Relative K, Of The Integral Group Ring Of Finite Commutative Groups <sub> 1 </ Sub> Group

The integral group rings of finite abelian groups are of great importance in algebraic k-theory. Suppose G is a finite group, then QG is a semisimple Q-algera, obviously ZG is a Z-order of QG,we denote the maximal Z-order by ? .If G is not abelian, it is not an easy thing to determine г; if G is an abelian group, then QG is isomorphic to the direct sum of a finite number of number fields, and r is the direct sum of these rings of algebraic integers of those number fields, but which elements of QG belong to r is not clear. The main results in this paper are listed as follows:1. When G is a cyclic group of order p(a prime number), the explicitelements of г are determined. When G is the nonabelian group D₃,the maximal Z-order of QG which contains ZG is also given .2. When G is a finite abelian group, we successfully provedK₁ (ZG,|G| г) = K₁ (T,|G| г) after getting a structure theorem for QG.3. When G is a finite abelian group, the abelian group K₁,(г,|G|г) iscalculated completely mainly using the Bass-Milnor-Serre theorem, then the structure of K₁{ZG,|G|г) is clear. When G is a cyclic group of order 2 or 3, K₂(ZG/|G|г ) = 1 is proved, and the result needs to beimproved to be more helpful to the calculation of the K₂ groups of some other rings.