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R. Esteban-Romero

Publications and source records attributed to R. Esteban-Romero.

12 recordsLinked to original sources
arXiv2026

Analogues of Grün's lemma and Baer's theorem for skew left braces

We prove in this paper some analogues of the well-known group-theoretical Grün's lemma, stating that in a perfect group the first and the second centre coincide, and Baer's theorem, stating that if the quotient by the nth centre of a group is finite, then so is the $(n + 1)$th term of the lower central series, in the scope of nfinite slew left braces. These results represent significant improvements over previous work. The trifactorised group associated with a skew left brace will be crucial for our proofs.

math.GR↗
arXiv2026

Representations of finite skew braces

One of the classical open problems in the theory of skew left braces is the study of their representation theory. We propose in this paper a definition of representation of a skew left brace and study its properties. Representations of the trifactorised groups associated with skew left braces play a fundamental role.

math.GR↗
arXiv2026

Addendum/Corrigendum to "On solubility of skew left braces and solutions of the Yang-Baxter equation"

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.

math.GR↗
arXiv2026

On left braces in which every subbrace is an ideal II

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 $2$.

math.GR↗
arXiv2025

On products of abelian skew braces

The main objective of this paper is to study factorisations of skew left braces through abelian subbraces. We prove a skew brace theoretical analog of the classical Itô's theorem about product of two abelian groups: if $B = A_1A_2$ is a skew brace which is the product of two abelian skew subbraces $A_1$ and $A_2$, and $A_1$ is a left and right ideal of $B$, then the commutator ideal $[B, B]^B$ of $B$ is an abelian brace. If $A_1$ is a left (non-necessarily right) ideal of $B$, we show that there exists a strong left ideal of $B$ contained in $A_1$ or $A_2$. We also show factorisations of relevant ideals of factorised braces that are sums and products of abelian subbraces.

math.GR↗
arXiv2025

Categories of skew left braces and trifactorised groups

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.

math.GR↗
arXiv2024

On left braces in which every subbrace is an ideal

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.

math.GR↗
arXiv2024

Constructing skew left braces whose additive group has trivial centre

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.

math.GR↗
arXiv2022

Maximal subgroups of small index of finite almost simple groups

We prove in this paper that a finite almost simple group $R$ with socle the non-abelian simple group $S$ possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index $\operatorname{l}(S)$ of a maximal group of $S$ or a conjugacy class of core-free maximal subgroups with a fixed index $v_S \leq {\operatorname{l}(S)^2}$, depending only on $S$. We show that the number of subgroups of the outer automorphism group of $S$ is bounded by $\log^3 {\operatorname{l}(S)}$ and $\operatorname{l}(S)^2 < |S|$.

math.GR↗
arXiv2022

Bounds on the number of maximal subgroups of finite groups

In this paper we obtain significant bounds for the number of maximal subgroups of a given index of a finite group. These results allow us to give new bounds for the number of random generators needed to generate a finite $d$-generated group with high probability.

math.GR↗