Related papers: Skew braces with no proper left ideals
Let $n\geq 1$ be an integer, $p$, $q$ be distinct odd primes. Let ${G}$, $N$ be two groups of order $p^nq$ with their Sylow-$p$-subgroups being cyclic. We enumerate the Hopf-Galois structures on a Galois ${G}$-extension, with type $N$. This…
We discuss the (first) Sylow theorem for certain classes of finite skew braces, proving it to hold true when the skew brace is two-sided, bi-skew, right nilpotent, $\lambda$-homomorphic or supersoluble. We also show it to hold true for…
The study of non-degenerate set-theoretic solutions of the Yang-Baxter equation calls for a deep understanding of the algebraic structure of a skew left brace. In this paper, the skew brace theoretical property of solubility is introduced…
Following recent work by Ghomi, Solomon and Tabachnikov, we study geometry and topology of skew branes. A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any pair of distinct points are not…
We give a self-contained proof that a skew left brace yields a solution of the Yang-Baxter equation.
Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, $\mathrm{SupLat}$, of…
Let $G$ and $N$ be two finite groups of the same order. It is well-known that the existences of the following are equivalent: (a) a Hopf-Galois structure of type $N$ on any Galois $G$-extension; (b) a skew brace with additive group $N$ and…
L. Childs has defined a skew brace $(G, \cdot, \circ)$ to be a bi-skew brace if $(G, \circ, \cdot)$ is also a skew brace, and has given applications of this concept to the equivalent theory of Hopf-Galois structures. The goal of this paper…
In this paper we enumerate the skew braces of size $p^2q$ for $p,q$ odd primes by the classification of regular subgroups of the holomorph of the groups of size $p^2q$. In particular, we provide explicit formulas for the skew braces of…
In this paper we investigate left ideals as codes in twisted skew group rings. The considered rings, which are often algebras over a finite field, allows us to detect many of the well-known codes. The presentation, given here, unifies the…
Our primary focus is on the theory of skew braces, specifically exploring their connection with combinatorial solutions to the Yang-Baxter equation. Skew braces have recently emerged as intriguing algebraic structures, and their link to the…
Braces were introduced by Rump to study involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. A constructive method for producing all such finite solutions from a description of all finite left braces has been…
In this short note, we construct some nontrivial examples of topological biquandle. The key ingredient of the construction is the notion of skew brace.
The notion of a pre-truss, that is, a set that is both a heap and a semigroup is introduced. Pre-trusses themselves as well as pre-trusses in which one-sided or two-sided distributive laws hold are studied. These are termed near-trusses and…
The main objective of this article is to initiate a detailed structure theory of left nilpotent skew braces $B$ of class $2$, i.e. skew braces with $B^3 = 0$. We prove that if $B$ is of nilpotent type, then $B$ is centrally nilpotent. In…
Skew braces are algebraic structures related to the solutions of the set-theoretic quantum Yang-Baxter equation. We develop the central nilpotency theory for such algebraic structures in the sense of Freese-McKenzie \cite{comm} and we…
In order to study two-sided skew braces, we introduce the notion of weakly trivial skew braces. We give a classification of such skew braces and show that they are essential in the study of two-sided skew braces. As an application, we…
We define the twofold semidirect product of two skew left braces, in which both the additive and multiplicative groups are semidirect products of the corresponding groups of the given skew left braces. We consider an odd prime $p$ and an…
We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding left braces. Besides we determine the separable field extensions of degree twice an odd prime square…
In this paper, we answer negatively to a question posed in the context of the 2025 Oberwolfach Mini-Workshop ``The Yang-Baxter Equation and Representations of Braid Groups'' regarding the existence of split extensions classifiers in the…