交换p群的整群环以及它的极大序的K1群

The K1 group of integral group ring and its maximal order for a commutative p group

整群环是代数乃至许多数学分支中很重要的一类环,也是代数K理论主要的研究对象之一。对几类交换p(p为素数)群G的整群环ZG,作为半单代数QG的一个Z-序,通过将其嵌入到极大Z-序Γ之中,然后利用核群的性质研究K 1(ZG)在K1(Γ)中的指数问题。主要结果有:首先对几类交换p群G给出[(Γp)×∶(ZpG)×]的确切表达式,然后用来确定[K1(Γ)∶K1(ZG)]的具体数值。

Group rings are very important rings in algebra and many other branches of mathematics.It is also one of the main research subjects of algebraic K-theory.In this work,we mainly deal with integral group rings ZG for some abelian p(p is prime)groups G.We can regard ZG as a Z-order of the semi-simple algebra QG and embed it into the maximal Z-orderΓ.Then we use the properties of the kernel group to study the exponential problem of K 1(ZG)in K1(Γ).In this paper,there are two main results.First,the explicit formula of[(Γp)×∶(ZpG)×]is obtained for some abelian p groups.Secondly,by using the formula,we get the specific result of[K1(Γ)∶K1(ZG)]for some abelian p groups.