一类p~3阶群的Burnside环之增广商群

群环理论将群论和环论有机地结合了起来,是代数学中的重要分支之一,其中增广理想和增广商群是群环理论中的一个经典课题.设G有限群,分别记的Burnside环及其增广理想为Ω(G)和Δ(G).本文对任意正整数n,具体构造了Δ~n(I_p)作为自由交换群的一组基,并确定了商群Δ~n(I_p)/Δ~(n+1)(I_p)的结构,其中I_p=〈a,b|a~(p~2)=b~p=1,b~(-1)ab=a~(p+1)〉,p为奇素数.

Augmentation Quotients for Burnside Rings ofSome Finite Groups of Order p3

Group ring theory is an important branch of algebra, which is an important branch of algebra.It is a classical problem in the theory of group rings.Let G be a finite group of the order p3, denote the Burnside ring of G and its augmentation ideal by Ω(G) and Δ(G), respectively.This paper constructs an explicit Z-basis of Δn(Ip) and determines the isomorphism class of the n-th quotient group Δn(Ip)/Δn+1(Ip) for each positive integer n, where Ip=〈a,b|ap2=bp=1, b-1ab=ap+1〉, p is an odd prime.