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

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