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Skew brace extensions, second cohomology and complements

Nishant Rathee, Manoj K. Yadav

TL;DR

The paper develops a cohomological and extension-theoretic framework for skew left braces by exploiting the natural semi-direct product Λ_H, establishing a map $H^2_{Sb}(H,I) \to H^2_{Gp}(\Lambda_H, I\times I)$ that is injective under specified action conditions and thereby embedding brace cohomology into group cohomology. It further connects brace cohomology with that of the opposite brace and derives several results on the existence and structure of complements, supplements, Hall ideals, Frattini sub-skew braces, and minimal/split extensions, including several Schur–Zassenhaus-type statements in particular cases. The work also develops a robust set of reduction techniques for split extensions, including Dedekind-like modular laws and Sylow-type decompositions, and shows that cohomological information translates across the Λ_H construction to yield concrete constraints on brace extensions and their generators. Together, these results enhance the understanding of the cohomology, extensions, and decomposition properties of finite skew left braces and their semidirect-product realizations, with implications for classification and structural analysis.

Abstract

We study extensions and second cohomology of skew left braces via the natural semi-direct products associated with the skew left braces. Let $0 \to I \to E \to H \to 0$ be a skew brace extension and $Λ_H$ denote the natural semi-direct products associated with the skew left brace $H$. We establish a group homomorphism from ${\rm H}_{Sb}^2(H, I)$ into ${\rm H}_{Gp}^2(Λ_H, I \times I)$, which turns out to be an embedding when $I \le {\rm Soc}(E)$. In particular the Schur multiplier of a skew left braces $H$ embeds into the Schur multiplier of the group $Λ_H$. Analog of the Schur-Zassenhaus theorem is established for skew left braces in several specific cases. We introduce a concept called minimal extensions (which stay at the extreme end of split extensions) of skew left braces and derive many fundamental results. Several reduction results for split extensions of finite skew left braces by abelian groups (viewed as trivial left braces) are obtained.

Skew brace extensions, second cohomology and complements

TL;DR

The paper develops a cohomological and extension-theoretic framework for skew left braces by exploiting the natural semi-direct product Λ_H, establishing a map that is injective under specified action conditions and thereby embedding brace cohomology into group cohomology. It further connects brace cohomology with that of the opposite brace and derives several results on the existence and structure of complements, supplements, Hall ideals, Frattini sub-skew braces, and minimal/split extensions, including several Schur–Zassenhaus-type statements in particular cases. The work also develops a robust set of reduction techniques for split extensions, including Dedekind-like modular laws and Sylow-type decompositions, and shows that cohomological information translates across the Λ_H construction to yield concrete constraints on brace extensions and their generators. Together, these results enhance the understanding of the cohomology, extensions, and decomposition properties of finite skew left braces and their semidirect-product realizations, with implications for classification and structural analysis.

Abstract

We study extensions and second cohomology of skew left braces via the natural semi-direct products associated with the skew left braces. Let be a skew brace extension and denote the natural semi-direct products associated with the skew left brace . We establish a group homomorphism from into , which turns out to be an embedding when . In particular the Schur multiplier of a skew left braces embeds into the Schur multiplier of the group . Analog of the Schur-Zassenhaus theorem is established for skew left braces in several specific cases. We introduce a concept called minimal extensions (which stay at the extreme end of split extensions) of skew left braces and derive many fundamental results. Several reduction results for split extensions of finite skew left braces by abelian groups (viewed as trivial left braces) are obtained.
Paper Structure (5 sections, 52 theorems, 179 equations)

This paper contains 5 sections, 52 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. From skew braces to groups : extensions and cohomology
  4. Complements and supplements
  5. Reductions for split extension

Key Result

Theorem 1.1

Let a skew left brace $H$ act on an abelian group via an action $(\nu, \mu, \sigma)$. There exists a group homomorphism $\Psi : \operatorname{H} ^2_{Sb}(H, I) \to \operatorname{H} ^2_{Gp}(\Lambda_H, I \times I)$. Moreover, if $\mu$ is trivial and $\nu^{-1}_h = \sigma_h$ for all $h \in H$, then $\Psi

Theorems & Definitions (91)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Remark 3.3
  • ...and 81 more