  Braces were introduced by Rump to study involutive non-degenerate
set-theoretic solutions of the Yang-Baxter equation. A constructive method for
producing all such finite solutions from a description of all finite left
braces has been recently discovered. It is thus a fundamental problem to
construct and classify all simple left braces, as they can be considered as
building blocks for the general theory. This program recently has been
initiated by Bachiller and the authors. In this paper we study the simple
finite left braces such that the Sylow subgroups of their multiplicative groups
are abelian. We provide several new families of such simple left braces. In
particular, they lead to the main, surprising result, that shows that there is
an abundance of such simple left braces.
