| Let G be a finite nonabelian group. ZG its associated integral group ring, and△(G) its augmentation ideal. This thesis deals with for the finite nonabelian group of order pk which have a cyclic subgroup of index p and the classical group over fi-nite fields the problem of their nth augmentation ideals△n(G) and quotient groups Qn(G)=△n(G)/△n+1(G).The thesis is composed of five chapters:In Chapter1, we summarize the background of the related issues and state the outline of the present thesis.In Chapter2, we solve for the semidihedral group and another nonabelian2-group the problem of their augmentation ideals△n(G) and quotient groups Qn(G). We obtained a recursive relation between△n(G) and An+1(G) and used this recursive relation as well as inductive method to get an explicit basis of△n(G). Based on this explicit basis of△n(G), we have determined the structure of Qn(G).In Chapter3, we consider the finite nonabelian p-group of order pκ which have a cyclic subgroup of index p, where p≠2, κ≥3. An concrete basis for the augmentation ideal is obtained, so the structure of its quotient groups can be obtained.In Chapter4, we discuss the consecutive quotients Qn(G) of the general linear group, the special linear group, the unitary group, the symplectic group and the orthogonal group over finite fields, therefore completely determine the structure of such consecutive quotient groups.In Chapter5, we describe the explicit generator of the ring of modular vector invari-ants of F[V]Z2. |
