Related papers: Finite skew braces with characteristically simple …

A skew-morphism of a finite group $G$ is a permutation $\s$ on $G$ fixing the identity element, and for which there exists an integer function $\pi$ on $G$ such that $\s(xy)=\s(x)\s^{\pi(x)}(y)$ for all $x,y\in G$. It has been known that…

Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $G=BY$, where $Y$ is cyclic and core-free…

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over $\Z$ cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers of an infinite order…

We prove two results on some special generators of finite simple groups and use them to prove that every non-abelian finite simple group $S$ admits a non-congruence presentation (as conjectured in [CLT24]), and that if $S$ has a non-trivial…

We introduce two common divisor graphs associated with a finite skew brace, based on its $\lambda$- and $\theta$-orbits. We prove that the number of connected components is at most two and the diameter of a connected component is at most…

We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group $G$ isospectral to a finite simple group has a unique nonabelian composition factor, that is, the…

We investigate two sub-classes of skew bracoids, the first consists of those we term almost a brace, meaning the multiplicative group decomposes as a certain semi-direct product, and then those that are almost classical, which additionally…

According to Letourmy and Vendramin, a representation of a skew brace is a pair of representations on the same vector space, one for the additive group and the other for the multiplicative group, that satisfies a certain compatibility…

A skew brace $A = (A,\cdot,\circ)$ is said to be \textit{left-simple} if $A\neq1$ and it has no left ideal other than $1$ and $A$. The purpose of this paper is to give a partial classification of the finite left-simple skew braces. A result…

One of the major problems in the structural theory of skew braces consists in the classification of skew braces of finite order up to isomorphism. In this light, the open question of the existence of a Cauchy theorem for finite skew braces…

We construct a family of infinite simple groups that we call \emph{twisted Brin-Thompson groups}, generalizing Brin's higher-dimensional Thompson groups $sV$ ($s\in\mathbb{N}$). We use twisted Brin-Thompson groups to prove a variety of…

The main result of the paper is that if $A$ is an abelian variety over a subfield $F$ of ${\bold C}$, and $A$ has purely multiplicative reduction at a discrete valuation of $F$, then the Hodge group of $A$ is semisimple. Further, we give…

Inspired by a conjecture by Guarnieri and Vendramin concerning the number of braces with a generalized quaternion adjoint group, many researchers have studied braces whose adjoint group is a non-abelian $2$-group with a cyclic subgroup of…

We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let $n\geq 5$ be an integer, $G$ a finite group, and let $\AAA$ and $\SSS^\pm$ denote the double…

The aim of this article is to advance the knowledge on the theory of skew left braces. We introduce a subclass of skew left braces, which we denote by $\mathcal{I}_n$, $n \ge 1$, such that elements of the annihilator and lower central…

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 $\Lambda_H$ denote the natural…

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…

This paper examines the connections between (relative) Rota--Baxter groups, skew left braces, and enlargements of these structures on naturally associated semi-direct products. Given a skew left brace, we define a new skew left brace,…

The famous theorem of It\^{o} in group theory states that if a group $G=HK$ is the product of two abelian subgroups $H$ and $K$, then $G$ is metabelian. We shall generalize this to the setting of a skew brace $(A,{\cdot\,},\circ)$. Our main…

We are interested in finite groups acting orientation-preservingly on 3-manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point…