Finite Group On The Integral Group Ring Zg Relative K <sub> 1 </ Sub> Group

Integral group ring is one of the most important rings . Its K- groups is an attractive topic in algebraic K -theory. When people talk about the structure of ZG and its K -groups, they often connect them with ZG 's maximal Z -order г inQG. In Chapter I , some of the knowing results about г have been mentioned,however all these results haven't given a detailed form of the elements in ZG. In Chapter II and Chapter III , we have found the detailed form for г when G is an finite cyclic group or G is an elementary p-group. In Chapter IV , based on the results of Chapter II and Chapter III, we have proved that the classicalhomomorphism ε₁ : K₁ (ZG, nг)â†' K₁ (г, nг) is an isomorphism.