Extensions, matched products, and simple braces
Braces have been introduced by \textit{W. Rump} [J. Pure Appl. Algebra 209, No. 3, 671--685 (2007; Zbl 1170.16031)] as generalisations of radical rings, to produce solutions to the Yang-Baxter equation of theoretical physics. A left brace is a triple \((B, +, \cdot)\) such that \((B, +)\) is an abelian group, \((B, \cdot)\) is a group, and \(a \cdot (b + c) + a = a \cdot b + a \cdot c\) for \(a, b, c \in B\). A subset \(L\) which is a subgroup of \((B, \cdot)\) and such that \(b \cdot x - b \in L\) for \(b \in B\) and \(x \in L\) is called a left ideal of the brace \(B\). If the left ideal \(L\) is a normal subgroup of \((B, \circ)\), it is called an ideal. The goal of the paper under review is to give a construction of the left braces which are extensions of left braces by ideals with a trivial brace structure. Simple left braces enter naturally in this context, very much as they do in ring theory. The author develops techniques that lead to the construction of the first non-trivial examples of finite, simple left braces. These are obtained with the the matched product construction, which allows one to put together two Sylow subgroups for different primes.
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