Every finite abelian group is a subgroup of the additive group of a finite simple left brace☆
Introduction
In order to investigate a question posed by Drinfeld [9], Rump, in [11], introduced a new algebraic structure, called a left brace, to study non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation. Using an equivalent formulation, given in [7], a left brace is a set B equipped with two operations + and ⋅ such that is an abelian group, is a group and for all . In [2] it was shown that all non-degenerate involutive solutions on a finite set can be explicitly determined from finite left braces. Hence the study of left braces becomes essential to classify all solutions. Moreover, intriguingly, braces have shown up in several areas of mathematics (see for example the surveys [6], [12]). One of the fundamental problems is to classify the building blocks of all finite left braces, that is, describe all finite simple left braces. It is well-known that the additive Sylow p-subgroup of a finite brace B is a left subbrace of B and, in [3], it has been shown that B is an iterated matched product of all . Rump, in [11], has shown that if , then B is a simple left brace precisely when B has order p; more generally, the conclusion holds if is a finite nilpotent group. In particular, such a brace is trivial, i.e. the two operations + and ⋅ coincide. Recall that Etingof, Schedler and Soloviev [10] have shown that if B is finite, then is a solvable group. Also if , then there are some natural constraints on the order of a finite simple left brace B. For example, Smoktunowicz, in [13], showed that if (with p and q different prime numbers and n, m positive integers), then and for some and . However, these conditions are not sufficient for simplicity of B. For example, there is no simple left brace of order , where p and q are distinct primes and n is the multiplicative order of p in the unit group of (see [3, Remark 5.3]). Bachiller, in [1, Theorem 6.3 and Section 7], produced the first example of a non-trivial finite simple left brace. This initiated a program of constructing and describing finite simple left braces [3], [4]. One of the structural approaches is via the matched product of the Sylow subgroups; this allowed to construct new classes of examples and provided some necessary conditions for simplicity. In [8], it has been proven that there is an abundance of finite simple left braces, indeed, for any positive integer and distinct prime numbers , , there exist positive integers , such that, for each n-tuple of integers , there exists a simple left brace of order that has a metabelian multiplicative group with abelian Sylow subgroups. The construction of these simple braces is via asymmetric products, as introduced by Catino, Colazzo and Stefanelli in [5]. This not only provided constructions of new classes of simple left braces, but also all previously known constructions have been interpreted as asymmetric products. Furthermore, in [4], a construction is given of finite simple left braces with a multiplicative group that is solvable of arbitrary derived length. In this paper, we focus on the additive group and we discover new examples of finite simple left braces. Our main result reads as follows:
Theorem 1.1 Let A be a finite abelian group. Then A is a subgroup of the additive group of a finite simple left brace B with metabelian multiplicative group with abelian Sylow subgroups. Moreover, the additive group is isomorphic to where , are any distinct prime numbers, are any positive integers, and .
Recall that solvable groups with abelian Sylow subgroups are also called A-groups [14]. In particular, we thus also construct examples that have elements of additive order (for any ). None of the known previous constructions included such elements.
Section snippets
Preliminaries
A left brace is a set B with two binary operations, + and ⋅, such that is an abelian group, is a group, and for every , In any left brace B there is an action , called the lambda map of B, defined by and , for . A trivial brace is a left brace B such that , for all , i.e. all .
A left ideal of a left brace B is a subgroup L of the additive group of B such that , for all and all . An ideal
The construction
Let be an integer. For , let be a prime number and let and be positive integers. We assume that for . Consider the rings and the polynomials for . Note that . Put . Let be the companion matrix of . Note that is invertible in and has order . Let . We define the symmetric bilinear form by for
References (14)
Extensions, matched products and simple braces
J. Pure Appl. Algebra
(2018)- et al.
Solutions of the Yang–Baxter equation associated with a left brace
J. Algebra
(2016) - et al.
Regular subgroups of the affine group and asymmetric product of radical braces
J. Algebra
(2016) Braces, radical rings, and the quantum Yang–Baxter equation
J. Algebra
(2007)A note on set-theoretic solutions of the Yang–Baxter equation
J. Algebra
(2018)- et al.
Iterated matched products of finite braces and simplicity; new solutions of the Yang–Baxter equation
Trans. Am. Math. Soc.
(2018) - et al.
Asymmetric product of left braces and simplicity; new solutions of the Yang–Baxter equation
Commun. Contemp. Math.
(2019)
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The first author was partially supported by the grants MINECO-FEDER MTM2017-83487-P and AGAUR 2017SGR1725 (Spain). The second author is supported in part by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Belgium) G016117. The third author is supported by the National Science Centre grant 2016/23/B/ST1/01045 (Poland).