Carter-like theorems for skew braces

Abstract

Analogues of the Sylow, Hall and Schur-Zassenhaus theorems for finite groups have recently been generalised to finite skew braces, providing a powerful framework for their structural analysis. Building on this foundation, the natural next step of skew braces is the study of Carter sub-skew braces, namely, centrally nilpotent, self-idealising sub-skew braces. However, while every finite soluble group possesses Carter subgroups, not every finite soluble skew brace possesses Carter sub-skew braces. We introduce Carter sub-skew braces, analyse their structural limitations and establish Carter-like theorems. We prove that finite soluble skew braces have Carter sub-skew braces if every sub-skew brace of prime-power order is a centrally nilpotent left ideal. We also prove their existence within finite multipermutational skew braces.

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BibTeX RIS

BibTeX

@misc{campmora2026carterlike,
  title = {{Carter-like theorems for skew braces}},
  author = {Sergio Camp-Mora and J. Pérez-Alemany and P. Pérez-Altarriba},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39430},
  eprint = {2609.39430},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39430},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Carter-like theorems for skew braces
AU  - Sergio Camp-Mora
AU  - J. Pérez-Alemany
AU  - P. Pérez-Altarriba
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39430
UR  - https://arxiv.org/abs/2609.39430
AB  - Analogues of the Sylow, Hall and Schur-Zassenhaus theorems for finite groups have recently been generalised to finite skew braces, providing a powerful framework for their structural analysis. Building on this foundation, the natural next step of skew braces is the study of Carter sub-skew braces, namely, centrally nilpotent, self-idealising sub-skew braces. However, while every finite soluble group possesses Carter subgroups, not every finite soluble skew brace possesses Carter sub-skew braces. We introduce Carter sub-skew braces, analyse their structural limitations and establish Carter-like theorems. We prove that finite soluble skew braces have Carter sub-skew braces if every sub-skew brace of prime-power order is a centrally nilpotent left ideal. We also prove their existence within finite multipermutational skew braces.
ER  -

Abstract

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