整群環的 Jordan 分解與冪零分解

Multiplicative Jordan Decomposition and Nilpotent Decomposition in Integral Group Rings

孫維良(Wei-Liang Sun)

指導教授 ： 劉家新

國立臺灣師範大學/理學院/數學系/博士(2020年)

https://doi.org/10.6345/NTNU202000897

摘要

none

關鍵字

none

並列摘要

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.

並列關鍵字

multiplicative Jordan decomposition ； integral group ring ； rational group algebra ； Wedderburn component ； Shoda pair ； strong Shoda pair ； nilpotent decomposition ； SN group ； SSN group

參考文獻

[AA69] R.G. Ayoub and C. Ayoub. On the group ring of a finite abelian group. Bull. Austral. Math. Soc., 1(2):245–261, 1969. doi:10.1017/S0004972700041496.

[AHP93] S.R. Arora, A.W. Hales, and I.B.S. Passi. Jordan decomposition and hypercentral units in integral group rings. Comm. Algebra, 21(1):25–35, 1993. doi:10.1080/00927879208824548.

[AHP98] S.R. Arora, A.W. Hales, and I.B.S. Passi. The multiplicative Jordan decomposition in group rings. J. Algebra, 209(2):533–542, 1998. doi:10.1006/jabr.1998.7557.

[Ami55] S.A. Amitsur. Finite subgroups of division rings. Trans. Amer. Math. Soc., 80:361–386, 1955. doi:10.1090/S0002-9947-1955-0074393-9.

[Aro94] S.R. Arora. A study of Jordan decomposition in group rings. PhD thesis, Panjab University, Chandigarh, 1994.