On the structure of a character ring of a finite group

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The objective of this research project is to study the structure of a chracter ring of a finite group. In the following we summarize our results.First we studied the structure of the units group of a character ring of a finite group. Since the units of finite order in a chracter ring are well known, we considered the units of infinite order in a character ring. Actually we were able to construct units of infinite order in a character ring of a solvable group. We are going to write a paper on this result.Secondly we studied induction theorems in ordinary representations of finite groups. By using Mackey decomposition theorem we have obtained another proof of a theorem of J.A.Green and written a paper. (K.Yamauchi, Another proof of a theorem of J.A.Green, Journal of Algebra 235 (2000), 829-832)Thirdly we considered a generalization of a theorem of Weidman and Saksonov with respect to a Brauer character ring of a finite group. We have obtained useful and new results and written a paper. (K.Yamauchi, On isomorphisms of a Brauer character ring onto another, J.Alg. (to appear))Finally we considered a generalization of a theorem of I.M.Isaacs about the extensions of representations of finite groups. We have obtained some new results and written a paper. (K.Yamauchi, On the Extensions of Group Representations over Arbitrary Fields II, The Bulletin of the Faculty of Education, Chiba Univerrsity 48 (2000), 11-19)Other co-workers also have made contribution to this research project and obtained excellent results on their field as well.