Classification Of Right Braces With Additive Group Isomorphic To Z_p~2×Z_p~2

A right brace is a non-empty set H with two operations+and o such that(H,+)is an abelian group,(H,o)is a group and(a+b)o c=aoc+boc-c,for all a,b,c ∈ H.It is denoted as(H,+,o).Braces were introduced by Rump as a tool in the study of the set-theoretic solutions of the Yang-Baxter equation.The Yang-Baxter equation is an important equation from theoretical physics.It first appeared in the papers of famous physicists Yang Zhenning and Baxter.It is very difficult and meaningful to study the solution of the Yang-Baxter equation,and classification of braces is very important to study the solutions of the Yang-Baxter equation.Therefore,it is very important to study the classification of braces.In this paper,we classify all the right braces with additive group isomorphic to Z_p~2 × Z_p~2,where p is an odd prime number.This article consists of five 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 right braces with additive group isomorphic to Z_p~2 × Z_p~2.The fourth chapter gives the classification of the right braces with additive group isomorphic to Z_p~2 × Z_p~2.The fifth chapter is the summary and prospect.