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研究生： 孫維良
Wei-Liang Sun
論文名稱： 整群環的 Jordan 分解與冪零分解
Multiplicative Jordan Decomposition and Nilpotent Decomposition in Integral Group Rings
指導教授： 劉家新
Liu, Chia-Hsin
學位類別： 博士
Doctor
系所名稱： 數學系
Department of Mathematics
論文出版年： 2020
畢業學年度： 108
語文別： 英文
論文頁數： 104
英文關鍵詞： multiplicative Jordan decomposition, integral group ring, rational group algebra, Wedderburn component, Shoda pair, strong Shoda pair, nilpotent decomposition, SN group, SSN group
DOI URL： http://doi.org/10.6345/NTNU202000897
論文種類： 學術論文
相關次數： 點閱：196  下載：25 
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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.

    Acknowledgment i Abstract ii Contents iii 0. Introduction 1 I. Jordan Decomposition in Integral Group Rings 6 I.1 Prerequisite 7 I.2 Jordan Decomposition 10 I.2.1 Matrix Rings 13 I.2.2 Group Rings 18 I.3 Additive Jordan Decomposition in Integral Group Rings 23 I.4 Multiplicative Jordan Decomposition in Integral Group Rings 25 I.4.1 2-Groups 27 I.4.2 3-Groups and {2,3}-Groups 29 I.4.3 Groups Cp⋊Cnk 29 I.5 Multiplicative Jordan Decomposition in Z[Q8xCp] 31 I.5.1 Nilpotent Elements in Q[Q8xCp] 31 I.5.2 Semisimple and Nilpotent Parts 33 I.5.3 The Main Theorem 37 I.5.4 A Proof for p = 5 41 I.5.5 The Density of Primes 44 I.6 Multiplicative Jordan Decomposition in Z[Q8xCp], II 46 I.7 Remaining Problems and Wedderburn Components 52 II. Nilpotent Decomposition in Integral Group Rings 54 II.1 Prerequisite 55 II.2 Nilpotent Decomposition Property, SN and SSN Groups 59 II.2.1 Nilpotent Decomposition 59 II.2.2 SN Groups and SSN Groups 61 II.2.3 Examples 63 II.3 Nilpotent Decomposition for Nilpotent SSN Groups 67 II.3.1 p-Groups 67 II.3.2 Non-p-Groups 75 II.4 Nilpotent Decomposition for Non-nilpotent SSN Groups 78 II.4.1 Solvable SSN Groups with ND 78 II.4.2 Solvable SSN Groups with ND, II 82 II.4.3 Non-solvable SSN Groups 89 III. Concluding Remark and Future Work 91 Bibliography 96 Index 102

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