Kernel--wreath constructions and infinite families of finite simple skew braces

@misc{damele2026kernelwreath,
  title = {{Kernel--wreath constructions and infinite families of finite simple skew braces}},
  author = {Marco Damele},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20401},
  eprint = {2609.20401},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20401},
  note = {CC0},
}

TY  - UNPB
TI  - Kernel--wreath constructions and infinite families of finite simple skew braces
AU  - Marco Damele
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20401
UR  - https://arxiv.org/abs/2609.20401
AB  - We introduce a kernel--wreath construction for finite skew braces. Let C be a finite skew brace, write (C,+)=A and (C,∘)=R, and let χ:R↠ T be an epimorphism onto a non-trivial finite abelian group. We show that a natural index-|T| kernel in the permutational wreath product Rwr T acts regularly on A^T, and hence defines a new skew brace KT_T(C,χ) with additive group A^|T|. The construction preserves every finite abelian quotient of the multiplicative group, as well as solvability, and contains the seed brace C as a diagonal subbrace. It can therefore be iterated indefinitely. More precisely, if Q is a non-trivial finite abelian quotient of R and p∈π(Q), then one obtains an infinite tower C=C_0↪ C_1↪ C_2↪⋯ with (C_m,+)≅ A^p^m for every m≥0. Our main permanence result shows that if C is simple and is not a trivial skew brace, then KT_T(C,χ) is again simple. Consequently, a single simple seed whose multiplicative group has a non-trivial finite abelian quotient gives rise to infinite families of finite simple skew braces. In particular, starting from suitable solvable regular subgroups of the holomorph of a finite non-abelian simple group S, we obtain infinite families of simple skew braces with additive groups S^p^m and solvable multiplicative groups. Further applications are given to the simple skew braces of order $12$ and to Byott's family.
ER  -