REVIEW 5 minor 1 cited by
On left nilpotent skew braces of class 2
T0 review · 0 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read Left nilpotency of class 2 forces central nilpotency in nilpotent-type skew braces.
desk verdict Answers Smoktunowicz's open question with a sound bound 2+mr; the proof is solid and the only real issues are expository gaps in the examples. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing identity is equation (1): for $c\in B^2$ and $b\in B$, $$c*b = [-c,b]_+ + [b,[$c^{{-1}}$,$b^{{-1}}$]_{\cdot}]_+ + [$c^{{-1}}$,$b^{{-1}}$]_{\cdot},$$ where $[\ ,\ ]_+$ and $[\ ,\ ]_{\cdot}$ are additive and multiplicative commutators. The hypothesis $B^3=0$ supplies Lemma 5, which makes $B^2$ a trivial skew brace and gives identities such as $(ab)*x = b*x + a*x$. The proof combines these identities with the chain of ideals $S_n = \ker\lambda^{(n)}\cap B^2$, where $\lambda^{(n)}$ is the action of $B$ on $B/Z_n(B^2,+)$. Each induction step uses Eq. (1) to show that an additional $m$ right multiplications move an element of $B^{(2+m(k-1))}$ into $S_{r-k}$; after $r$ steps the element lies in the kernel of the full action, forcing the next right multiplication to give $0$.
What would settle it
Find a skew brace of nilpotent type with $B^3=0$ whose right nilpotent class exceeds $2+mr$, where $m$ and $r$ are the nilpotency classes of $(B,+)$ and $(B^2,+)$. The first place to look is a brace constructed from a bijective derivation $\delta:G\to B$ in which the commutator $[c^{-1},b^{-1}]_{\cdot}$ for some $c\in B^{(2+m(k-1))}$ fails to lie in $S_{r-k}$; exhibiting such a brace would falsify Theorem A.
Extended reading notes
Core claim
The central result, Theorem A, states: if $B$ is a skew brace of nilpotent type with $B^3=0$ (left nilpotent of class 2), and if $m$ and $r$ are the nilpotency classes of the additive group of $B$ and of $B^2$ respectively, then $B$ is right nilpotent of class at most $2+mr$, i.e. $B^{(2+mr+1)}=0$. In particular $B$ is centrally nilpotent. The proof shows by induction that $B^{(2+mk)}\subseteq S_{r-k}$ for a chain of ideals $S_n$ inside $B^2$; the chain terminates because the additive group of $B^2$ has nilpotency class $r$. In the abelian-type case ($m=r=1$) this yields right nilpotency class at most $3$, and an explicit 8-element example shows the bound is attained.
Load-bearing premise
The argument stands on a single rewriting formula for $c*b$, and on the fact that the leftover commutator term always falls into the next-lower level of the chain used for the induction; if that landing condition fails, the bound $2+mr$ collapses.
Editorial extensions
If this is right
- Corollary 1: every Yang-Baxter solution whose associated skew brace has nilpotent type and left nilpotent class 2 is a multipermutation solution.
- Corollary 2: in the abelian-type case the right nilpotency class is at most 3, and Example 7 shows this is best possible.
- The brace is centrally nilpotent, so it admits a central series of ideals and falls under the structural theory used to describe finitely generated skew braces.
- If the multiplicative group is abelian, the bound improves to $2+m+1$ (Corollary 10).
Reading between the lines
- The bound $2+mr$ is probably not sharp in general; the paper's own Proposition 9 already improves it when a higher right ideal lies in the multiplicative centre, and the authors ask in Question 8 whether $2+mr$ is ever attained. A natural test is to compare the bound with explicit small braces of additive class $m>1$.
- The method hinges on the additive upper central series of $B^2$; one might try to generalise the induction to left nilpotent class 3 by replacing Lemma 5 with the corresponding identities, though Eq. (1) would need a new analogue.
- Since Corollary 1 ties the result to multipermutation solutions, Theorem A gives a sufficient condition for a finite nilpotent-type solution to be retractable to the trivial solution after finitely many retractions, with the number of steps bounded by the right class and hence by $2+mr$.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper studies skew braces B that are left nilpotent of class 2, i.e. B^3=0. Theorem A states that if B is of nilpotent type, with (B,+) nilpotent of class m and B^2 nilpotent of class r, then B is right nilpotent of class at most 2+mr, so B^{(2+mr+1)}=0; consequently B is centrally nilpotent. The proof introduces the ideals S_n = Ker λ^(n) ∩ B^2, derives the reduction identity (1), and proves by induction that B^{(2+mk)} ⊆ S_{r-k} for all 1≤k≤r. Corollary 1 asserts that the associated Yang-Baxter solutions are multipermutation, and Corollary 2 specializes to abelian type with right nilpotency class at most 3. Example 6 shows the nilpotent-type hypothesis is necessary, and Example 7 shows the abelian-type bound is sharp.
Significance. The result answers a natural open question left by Smoktunowicz's examples, and the explicit bound 2+mr is new and falsifiable; the abelian-type bound 3 is shown best possible. I verified the central identity (1) under the paper's commutator convention [x,y]_· = xyx^{-1}y^{-1}; the stress-test concern about Eq. (1) does not land. The proof is self-contained modulo standard cited facts and contains no fitted parameters or target-built assumptions. The main weaknesses are expository: one step of the induction in Theorem A is omitted (all ingredients for the repair are present), the B^2=0 edge case is not separated, and the two examples leave key computations to the reader. These do not affect the validity of the theorem.
minor comments (5)
- [Section 3, proof of Theorem A] The induction step from B^{(2+m(k-1))} ⊆ S_{r-k+1} to B^{(2+mk)} ⊆ S_{r-k} omits the verification that allows Eq. (1) to be iterated m times. From d ∈ S_{r-k+1}, Eq. (1) gives d*b = [-d,b]_+ + [b,z]_+ + z with z ∈ S_{r-k}; one should add that [-d,b]_+ ∈ S_{r-k+1} because S_{r-k+1} is additively normal, and that all intermediate star products lie in B^2 so Lemma 5(4) applies at each step. With this one line the displayed 'therefore' is justified; as written, the proof has a gap but not an error.
- [Section 3, proof of Theorem A] The proof implicitly assumes r ≥ 1 (and hence B^2 ≠ 0), since S_{r-1} and the induction over k = 1,...,r are otherwise undefined. If left nilpotency class 2 is taken to mean B^3 = 0 with B^2 possibly zero, the trivial case B^2 = 0 should be separated; the theorem is immediate there.
- [Examples 6 and 7] The computations of B^2, B^3, B^{(3)}, and B^{(4)} are left as routine checks, but Example 7 is used to prove that the bound 3 in Corollary 2 is best possible and Example 6 supports the necessity of the nilpotent-type hypothesis. The authors should include at least the key steps establishing B^2, B^3, B^{(3)}, and B^{(4)} for both examples.
- [Section 2, Eq. (1)] The derivation of Eq. (1) depends on the convention [x,y]_· = xyx^{-1}y^{-1} (and similarly [x,y]_+ = x+y-x-y); this convention is not stated explicitly. Please state it when the commutators are introduced, since the displayed identity is otherwise easy to misread.
- [Abstract/Keywords] The keyword 'multipermutational level' appears to be a typo for 'multipermutation level'.
Circularity Check
No circularity: Theorem A is derived self-containedly from the skew-brace identities, with external citations used only as context.
full rationale
The central claim, Theorem A, is not circular. The proof begins from the hypothesis B^3 = 0 and derives identity (1) using Lemma 5, which itself follows from B^3 = 0 and the definition of the star product. No fitted parameter or assumed conclusion is hidden in the argument: the quantities m and r are the intrinsic nilpotency classes of the additive groups of B and B^2, and the bound 2+mr is an output, not an input. The induction B^(2+mk) subset S_{r-k} uses only that the S_n are ideals, that upper central series quotients of B^2 are abelian, and that the additive group of B has nilpotency class m; each step is a direct consequence of identity (1) and standard ideal properties, not of the right-nilpotency conclusion being proved. The final inference from right nilpotency plus nilpotent type to central nilpotency invokes Corollary 2.15 of Jespers, Van Antwerpen, and Vendramin [7], an external prior result, not a self-citation. The self-citations [1], [2], [6], and [8] appear only in the introduction as references for related background and are not load-bearing in the proof of Theorem A. The only expository weaknesses are the omitted routine computations in Examples 6 and 7 and a slightly compressed induction step, but these affect self-containedness of the examples, not the logical dependence of the main theorem. Thus the derivation chain does not reduce to its inputs.
Assumptions & free parameters
assumptions (5)
- domain assumption Skew brace axioms and the associated lambda action
- domain assumption B^3 = 0 implies B2 is a trivial skew brace and yields Lemma 5
- domain assumption For nilpotent type, central nilpotency is equivalent to left and right nilpotency
- standard math Standard group theory facts about characteristic subgroups, normal subgroups, and upper/lower central series
- standard math A bijective derivation from a group to an abelian group yields a skew brace structure
Cite this review
Pith. "Pith review of On left nilpotent skew braces of class 2." pith.science (2026). https://pith.science/paper/HLUH5U73
@misc{pith2026250507115,
author = {Pith},
title = {Pith review of: On left nilpotent skew braces of class 2},
year = {2026},
howpublished = {\url{https://pith.science/paper/HLUH5U73}},
note = {Machine review of arXiv:2505.07115}
}
abstract
The main objective of this article is to initiate a detailed structure theory of left nilpotent skew braces $B$ of class $2$, i.e. skew braces with $B^3 = 0$. We prove that if $B$ is of nilpotent type, then $B$ is centrally nilpotent. In fact, we show that $B$ is right nilpotent of class at most $2+mr$, i.e. $B^{(2+mr+1)} = 0$, where $m$ and $r$ are the nilpotency classes of the additive group of $B$ and $B^2$, respectively. If $B$ is of abelian type, then $B$ is actually right nilpotent of class $3$, i.e. $B^{(4)} = 0$, and this bound is best possible.
Forward citations
Cited by 1 Pith paper
-
Powerful multiplicative groups do not force right nilpotence in finite braces
For every odd prime p, a uniform family of finite left braces of order p^{2p+1} has powerful multiplicative group of class two but nonterminating right series, disproving the Shalev-Smoktunowicz conjecture in every od...
Reference graph
Works this paper leans on
-
[1]
A. Ballester-Bolinches, R. Esteban-Romero, M. Ferrara, V. Pérez- Calabuig, and M. Trombetti. Central nilpotency of left skew braces and solutions of the Yang-Baxter equation.Pac. J. Math. , 335(1):1–32, 2025
work page 2025
-
[2]
A. Ballester-Bolinches, R. Esteban-Romero, L. A. Kurdachenko, and V. Pérez-Calabuig. From actions of an abelian group on itself to left braces. Math. Proc. Camb. Philos. Soc. , 178(1):65–79, 2025
work page 2025
-
[3]
M. Bonatto and P. Jedlička. Central nilpotency of skew braces. J. Algebra Appl., 22(12):2350255, 2023
work page 2023
-
[4]
F. Cedó, E. Jespers, Ł. Kubat, A. Van Antwerpen, and C. Verwimp. On various types of nilpotency of the structure monoid and group of a set- theoretic solution of the Yang-Baxter equation.J. Pure Appl. Algebra , 227(2):107194, 2023
work page 2023
-
[5]
F. Cedó, A. Smoktunowicz, and L. Vendramin. Skew left braces of nilpotent type. Proc. London Math. Soc. , 118(6):1367–1392, 2019
work page 2019
-
[6]
M. R. Dixon, L. A. Kurdachenko, and I. Ya. Subbotin. On the structure of some one-generator nilpotent brace. arXiv preprint arXiv:2501.04567, 2025
arXiv 2025
-
[7]
E. Jespers, A. Van Antwerpen, and L. Vendramin. Nilpotency of skew braces and multipermutation solutions of the Yang–Baxter equation. Commun. Contemp. Math. , 25(09):2250064, 2023. 8
work page 2023
-
[8]
L.A. Kurdachenko and I. Ya. Subbotin. On the structure of some one- generator braces. Proc. Edinb. Math. Soc. , 67:566–576, 2024
work page 2024
Show all 10 references
-
[9]
W. Rump. Braces, radical rings, and the quantum Yang-Baxter equa- tion. J. Algebra, 307:153–170, 2007
2007
-
[10]
Smoktunowicz
A. Smoktunowicz. On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation.Trans. Amer. Math. Soc., 370(9):6535–6564, 2018. 9
2018
Reviewed August 15, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.