arXiv subjects
Publications and source records attributed to V. Pérez-Calabuig.
The aim of this short note is to show that the Sylow theorem (respectively Hall theorem) for finite skew braces proved by Truman in arXiv:2606.18414 is a direct consequence of the Sylow structure (resp. Hall structure) of a finite trifactorised group. A Cauchy theorem for finite skew braces naturally emerges.
In our previous work: Adv. Math. 455 (2024), no. 109880, solubility of solutions was introduced as an extension of solubility of skew braces in the classification context of non-degenerate solutions of the Yang-Baxter equation. One of our main results (Theorem C) proved that a skew brace is soluble if, and only if, its associated solution is soluble. A minor step depending on the definition of homomorphism of solutions was overlooked. In this work, proof of Theorem C is repaired by means of a new class of homomorphisms of solutions: i-homomorphisms of solutions. The importance of this new class is twofold: indecomposable solutions are characterised by means of i-simplicity of solutions, and i-kernels of i-homomorphisms generate ideals in structure skew braces of solutions. Hence, solubility of solutions is redefined as an opposite class of indecomposable solutions. The results obtained with this definition improve our previous outcomes: every soluble solution is proved to have a soluble structure skew brace, and consequently, Theorem C still holds. Several results stemming from this new analysis are outlined.
The aim of this paper is to take the study of Dedekind braces, that is, left braces for which every subbrace is an ideal, started in a previous paper, further. Dedekind braces $A$ whose additive group is non-periodic are analysed. We prove sufficient conditions for $A$ to be abelian: it is enough that every element is $2$-nilpotent for the star operation; and, if $A$ is hypermultipermutational, it suffices that the additive group of the socle is torsion-free. Both conditions can be translated in terms of set-theoretical solutions of the Yang-Baxter equation. In addition, we prove a structural theorem for the case of $A$ to be a multipermutational brace of level $2$.
A description of finitely generated left nilpotent braces of class at most two is presented in this paper. The description heavily depends on the fact that if $B$ is left nilpotent of class at most $2$, that is $B^3 = 0$, then $B$ is right nilpotent of class at most $3$, that is $B^{(4)} = 0$. In addition, we construct a free object in the category of finitely generated left nilpotent braces of class at most $2$.
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 fact, we show that $B$ is right nilpotent of class at most $2+mr$, i.e. $B^{(2+mr+1)} = 0$, where $m$ and $r$ are the nilpotency classes of the additive group of $B$ and $B^2$, respectively. If $B$ is of abelian type, then $B$ is actually right nilpotent of class $3$, i.e. $B^{(4)} = 0$, and this bound is best possible.
The main objective of this paper is to deepen the relationship between skew left braces and trifactorised groups that encodes the information about skew left braces, their structure, their quotients, and their homomorphisms.
The aim of this paper is to introduce and study the class of all left braces in which every subbrace is an ideal. We call them Dedekind left braces. It is proved that every finite Dedekind left brace is centrally nilpotent. Structural results about Dedekind left braces and a complete description of those ones whose additive group is elementary abelian are also shown. As a consequence, every hypermultipermutational Dedekind left brace whose additive group is elementary abelian is multipermutational of level $2$. A new class of left braces, the extraspecial left braces, is introduced and plays a prominent role in our approach.
A complete description of all possible multiplicative groups of finite skew left braces whose additive group has trivial centre is shown. As a consequence, some earlier results of Tsang can be improved and an answer to an open question set by Tsang at Ischia Group Theory 2024 Conference is provided.
We show that the number of isomorphism classes of left braces of order 64 with additive group isomorphic to $C_4 \times C_4 \times C_4$ is 1\,515\,429.
We show that the number of isomorphism classes of left braces of order~$64$ with additive group isomorphic to $C_2\times C_2\times C_4\times C_4$ is $10\,326\,821$. This completes the classification of left braces of order~$64$, that turn out to fall into $15\,095\,601$ isomorphism classes.