几类非Abel群之增广商群的结构(英文)

On the Structure of the Augmentation Quotient Groups for Some Nonabelian Groups

记ZG为有限群G的整群环,△~n(G)为增广理想△(G)的n次幂,Q_n(G)=△~n(G)/△^(n+1)(G)为G的增广商群.本文考虑了二面体群D_(2tk)(k奇)和m次对称群S_m,证明了Q_n(D_(2tk))为秩不超过2t+1的基本2-群以及Q_n(S_m)≌Z_2.

Let G be a finite group, ZG its integral group ring and An（G） the nth power of the augmentation ideal △（G）, denote Qn（G） = △^n（G）/△^n＋1（G） the augmentation quotient groups of G. In this paper, dihedral group D2^tk （kodd） and m＇ th symmetric group Sm are considered. We show Qn（D2tk） is an elementary 2-group and its rank is no more than 2t ＋ 1. As for Qn（Sm）, we have Qn（Sm） Z2.