整群環的 Jordan 分解與冪零分解

Abstract

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We study the multiplicative Jordan decomposition property in integral group rings. The aim of this study is to find out which integral group rings have this property. This problem was proposed by A.W. Hales and I.B.S. Passi in 1991 and it is still open now. In this dissertation, we prove that this property holds when the group is the direct product of a quaternion group of order 8 and a cyclic group of certain prime order p. We also show negative statements for some different prime numbers p. These results give a great advance of this problem. Additionally, we study the nilpotent decomposition property in integral group rings where this concept comes from the multiplicative Jordan decomposition property. Moreover, this research leads us to another problem that when a rational group algebra of a finite group has only one Wedderburn component which is not a division ring. We classify these rational group algebras for finite SSN groups. Two related conjectures are presented in the content.