Induced modules over group algebras and induced characters of finite groups

本研究的目的是研究群代数上的诱导模和有限群的诱导特征。首先，我们利用B.Banaschewski的一个定理证明了有限群的特征标环的Jacobson根为零。(B.Banaschewski，关于有限群的特征标环，加拿大数学出版社.15(1963)，605-612。)我们就这一结果写了一篇论文。其次，利用Z[ω]中的单位，具体构造了p阶循环群的特征标环中的无限级单位，其中Z为有理整数环，ω为本原p次单位根，p(大于或等于5)为素数，并证明了R(G)有无限级单位。如果G/G‘的阶可被素数p([大于或等于]5)整除，其中R(G)是有限群G的特征标环，G’是有限群G的换位子群.我们就这些结果写了一篇论文，并提交出版.最后，我们考虑了对两个有限群G和H，R(G)到R(H)的同构的存在程度.可以影响G和H的模表示。我们写了一篇关于这一主题的论文，将提交到其他地方发表。其他同事也为这一研究项目做出了贡献，并在他们的领域取得了出色的成果。

The objective of this research project is to study the induced modules over group algebras and the induced characters of finite groups. In the following we summarize our results.First we proved that the Jacobson radical of the character ring of a finite group is zero, by making use of a theorem of B.Banaschewski. (B.Banaschewski, On the character rings of finite groups, Canad.J.Math. 15 (1963), 605-612.) We wrote a paper on this result.(K.Yamauchi, The bulletin of the faculty of education, Chiba Univ. Vol.51 (2003),315-317.)Secondly we concretely constructed the units of infinite order in the character ring of a cyclic group of order p, by making use of units in Z[ω] where Z is the ring of rational integers and ω is a primitive p-th root of unity and p(【greater than or equal】 5) is a prime number, and we also proved that R(G) has units of infinite order, if the order of G/G' is divisible by a prime number p(【greater than or equal】5) where R(G) is the character ring of a finite group G and G' is the commutator subgroup of a finite group G. We wrote a paper on these results which is submitted for publication.Finally we considered to what extent the existence of an isomorphism of R(G) onto R(H) for two finite groups G and H, can influence the modular representations of G and H. We wrote a paper on this theme which will be submitted for publication elsewhere.Other co-workers also have made contribution to this research project and obtained excellent results on their field as well.