Powerful multiplicative groups do not force right nilpotence in finite braces
Abstract
For every odd prime p, we construct a finite left brace Ap of order p2p+1 whose additive group is elementary abelian and whose multiplicative group Gp is a powerful p-group, but such that Ap is not right nilpotent. More precisely, Gp'=Gpp≅ Cp2, cl(Gp)=2, exp(Gp)=p2, and Soc(Ap)=0. Thus powerfulness does not force right nilpotence even for multiplicative groups of class two with derived subgroup of order p2. The obstruction is explicit: Ap contains a three-dimensional trivial ideal T satisfying T*Ap=T, whereas its left series is the ideal-power filtration of a nilpotent commutative algebra. Both T and Ap/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 group homomorphism, yielding a regular affine subgroup as the kernel of a homomorphism on a semidirect product. It disproves the Shalev--Smoktunowicz conjecture in every odd characteristic and yields finite non-degenerate irretractable involutive set-theoretic solutions of the Yang--Baxter equation whose permutation groups are powerful p-groups of class two.
Related
- Determinant characters and multiplicative solubility of finite skew braces
- Solvability and nilpotence of ind-groups via tangent curves
- Solvability and nilpotence of ind-groups via tangent curves
- Cofinite integral braces and crystallographic groups
- Bisoluble and Binilpotent Skew Braces of Finite Morley Rank
BibTeX
@misc{verbeken2026powerful,
title = {{Powerful multiplicative groups do not force right nilpotence in finite braces}},
author = {Brecht F. Verbeken},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.01884},
eprint = {2608.01884},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01884},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Powerful multiplicative groups do not force right nilpotence in finite braces
AU - Brecht F. Verbeken
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2608.01884
UR - https://arxiv.org/abs/2608.01884
AB - 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≅ C_p^2, cl(G_p)=2, exp(G_p)=p^2, and 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 group homomorphism, yielding a regular affine subgroup as the kernel of a homomorphism on a semidirect product. It disproves the Shalev--Smoktunowicz conjecture in every odd characteristic and yields finite non-degenerate irretractable involutive set-theoretic solutions of the Yang--Baxter equation whose permutation groups are powerful p-groups of class two.
ER -