Every finite abelian group is a subgroup of the additive group of a finite simple left brace

https://doi.org/10.1016/j.jpaa.2020.106476Get rights and content

Abstract

Left braces, introduced by Rump, have turned out to provide an important tool in the study of set-theoretic solutions of the quantum Yang–Baxter equation. In particular, they have allowed to construct several new families of solutions. A left brace (B,+,) is a structure determined by two group structures on a set B: an abelian group (B,+) and a group (B,), satisfying certain compatibility conditions. The main result of this paper shows that every finite abelian group A is a subgroup of the additive group of a finite simple left brace B with metabelian multiplicative group with abelian Sylow subgroups. This result complements earlier unexpected results of the authors on an abundance of finite simple left braces.

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 (B,+) is an abelian group, (B,) is a group anda(b+c)+a=ab+ac, for all a,b,cB. 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 Bp 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 Bp. Rump, in [11], has shown that if B=Bp, then B is a simple left brace precisely when B has order p; more generally, the conclusion holds if (B,) 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 (B,) is a solvable group. Also if BpB, then there are some natural constraints on the order of a finite simple left brace B. For example, Smoktunowicz, in [13], showed that if |B|=pnqm (with p and q different prime numbers and n, m positive integers), then p|(qt1) and q|(ps1) for some 0<tm and 0<sn. However, these conditions are not sufficient for simplicity of B. For example, there is no simple left brace of order pnq, where p and q are distinct primes and n is the multiplicative order of p in the unit group of Z/(q) (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 n>1 and distinct prime numbers p1, p2,,pn, there exist positive integers l1,l2,,ln, such that, for each n-tuple of integers m1l1,m2l2,,mnln, there exists a simple left brace of order pm1pm2pmn 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 (B,) that is solvable of arbitrary derived length. In this paper, we focus on the additive group (B,+) 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 (B,+) is isomorphic toiZ/(m)(Z/(pini))2sili1+1, where m>1, p1,,pm are any distinct prime numbers, n1,,nm,s1,,sm are any positive integers, and li=pinipini1.

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 2n (for any n>1). 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 (B,+) is an abelian group, (B,) is a group, and for every a,b,cB,a(b+c)+a=ab+ac. In any left brace B there is an action λ:(B,)Aut(B,+), called the lambda map of B, defined by λ(a)=λa and λa(b)=aba, for a,bB. A trivial brace is a left brace B such that ab=a+b, for all a,bB, i.e. all λa=id.

A left ideal of a left brace B is a subgroup L of the additive group of B such that λa(b)L, for all bL and all aB. An ideal

The construction

Let m>1 be an integer. For iZ/(m), let pi be a prime number and let ni and si be positive integers. We assume that pipj for ij. Consider the rings Ri=Z/(pini) and the polynomialsqi(x)=k=0pi1xkpini1Ri+1[x], for iZ/(m). Note that (xpini11)qi(x)=xpini1. Put li=deg(qi(x)). Let CiMli1(Ri) be the companion matrix of qi1(x). Note that Ci is invertible in Mli1(Ri) and has order pi1ni1. Let Ti=Ri2li1. We define the symmetric bilinear form bi:Ti×TiRi bybi(u,v)=u(0Ili1Ili10)vt, for

References (14)

There are more references available in the full text version of this article.

Cited by (0)

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).

View full text