Induced modules over group algebras and induced characters of finite groups

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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.