Research On Idempotent Elements In Burnside Ring

Let G be a finite group,we denote the Burnside ring of group G by B(G).This paper mainly focuses on the idempotent element of the Burnside ring.Through the primitive idempotent element formula,(?)we can get the primitive idempotent element of the Burnside ring and we know that the most important thing is to solve the M¨obius functionμ(K,H).The paper is mainly divided into two parts:The first part is the introduction about the group action and G-set,the proof of related theorems,and the definition of the category.This is the preparatory knowledge of the paper,which provides theoretical basis for the proof and calculation in the later parts.Moreover,we reviewed the definition and properties of the Burnside ring B(G).The second part is the main part of the paper,we calculate the primitive idempotents of some special groups,such as Abelian groups,low-order permutation groups,dihedral groups,quaternion group and so on.To calculate the primitive idempotent of B(G),first we find out all the subgroup lattice of these groups.Let X={H|H≤G},thenζ(K,H)=1if K≤H andζ(K,H)=0 if else,then we arrange all subgroups in a matrix in partial order and get an upper triangular matrix.Finally,we find the M¨obius function by using the inverse matrix of this upper triangular matrix.