Skew Braces of Size p2 q I: Abelian Type

Abstract

This is the first part of a series of two articles. In this paper we enumerate and classify the left braces of size [Formula: see text], where[Formula: see text] and [Formula: see text] are distinct prime numbers, by the classification of regular subgroups of the holomorph of the abelian groups of the same order. We also provide the formulas that define the constructed braces.

Indecomposable solutions of the Yang-Baxter equation with permutation group of sizes $pq$ and $p^2q$

“…In order to classify all solutions with a given permutation group, we must then understand all possible braces with that multiplicative structure. For this we refer to recent work classifying braces, and more general skew-braces, of fixed sizes [1,2,15,4]. We are going to be refering mainly to the results of Acri and Bonato in [1] and [2], as they give explicit formulas for the structure of the all the braces we require.…”

“…For this we refer to recent work classifying braces, and more general skew-braces, of fixed sizes [1,2,15,4]. We are going to be refering mainly to the results of Acri and Bonato in [1] and [2], as they give explicit formulas for the structure of the all the braces we require.…”

“…Now we turn to study braces of size p 2 q and start by considering those whose multplicative group is cyclic. We refer to [2] for the enumaration and explicit descriptions of the braces. 4.1.…”

“…Case p = 2. In [2] the analysis is split into the cases q ≡ 1 (mod 4) and q ≡ 3 (mod 4), however the braces with abelian permutation group can be described uniformly for both cases so we make no such distinction here.…”

“…By Theorem 3.1 there is a single indecomposable solution for this brace. For the other solution we identify the brace with its additive group Z 2 2 × Z q . The multiplicative structure is given by…”