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K1（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-order Γ. Then we use the properties of the kernel group to study the exponential problem of K1(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.
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