关于有限群环与它的因子群环之间的类和对应

设N是有限群G的一个正规子群，γ：G→G是自然满同态以及γ：RG→RG是由γ经过线性扩张得到的一个R-代数满同态，其中R是一个代数整数环。首先证明了γ在Z（RG）上限制，仍是Z（RG）到Z（RG）之间的代数同态。进一步，确定了RG中的类和在γ下的像，同时给出了RG中的类和与RG中的类和之间的一个对应。最后，作为这个对应的应用，得到了有限群G的共轭类与N的陪集之间一个数量关系。

On Class Sum Correspondence of Group Rings of a Finite Group and Its Factor Group

Let N be a normal subgroup of finite group G and γ：G→G/N be the natural homomorphism, and let γ：RG→R（G/N） be the R-algebra homomorphism obtained by extending γ linearly to group ring RG, where R is an algebraic integer ring. In this paper, it is proven that the restriction of γ on Z（RG） is still a R-algebra homomorphism between Z（RG） and Z（RG）. Furthermore, the images of the class sums in RG under y are determined and a correspondence between the class sums in RG and that in G is given. Finally, as an application of this correspondence, a numerical relationship between cosets of N and any conjugacy class C of G is obtained.