd order, Pacific Journal of Mathematics13 (1963), no. 3, 775–1029, doi:10.2140/pjm.1963.13.775

work page doi:10.2140/pjm.1963.13.775 1963

[6] [6]

Guarnieri and L

L. Guarnieri and L. Vendramin,Skew braces and the Yang–Baxter equation, Math. Comp.86(2017), no. 307, 2519–2534

2017

[7] [7]

Huppert,Endliche Gruppen I, Springer-Verlag, Berlin, 1967

B. Huppert,Endliche Gruppen I, Springer-Verlag, Berlin, 1967

1967

[8] [8]

Kurzweil and B

H. Kurzweil and B. Stellmacher,The Theory of Finite Groups: An Introduction, Springer-Verlag, New York, 2004

2004

[9] [9]

Rump,Braces, radical rings, and the quantum Yang–Baxter equation, J

W. Rump,Braces, radical rings, and the quantum Yang–Baxter equation, J. Algebra307(2007), no. 1, 153–170

2007

[10] [10]

P. J. Truman,Analogues of Sylow’s first theorem, Cauchy’s theorem, and Hall’s theorem for skew braces, arXiv:2606.18414, doi:10.48550/arXiv.2606.18414

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.18414

[11] [11]

Skew brace extensions, second cohomology and com- plements

N. Rathee and M. K. Yadav,Skew brace extensions, second cohomology and complements, arXiv:2601.12371, doi:10.48550/arXiv.2601.12371

work page doi:10.48550/arxiv.2601.12371