Inverse Baer deformations and finite simple skew braces of nilpotent type
Abstract
Finite simple skew braces with non-abelian additive group remain largely unexplored. In particular, to the best of our knowledge, no finite simple skew brace with non-abelian nilpotent additive group had previously been constructed. We introduce an inverse Baer deformation which produces skew braces of additive nilpotency class at most two from braces of abelian type. Starting from a finite left brace C=(A,⊕,∘) of odd order and a suitable biadditive alternating map κ:A× A→ A, we define x+y=x⊕ y⊕1/2κ(x,y). The resulting skew brace IBκ(C) satisfies Br(IBκ(C))=C and [x,y]+=κ(x,y), and simplicity passes from C to IBκ(C). Applying this construction to simple left braces arising from asymmetric products, for every pair of odd primes p,q with p|(q−1) and every m≥1 we construct a finite simple skew brace Xp,q(m) of order p2m(q−1)+1q such that (Xp,q(m),+)≅ Ep,q(m)× Cq, where Ep,q(m) is an extraspecial p-group of order p2m(q−1)+1 and exponent p. Thus every fixed admissible pair (p,q) yields infinitely many pairwise non-isomorphic finite simple skew braces with non-abelian nilpotent additive group of class two.
Related
- The Left-Nilpotent Residual and Its Supplements in Finite Skew Braces
- Classification of finite left-simple skew braces
- Kernel--wreath constructions and infinite families of finite simple skew braces
- Finite groups with simple or p-nilpotent maximal subgroups
- Holomorphic isomonodromic deformations of Higgs bundles: absolute lifting, spectral flatness, and nilpotent rigidity
BibTeX
@misc{damele2026inverse,
title = {{Inverse Baer deformations and finite simple skew braces of nilpotent type}},
author = {Marco Damele},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24640},
eprint = {2609.24640},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24640},
note = {CC0},
}
RIS
TY - UNPB
TI - Inverse Baer deformations and finite simple skew braces of nilpotent type
AU - Marco Damele
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24640
UR - https://arxiv.org/abs/2609.24640
AB - Finite simple skew braces with non-abelian additive group remain largely unexplored. In particular, to the best of our knowledge, no finite simple skew brace with non-abelian nilpotent additive group had previously been constructed. We introduce an inverse Baer deformation which produces skew braces of additive nilpotency class at most two from braces of abelian type. Starting from a finite left brace C=(A,⊕,∘) of odd order and a suitable biadditive alternating map κ:A× A→ A, we define x+y=x⊕ y⊕1/2κ(x,y). The resulting skew brace IB_κ(C) satisfies Br(IB_κ(C))=C and [x,y]_+=κ(x,y), and simplicity passes from C to IB_κ(C). Applying this construction to simple left braces arising from asymmetric products, for every pair of odd primes p,q with p|(q−1) and every m≥1 we construct a finite simple skew brace X_p,q^(m) of order p^2m(q−1)+1q such that (X_p,q^(m),+)≅ E_p,q^(m)× C_q, where E_p,q^(m) is an extraspecial p-group of order p^2m(q−1)+1 and exponent p. Thus every fixed admissible pair (p,q) yields infinitely many pairwise non-isomorphic finite simple skew braces with non-abelian nilpotent additive group of class two.
ER -