Powerful multiplicative groups do not force right nilpotence in finite braces

Brecht Verbeken

Powerful multiplicative groups do not force right nilpotence in finite braces

T0 review · 0 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read For every odd prime p, there is a finite left brace A_p with powerful multiplicative group of class two that is nevertheless not right nilpotent, disproving a conjecture that powerfulness forces right nilpotence.

desk verdict Uniform counterexample family for every odd prime; verifies cleanly; a few compressed proofs but no load-bearing gaps. read the letter →

arxiv 2608.01884 v1 pith:XHZPKBS5 submitted 2026-08-03 math.GR

Brecht Verbeken This is my paper · ORCID

classification math.GR MSC 16T2520D1516N40

keywords leftbracepowerfulp-grouprightnilpotenceYang–Baxterequationregularsubgroupskewnilpotencyclasselementaryabeliangroup

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

Powerful p-groups behave like abelian p-groups in many ways, so a natural question was whether a brace whose multiplicative group is powerful must be right nilpotent. This paper answers that question negatively for every odd prime p by constructing a left brace A_p of order p^{2p+1} with elementary abelian additive group. Its multiplicative group G_p is powerful of nilpotency class two, with G_p'=G_p^p≅C_p^2 and exponent p^2, yet the right series A^{(n)}_p never reaches zero. The obstruction is an explicit three-dimensional trivial ideal T inside A_p for which T*A_p=T, while the left series is just the ideal-power filtration of a nilpotent commutative algebra and terminates. The construction is uniform and yields finite irretractable Yang–Baxter solutions with powerful permutation groups, so the failure is not an isolated curiosity.

What carries the argument

The key mechanism is the kernel-graph construction (Proposition 2.2): given a finite-dimensional nilpotent commutative F_p-algebra J, a derivation D with D^2=0 and (D(J))^2=0, and a (1+D)-invariant character ℓ on the principal-unit group U=1+J, the set G_ℓ={M_u Φ^{-ℓ(u)} : u∈U} is a subgroup of GL(B) that acts regularly on J; identifying U with J yields a left brace with λ_a(b)=(1+a)(1-ℓ(1+a)D)(b). Together with Lemma 2.3, the powerfulness criterion, this gives control over the multiplicative group: if φ=1+D acts trivially on U/U^p and the generators of U^p lie in the kernel of ℓ, then G_ℓ is powerful. The specific algebra B then supplies the persistent ideal T, whose products under ∗ reprod

What would settle it

Compute a Gröbner basis for the ideal (xy, x^{p+1}-y^{p+1}) in F_p[x,y] for a small odd prime such as p=3, and check whether the standard monomials are exactly 1,x,...,x^p,y,...,y^p,z and whether the product x^p*x equals z in the quotient. If z can be expressed as a combination of x^p and y^p, or if J^{p+2}≠0, then the dimension claim fails and with it the entire construction; conversely verifying these identities for p=3 would confirm the paper's core computational claim.

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Extended reading notes

Core claim

The paper establishes a uniform counterexample to the conjecture that finite braces of abelian type with powerful multiplicative group must be right nilpotent. For each odd prime p, the brace A_p is built from the local algebra B = F_p[x,y]/(xy, x^{p+1}-y^{p+1}) with radical J. The derivation D defined by D(x)=y^p, D(y)=-x^p satisfies D^2=0 and (D(J))^2=0, so φ=1+D is an automorphism of order p. A carefully chosen φ-invariant character ℓ of the principal-unit group U=1+J, taking value 1 on 1+z with z=x^{p+1}=y^{p+1}, produces a regular affine subgroup via the kernel-graph construction; the resulting brace has operation a∘b = a+(1+a)φ^{-ℓ(1+a)}(b). The paper shows that G_p=(A_p,∘) satisfies G

Load-bearing premise

The whole construction rests on the claim in Lemma 3.1 that the quotient algebra B has basis 1,x,...,x^p,y,...,y^p,z with z=x^{p+1}=y^{p+1} a new element independent of x^p and y^p, so that dim J=2p+1 and J^{p+2}=0; if that dimension were wrong or z collapsed into the span of x^p and y^p, the brace's order, the ideal T, and the invariant character would all fail.

Editorial extensions

If this is right

The conjecture that a finite skew brace of abelian type with powerful multiplicative group must be right nilpotent is false in every odd characteristic.
Right nilpotence is not closed under extensions: the brace A_p contains an ideal T such that both T and A_p/T are right nilpotent but A_p is not.
The simultaneous conditions cl(G)=2, exp(G)=p^2, and |G'|=p^2 are insufficient to force right nilpotence.
For every odd prime p there exists a finite irretractable involutive set-theoretic solution of the Yang–Baxter equation of cardinality p^{2p+1} whose permutation group is powerful of class two and exponent p^2.
The examples have order p^{2p+1}, so they lie outside the previously known positive range p>n+1.

Reading between the lines

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

If the monomial-basis computation is correct, the same kernel-graph mechanism may yield counterexamples in other families, for instance by varying the relation x^{p+1}-y^{p+1} to higher powers or adding more variables, potentially approaching the paper's open question of the minimal order of such a brace.
The explicit failure of extension closure suggests that right nilpotence is sensitive to the entire multiplicative action; testing whether stronger conditions on the brace's own products (such as requiring the right series to vanish after one step for all ideals) restore closure would clarify the boundary.
For p=2, where powerfulness means G'≤G^4, the construction does not directly apply; testing the analogous algebra over F_2 would either produce a dyadic counterexample or reveal a genuinely different obstruction.
The construction may be modifiable to yield braces with nonzero socle while retaining non-right-nilpotence, which would test the role of trivial socle in the irretractability of the associated Yang–Baxter solutions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The short version: this paper settles the Shalev–Smoktunowicz conjecture for odd primes in the negative, with a single uniform construction that is explicit enough to check by hand. I went through the main computations (the Gröbner basis in Lemma 3.1, the derivation identities, the invariant character, the regular subgroup, the persistent ideal T, the left series, Soc=0) and they hold. The stress-test note's verification of the S-polynomial for xy and x^{p+1}-y^{p+1} is right, and the later claims follow. The reader's ACCEPT verdict is fair.

What's new: previous non-right-nilpotent braces of order p^4 had non-powerful multiplicative groups for odd p; positive results in [8] and [14] live in complementary regimes (large characteristic, low class). This family has multiplicative group powerful of class two with derived subgroup C_p^2, exponent p^2, yet right series does not terminate. The obstruction is a three-dimensional trivial ideal T with T*A_p=T, and both T and A_p/T are right nilpotent, so extension closure fails. That's a clean and surprising package.

Soft spots, all minor. Lemma 3.1's Gröbner basis proof is a two-sentence sketch; it's correct, but a referee will want a sentence showing the remaining S-polynomials reduce to zero. Proposition 7.4's annihilation statement is also compressed. The paper explicitly leaves p=2 and minimal order open; that's honest, not a flaw. The AI-assistance declaration looks standard and the author states all claims were verified.

Citation pattern is appropriate. The Oberwolfach report and Vendramin's survey are the right places for the conjecture; the positive results are cited and contextualized. No self-citation inflation.

Who is this for: people working on braces, skew braces, powerful p-groups, and set-theoretic Yang–Baxter solutions. It also gives a nice example of extension failure for a nilpotence property, useful beyond the immediate topic.

Recommendation: send this to a serious referee. It deserves a regular review, not a desk reject. I'd ask the referee to expand the Gröbner basis verification and double-check the quotient brace in Corollary 7.5; otherwise accept.

Best,

Referee Report

0 major / 4 minor

Summary. For each odd prime p, the paper constructs a finite left brace A_p of order p^{2p+1} whose additive group is elementary abelian and whose multiplicative group G_p is powerful with G_p' = G_p^p ≅ C_p^2, cl(G_p) = 2, exp(G_p) = p^2, and Soc(A_p) = 0. The brace is left nilpotent but not right nilpotent: it contains a 3-dimensional trivial ideal T with T ∗ A_p = T, and both T and A_p/T are right nilpotent, so right nilpotence is not extension-closed. The construction starts from the local algebra B = F_p[x,y]/(xy, x^{p+1}-y^{p+1}), a square-zero derivation D, and an invariant character ℓ, producing a regular affine subgroup. This disproves the Shalev–Smoktunowicz conjecture in every odd characteristic and yields finite irretractable involutive Yang–Baxter solutions whose permutation groups are powerful p-groups of class two.

Significance. This is a substantial negative answer to a conjecture recorded in [1] and [10]. The proof is self-contained and highly explicit: the Gröbner basis computation, derivation identities, group invariants, and the persistent right-series ideal are all concrete and checkable by hand. The construction is uniform in p and uses no numerical or black-box steps. I verified the central computations: the basis and dimension in Lemma 3.1, D^2 = 0 and D(a)a = 0 in Lemma 3.2, U^p = 1+W and U ≅ C_{p^2}^2 × C_p^{2p-3} in Lemma 4.1, the invariant character in Lemma 4.2, and the commutator/power computations in Proposition 6.1. The paper is a strong contribution to the theory of braces and set-theoretic Yang–Baxter solutions.

minor comments (4)

[Lemma 3.1] The Gröbner basis argument is correct but compressed into two sentences. Since every later computation (dimension of J, J^{p+1} = F_p z, J^{p+2} = 0, and the basis of U) depends on this lemma, please expand the S-polynomial reduction: show that the S-polynomial of xy and x^{p+1}-y^{p+1} reduces to y^{p+2}, and that all remaining S-polynomials reduce to zero.
[§2.2, Proposition 2.2] Immediately after defining the affine action on J, the text says 'Since G_ℓ acts regularly, the restriction of this action to G_ℓ is faithful.' At that point regularity has not yet been proved; the following paragraph proves the orbit map is bijective. Reorder the argument so that regularity is established before the faithfulness statement, or phrase it as 'the orbit map below shows regularity, hence faithfulness.'
[§6, Proposition 6.1] In the commutator calculation, the notation g_x and g_y should be defined explicitly, via the identification a ↦ g_{1+a} in Definition 5.1. It is because ℓ(1+x) = ℓ(1+y) = 0 that g_x = M_{1+x} and g_y = M_{1+y}; this is worth stating to avoid ambiguity.
[§9, Corollary 9.1] The sentence 'Since the solution has more than one element and its first retraction is itself, it is not a multipermutation solution' is terse. A one-sentence reminder that a non-singleton irretractable solution is not a multipermutation solution would help readers not immersed in the Yang–Baxter terminology.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is a self-contained construction whose conclusions are proven from explicit algebra, not fitted or assumed.

full rationale

The paper builds an explicit counterexample rather than deriving a prediction from data or from an assumed ansatz. The central construction is transparent: B = F_p[x,y]/(xy, x^{p+1}-y^{p+1}) is fixed, Lemma 3.1 proves the monomial basis via a Gröbner-basis argument, Lemma 3.2 verifies the square-zero derivation D, and Lemma 4.2 constructs the invariant character ℓ by prescribing values on three classes shown to be linearly independent in U/U^p. The brace A_p is then defined by specializing the general kernel-graph Proposition 2.2. The non-right-nilpotence conclusion is a theorem: Proposition 7.2 and Theorem 7.3 prove that the three-dimensional ideal T satisfies T*A_p = T and that T is contained in every term of the right series, so the series cannot terminate. This is a proof, not a fitted claim. The Shalev–Smoktunowicz conjecture is used only as the external statement being disproved; no load-bearing argument cites the author's own prior work, and the reference list contains no self-citations. The free choices in the construction (the character values) are explicitly shown to be realizable from the algebra structure. Overall, the derivation chain is self-contained and exhibits no reduction of a conclusion to an input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The construction is explicit and self-contained: the algebra B, the derivation D, the automorphism phi = 1+D, and the character ell are all built in the paper, and the existence of ell with the needed prescribed values is proven in Lemma 4.2, not assumed. No numerical data are fitted, no computations are outsourced to a black box, and no new unproved objects are postulated. The central claim rests on the standard background results listed above plus the paper's own lemmas, all of which I read as correct.

assumptions (4)

standard math Burnside basis theorem: d(G) = dim_Fp G/Frat(G) = dim_Fp G/(G^p G') for finite p-groups
Used in Lemma 2.3 and Proposition 6.1 to compute d(G) = 2p-1 and to control G^p and G' through the quotient P/U^p =~ (U/U^p) x C_p.
standard math Groebner basis criterion: if all S-polynomials of a generating set reduce to zero, the set is a Groebner basis with the corresponding standard monomials
Invoked in Lemma 3.1 to justify the monomial basis of B, on which the order p^{2p+1}, the subspace W, the distinct element z, and J^{p+2} = 0 all depend. The S-polynomial verification is only sketched in the paper.
domain assumption Every left brace with abelian additive group determines a non-degenerate involutive set-theoretic solution of the Yang-Baxter equation, and Soc(A) = ker(lambda) controls retractability
Cited from Rump [7] and Guarnieri-Vendramin [3], used in Section 9 for Corollary 9.1 (irretractability, multipermutation level, permutation group identification).
domain assumption Standard brace background: lambda is a homomorphism (A,o) -> Aut(A,+); an ideal is a lambda-invariant additive subgroup that is normal in (A,o); left and right series behave as stated
Used throughout Sections 2.1, 7, and 8. The regular-subgroup-to-brace construction itself is proved inside the paper (Lemma 2.1), so it is not external.

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Pith. "Pith review of Powerful multiplicative groups do not force right nilpotence in finite braces." pith.science (2026). https://pith.science/paper/XHZPKBS5

@misc{pith2026260801884,
  author       = {Pith},
  title        = {Pith review of: Powerful multiplicative groups do not force right nilpotence in finite braces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XHZPKBS5}},
  note         = {Machine review of arXiv:2608.01884}
}

abstract

For every odd prime $p$, we construct a finite left brace $A_p$ of order $p^{2p+1}$ whose additive group is elementary abelian and whose multiplicative group $G_p$ is a powerful $p$-group, but such that $A_p$ is not right nilpotent. More precisely, $G_p'=G_p^p\cong C_p^2$, $\operatorname{cl}(G_p)=2$, $\operatorname{exp}(G_p)=p^2$, and $\operatorname{Soc}(A_p)=0$. Thus powerfulness does not force right nilpotence even for multiplicative groups of class two with derived subgroup of order $p^2$. The obstruction is explicit: $A_p$ contains a three-dimensional trivial ideal $T$ satisfying $T*A_p=T$, whereas its left series is the ideal-power filtration of a nilpotent commutative algebra. Both $T$ and $A_p/T$ are right nilpotent, so right nilpotence of finite left braces is not closed under extensions. The construction is uniform and arises from a finite local commutative algebra, a square-zero derivation, and an invariant character, yielding a regular affine subgroup as the kernel of a character on a semidirect product. It disproves the Shalev--Smoktunowicz conjecture in every odd characteristic and yields finite irretractable involutive set-theoretic solutions of the Yang--Baxter equation whose permutation groups are powerful $p$-groups of class two.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [8]

Shalev and A

A. Shalev and A. Smoktunowicz,Groups, Lie rings and braces, Bull. Belg. Math. Soc. Simon Stevin31(2024), no. 4, 422–433

2024

[2] [14]

Ballester–Bolinches, L

A. Ballester–Bolinches, L. A. Kurdachenko, and V. Pérez–Calabuig,On left nilpotent skew braces of class 2, arXiv preprint arXiv:2505.07115 [math.GR] (2025)

arXiv 2025

[3] [1]

Brzeziński, I

T. Brzeziński, I. Colazzo, A. Doikou, and L. Vendramin,Mini-workshop: Skew braces and the Yang–Baxter equation, Oberwolfach Rep.20(2023), no. 1, 537–563

2023

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Vendramin,Skew braces: a brief survey, Geometric Methods in Physics XL, Trends in Mathematics, Birkhäuser/Springer, Cham, 2024, pp

L. Vendramin,Skew braces: a brief survey, Geometric Methods in Physics XL, Trends in Mathematics, Birkhäuser/Springer, Cham, 2024, pp. 153–175

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[5] [2]

F. Cedó, T. Gateva-Ivanova, and A. Smoktunowicz,On the Yang–Baxter equation and left nilpotent left braces, J. Pure Appl. Algebra221(2017), no. 4, 751–756

2017

[6] [3]

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] [4]

Jespers, A

E. Jespers, A. Van Antwerpen, and L. Vendramin,Nilpotency of skew braces and multipermutation solutions of the Yang–Baxter equation, Commun. Contemp. Math. 25(2023), no. 9, Paper No. 2250064

2023

[8] [5]

Lubotzky and A

A. Lubotzky and A. Mann,Powerfulp-groups. I. Finite groups, J. Algebra105(1987), no. 2, 484–505

1987