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

2020/01/24 by Ferran Cedó, Cedo, F., Eric Jespers +3
Mathematics · #16T25 #20E22 #20F16 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · doi:10.48550/arxiv.2001.08905

openalex publication_date 2020/01/24 · openalex created_date 2020/01/30 · openalex updated_date 2026/07/28

Abstract

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

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