令G为一个有限交换群，它的整群环ZG为QG中的一个Z-序。令Γ为QG中包含ZG的极大Z-序，对几类有限交换群G，计算K1（ZG）在K1（Γ）中的指数。
Let G be a finite abelian group. Its integral group ring ZG is a Z-order in QG. Let Γ be the maximal Z-order in QG containing ZG. We calculate the index of K1(ZG) in K1(Γ) for some types of finite abelian groups.
[1] Fröhlich A. Locally free modules over arithmetic orders[J]. Journal für Mathematik. Band, 1975, 274(275):15.
[2] Oliver R. Whitehead groups of finite groups[M]. Cambridge:Cambridge University Press, 1988.
[3] Rosenberg J. Algebraic K-theory and its applications[M]. New York:Springer Science & Business Media, 1995.
[4] Reiner I. Maximal orders[M]. London:Academic press, 1975.
[5] Higman G. The units of group-rings[J]. Proceedings of the London Mathematical Society, 1940, 2(1):231-248.
[6] Fröhlich A. On the classgroup of integral grouprings of finite abelian groups[J]. Mathematika, 1969, 16(2):143-152.
[7] Weiss E. Algebraic number theory[M]. New York:McGraw-Hill, 1963.
[8] Rim D S. Modules over finite groups[J]. Annals of Mathematics, 1959(69):700-712.
[9] Reiner I, Ullom S. Class groups of integral group rings[J]. Transactions of the American Mathematical Society, 1972, 170:1-30.
[10] Reiner I, Ullom S. A Mayer-Vietoris sequence for class groups[J]. Journal of Algebra, 1974, 31(2):305-342.
[11] Cassou-Nogues P. Classes d'idéaux de l'algèbre d'un groupe abélien[J]. Mémoires de la Société Mathématique de France, 1974, 37:23-32.
[12] Galovich S. The class group of a cyclic p-group[J]. Journal of Algebra, 1974, 30(1-3):368-387.
