Extensions, matched products, and simple braces

TL;DR

This paper develops methods to construct all extensions of left braces by ideals, introduces the concept of simple left braces, and provides the first non-trivial examples of finite simple left braces using matched products.

Contribution

It introduces the first non-trivial finite simple left braces and defines the matched product to construct and analyze extensions of left braces.

Findings

Explicit construction of finite simple left braces

Introduction of matched product method

New examples of left braces

Abstract

We show how to construct all the extensions of left braces by ideals with trivial structure. This is useful to find new examples of left braces. But, to do so, we must know the basic blocks for extensions: the left braces with no ideals except the trivial and the total ideal, called simple left braces. In this article, we present the first non-trivial examples of finite simple left braces. To explicitly construct them, we define the matched product of two left braces, which is a useful method to recover a finite left brace from its Sylow subgroups.