Classification Of Braces With Additive Group Isomorphic To Z₂×Z₈

A right brace is a non-empty set H with two operations+and(?)such that(H,+)is an abelian group,(H,(?))is a group and(a+b)(?)c=a(?)c+b(?)c-c,for all a,b,c ∈ H.It is denoted as(H,+,(?)).If the above condition(a+b)(?)c=a(?)c+b(?)c-c is replaced by a(?)(b+c)=a(?)b+a(?)c-a,(H,+,(?))is said to be a left brace.For brevity,we call the left brace and right brace together brace.The Yang-Baxter equation is an important equation from theoretical physics.In recent years,The problem of constructing all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation has been reduced to the problem of describing all the braces.In 2015,Bachiller in paper[Classification of braces of order p3,J.Pure Appl.Algebra,219:8(2015),3568-3603.]classified the braces of order p2 and p3.For the brace of order p4,when p is an odd prime number,in 2020,Jie Li,xinyuan Zhang in their thesis classified the braces with additive group isomorphic to Zp2×Zp2 and Zp ×Zp3,respectively.In this thesis,we classify the braces with additive group isomorphic to Z2 × Z23.This article consists of four chapters.The first chapter is introduction,including the research background,research methods and main results of this article.The second chapter is preliminary knowledge,including some concepts and properties of braces.The third chapter introduces the properties of the braces with additive group isomorphic to Z2× Z8.The fourth chapter gives the classification of the braces with additive group isomorphic to Z2 × Z8.