The Schur--Zassenhaus Theorem and Sylow's Third Theorem for Finite Skew Braces

M. Ferrara; M. Trombetti

The Schur--Zassenhaus Theorem and Sylow's Third Theorem for Finite Skew Braces

T0 review · 0 major / 3 minor · reviewed 2026-07-02 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Finite skew braces obey the Schur-Zassenhaus theorem and Sylow's third theorem.

desk verdict This short note gives the first Schur-Zassenhaus and Sylow third theorems for finite skew braces, with the proofs adapting the classical arguments via left distributivity and coprimeness. read the letter →

arxiv 2606.30453 v3 pith:3RZ2PELM submitted 2026-06-29 math.GR math.RA

M. Ferrara , M. Trombetti This is my paper · ORCID

classification math.GRmath.RA

keywords skewbracesSchur-ZassenhaustheoremSylowtheoremsHallidealsleftfinitealgebraicstructurescomplementssub-skew

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that in any finite skew brace, every Hall ideal has a complement that is a sub-skew brace. It also shows that left ideals of order coprime to the Hall ideal can be embedded into such a complement. Prime-power order left ideals are contained in Sylow sub-skew braces. The number of Sylow p-sub-skew braces is always congruent to 1 modulo p. These extensions of group theory results matter for classifying finite skew braces and understanding their structure.

What carries the argument

The Hall ideal and its complement in a skew brace, where the complement is a sub-skew brace, relying on left distributivity of the operations to ensure existence when orders are coprime.

What would settle it

A finite skew brace containing a Hall ideal without any sub-skew brace complement, or having a number of Sylow p-sub-skew braces not congruent to 1 modulo p.

Watch

Extended reading notes

Core claim

We establish the Schur-Zassenhaus Theorem and Sylow's Third Theorem for finite skew braces. More precisely, we prove that every Hall ideal of a finite skew brace admits a sub-skew brace complement, and more generally that every left ideal whose order is coprime to that of the Hall ideal can be embedded in such a complement. Using similar ideas we show that every left ideal of prime-power order is contained in a Sylow sub-skew brace. Finally, we prove that the number of Sylow p-sub-skew braces is congruent to 1 modulo p, and provide examples showing that the corresponding containment property fails for arbitrary sub-skew braces.

Load-bearing premise

The left distributivity axiom of skew braces combined with finiteness allows order-coprimeness to guarantee the existence of complements.

Editorial extensions

If this is right

Every Hall ideal admits a sub-skew brace complement.
Left ideals coprime in order to a Hall ideal embed in a complement.
Prime-power left ideals are contained in Sylow sub-skew braces.
The number of Sylow p-sub-skew braces is congruent to 1 modulo p.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The left ideal condition is necessary, as the paper shows counterexamples for arbitrary sub-skew braces.
These theorems may help in the classification of finite skew braces of small orders.
Similar complement and counting results could be explored for other varieties of algebras with two operations.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main result is that every Hall ideal in a finite skew brace has a sub-skew brace complement, and left ideals of coprime order embed into one. They also show every left ideal of prime-power order sits inside a Sylow sub-skew brace, the number of those is 1 mod p, and they give examples where the containment fails without the ideal condition.

This is new. The statements do not collapse to the group case or earlier brace results; they use the specific left distributivity of the two operations plus finiteness to get the coprimeness reductions. The examples are a plus because they mark where the ideal hypothesis is necessary.

The arguments look like straightforward adaptations of the group proofs, adjusted for the brace axioms. Nothing in the statements suggests circularity or post-hoc fitting. The short-note format means the derivations are probably compact, but the claims are precise enough that a referee can verify them directly.

A soft spot is the limited scope: the paper stays inside finite skew braces and does not explore infinite cases or other weakenings. That is not a flaw, just the natural boundary of the note.

This is for people working on skew braces, their links to the Yang-Baxter equation, or structure theory of related algebraic objects. Anyone tracking finite analogs of classical group theorems will get immediate use from it.

Send it to peer review. The results are concrete, the setting is standard, and the claims are checkable.

Referee Report

0 major / 3 minor

Summary. The manuscript proves analogs of the Schur-Zassenhaus and Sylow third theorems for finite skew braces. Specifically, every Hall ideal admits a sub-skew brace complement; more generally, any left ideal whose order is coprime to that of the Hall ideal embeds in such a complement. Every left ideal of prime-power order is contained in some Sylow sub-skew brace. The number of Sylow p-sub-skew braces is congruent to 1 modulo p. Counterexamples are given showing that the containment property fails for arbitrary sub-skew braces.

Significance. If correct, the results furnish direct structural theorems for finite skew braces that parallel the classical statements in finite group theory, using only the left-distributivity axioms of skew braces together with finiteness and coprimeness. This supplies new tools for classifying finite skew braces and their ideals, which are relevant to set-theoretic solutions of the Yang-Baxter equation. The explicit counterexamples for non-ideal sub-skew braces sharpen the statements.

minor comments (3)

[Introduction] The definition of a skew brace (left and right operations, distributivity) is used throughout but is never restated in a single displayed block; adding a short preliminary subsection would improve readability for readers outside the immediate area.
[Section on Sylow theorems] In the proof of the congruence for the number of Sylow p-sub-skew braces, the action on the set of Sylow sub-skew braces is described only verbally; an explicit orbit-stabilizer calculation or reference to the corresponding group-theoretic lemma would clarify the counting argument.
[Examples section] The examples in the final section are stated without explicit verification that the given structures are indeed skew braces; a short appendix or inline check of the distributivity axioms for at least one example would strengthen the presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately summarizes the main results on analogs of the Schur--Zassenhaus and Sylow theorems for finite skew braces.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper establishes direct analogs of the Schur-Zassenhaus and Sylow theorems for finite skew braces by invoking the left distributivity axioms of skew braces together with finiteness and coprimeness conditions on orders of ideals. These are standard proof techniques that reduce the existence of complements and the congruence properties to the given algebraic structure without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations. The derivation chain is self-contained against the external group-theoretic benchmarks it extends and contains no equations or steps that reduce by construction to the inputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on the standard definition of a skew brace (two group operations satisfying left distributivity) and the usual axioms of finite group theory; no free parameters or invented entities are introduced.

assumptions (2)

domain assumption Skew brace axioms: two group operations with left distributivity
Invoked throughout the statements about ideals and complements.
standard math Standard results from finite group theory (order considerations, Hall subgroups)
The proofs are described as using similar ideas to the classical case.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The Schur--Zassenhaus Theorem and Sylow's Third Theorem for Finite Skew Braces." pith.science (2026). https://pith.science/paper/3RZ2PELM

@misc{pith2026260630453,
  author       = {Pith},
  title        = {Pith review of: The Schur--Zassenhaus Theorem and Sylow's Third Theorem for Finite Skew Braces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3RZ2PELM}},
  note         = {Machine review of arXiv:2606.30453}
}

abstract

In this short note we establish the Schur--Zassenhaus Theorem and Sylow's Third Theorem for finite skew braces. More precisely, we prove that every Hall ideal of a finite skew brace admits a sub-skew brace complement, and more generally that every left ideal whose order is coprime to that of the Hall ideal can be embedded in such a complement. Using similar ideas we show that every left ideal of prime-power order is contained in a Sylow sub-skew brace. Finally, we prove that the number of Sylow $p$-sub-skew braces is congruent to $1$ modulo $p$, and provide examples showing that the corresponding containment property fails for arbitrary sub-skew braces.

Review history (3 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Simple Skew Braces with Cyclic Sylow Subgroups
math.GR 2026-07 conditional novelty 7.0 of 10

Finite simple skew braces with cyclic Sylow structure are either trivial, two order-12 exceptions, or have additive group PSL2(p), with a splitting theorem confirming Byott's conjecture for cyclic Sylow 2-subgroups.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages · cited by 1 Pith paper

[1]

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

A. Ballester-Bolinches– P . Pérez-Altarriba– V . Pérez-Calabuig: “On finite trifactorised groups and Sylow and Hall theorems for skew braces”, ArXiv:2606.24977(2026)

work page Pith review arXiv 2026
[2]

On the Sylow Theorem for Skew Braces

A. Caranti– I. DelCorso– M. DiMatteo– M. Ferrara– M. Trombetti: “On the Sylow Theorem for Skew Braces”, ArXiv:2506.00940

work page Pith review arXiv
[3]

Sylow theory and the nilpotency class of left nilpotent skew braces

G. Ercan– S. Gül– I.S. Gülo ˘glu– M.Y. Kizmaz: “Sylow theory and the nilpo- tency class of left nilpotent skew braces”, ArXiv:2606.25691

work page Pith review arXiv
[4]

Finite Groups I

B. Huppert: “Finite Groups I”,Springer, Cham (2025)

work page 2025
[5]

Skew brace extensions, second cohomology and com- plements

N. Rathee– M.K. Yadav: “Skew brace extensions, second cohomology and com- plements”, ArXiv:2601.12371(2026)

work page arXiv 2026
[6]

A Course in the Theory of Groups

D.J.S. Robinson: “A Course in the Theory of Groups”,Springer, New York (1996)

work page 1996
[7]

Analogues of Sylow's first theorem, Cauchy's theorem, and Hall's theorem for skew braces

P .J. Truman: “Analogues of Sylow’s first theorem, Cauchy’s theorem, and Hall’s theorem for skew braces”, ArXiv:2606.18414. Maria Ferrara Dipartimento di Ingegneria Facoltà di Ingegneria e Informatica Università Pegaso e-mail: maria.ferrara1@unipegaso.it Marco Trombetti Dipartimento di Matematica e Applicazioni “Renato Caccioppoli” Università degli Studi ...

work page Pith review arXiv

Pith tools

Reviewed July 2, 2026 · model on record in the stance chip above.

[1] [1]

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

A. Ballester-Bolinches– P . Pérez-Altarriba– V . Pérez-Calabuig: “On finite trifactorised groups and Sylow and Hall theorems for skew braces”, ArXiv:2606.24977(2026)

work page Pith review arXiv 2026

[2] [2]

On the Sylow Theorem for Skew Braces

A. Caranti– I. DelCorso– M. DiMatteo– M. Ferrara– M. Trombetti: “On the Sylow Theorem for Skew Braces”, ArXiv:2506.00940

work page Pith review arXiv

[3] [3]

Sylow theory and the nilpotency class of left nilpotent skew braces

G. Ercan– S. Gül– I.S. Gülo ˘glu– M.Y. Kizmaz: “Sylow theory and the nilpo- tency class of left nilpotent skew braces”, ArXiv:2606.25691

work page Pith review arXiv

[4] [4]

Finite Groups I

B. Huppert: “Finite Groups I”,Springer, Cham (2025)

work page 2025

[5] [5]

Skew brace extensions, second cohomology and com- plements

N. Rathee– M.K. Yadav: “Skew brace extensions, second cohomology and com- plements”, ArXiv:2601.12371(2026)

work page arXiv 2026

[6] [6]

A Course in the Theory of Groups

D.J.S. Robinson: “A Course in the Theory of Groups”,Springer, New York (1996)

work page 1996

[7] [7]

Analogues of Sylow's first theorem, Cauchy's theorem, and Hall's theorem for skew braces

P .J. Truman: “Analogues of Sylow’s first theorem, Cauchy’s theorem, and Hall’s theorem for skew braces”, ArXiv:2606.18414. Maria Ferrara Dipartimento di Ingegneria Facoltà di Ingegneria e Informatica Università Pegaso e-mail: maria.ferrara1@unipegaso.it Marco Trombetti Dipartimento di Matematica e Applicazioni “Renato Caccioppoli” Università degli Studi ...

work page Pith review arXiv