Abstract: 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[(Γ_p)^×:(Z_pG)^×] 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.

Key words: integral group ring, Z-order, maximal order, kernel group, K₁ group