Classification Of Right Braces With Additive Group Isomorphic To Z_p×Z_p~3

A right brace is a non-empty set B with two operations+ and(?)such that(B,+)is an abelian group,(B,(?))is a group and(a+b)(?)c=a(?)c+b(?)c-c,for all a,b,c?B.It is denoted as(B,+,(?)).If the above condition(a+b)(?)c=a(?)c+b(?)c-c is replaced by a(?)(b+c)=a(?)b+a(?)c-a,(B,+,(?))is said to be a left brace.Braces were introduced by Rump as a tool in the study of the set-theoretic solutions of the Yang-Baxter equation.The problem of constructing all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation recently has been reduced to the problem of describing all the left braces.In particular,the classification of all finite left braces is fundamental in order to describe all finite such solutions of the Yang-Baxter equation.In fact,there is a one-to-one relationship between left braces and right braces.Therefore,the classification of right braces is of great significance for studying the solutions of Yang-Baxter equation.In this master's thesis,we classify all the right braces with additive group isomorphic to Zp×Zp3,where p is an odd prime number.A total of 3/2p2+6p+25/2 braces,which are not isomorphic to each other,are obtained.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.,The third chapter introduces the properties of the right braces with additive group isomorphic to Zp × Zp3.The fourth chapter gives the classification of the right braces with additive group isomorphic to Zp×Zp3.The fifth chapter is summary and prospect.