簡易檢索 / 詳目顯示

回結果列表
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：172  下載：25 
分享至:
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報

    • 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

    [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.

    [Bae33] R. Baer. Situation der untergruppen und struktur der gruppe. S.-B. Heidelberg. Akad. Math.-Nat. Klasse, 2:12–17, 1933.

    [BC74] N. Burgoyne and R. Cushman. The decomposition of a linear mapping. Linear Algebra Appl., 8(6):515–519, 1974. doi:10.1016/0024-3795(74)90085-8.

    [Ber08] Y. Berkovich. Groups of Prime Power Order, Vol. 1, volume 46 of De Gruyter Expositions in Mathematics. Walter de Gruyter GmbH & Co., KG, Berlin, 2008. With a foreword by Z. Janko. doi:10.1515/9783110208221.

    [BH62] A. Borel and Harish-Chandra. Arithmetic subgroups of algebraic groups. Ann. Math., 75(3):485–535, 1962. doi:10.2307/1970210.

    [BJ06] Y. Berkovich and Z. Janko. Structure of finite p-groups with given subgroups. In Ischia Group Theory 2004, volume 402 of Contemp. Math., pages 13–93. Amer. Math. Soc., Providence, RI, 2006. doi:10.1090/conm/402.

    [BJ09] Z. Božikov and Z. Janko. A complete classification of finite p-groups all of whose noncyclic subgroups are normal. Glas. Mat. Ser. III, 44(64)(1):177–185, 2009. doi:10.3336/gm.44.1.10.

    [Bor91] A. Borel. Linear Algebraic Groups, volume 126 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. doi:10.1007/978-1-4612-0941-6.

    [Bou58] N. Bourbaki. Éléments de Mathématique; Algèbre. Paris, 1958.

    [Ced18] F. Cedó. Left braces: solutions of the Yang-Baxter equation. Adv. Group Theory Appl., 5:33–90, 2018. doi:10.4399/97888255161422.

    [CR62] C.W. Curtis and I. Reiner. Representation Theory of Finite Groups and Associative Algebras. Interscience Publishers, New York, 1962. doi:10.1090/chel/356.

    [Dri92] V.G. Drinfeld. On some unsolved problems in quantum group theory. In Quantum Groups, volume 1510 of Lecture Notes in Mathematics., pages 1–8. Springer, Berlin, 1992. doi:10.1007/BFb0101175.

    [Dub93] H. Dubner. Generalized repunit primes. Math. Comp., 61(204):927–930, 1993. doi:10.1090/S0025-5718-1993-1185243-9.

    [Fer09] R.A. Ferraz. Units of ZCp. In Groups, Rings and Group Rings, volume 499 of Contemp. Math., pages 107–119. Amer. Math. Soc., Providence, RI, 2009. doi:10.1090/conm/499.

    [GAP20] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.11.0, 2020. URL: https://www.gap-system.org.

    [GH87] R. Gow and B. Huppert. Degree problems of representation theory over arbitrary fields of characteristic 0 - On theorems of N. Itô and J.G. Thompson. J. Reine Angew. Math., 381:136–147, 1987. doi:10.1515/crll.1987.381.136.

    [Gir06] C.R. Giraldo Vergara. Wedderburn decomposition of small rational group algebras. In Groups, Rings and Group Rings, volume 248 of Lect. Notes Pure Appl. Math., pages 191–200. Chapman & Hall/CRC, Boca Raton, FL, 2006. doi:10.1201/9781420010961.

    [GJP96] G. Goodaire, E. Jespers, and C. Polcino Milies. Alternative Loop Rings, volume 184 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1996.

    [Gol12] J.S. Golan. The Linear Algebra a Beginning Graduate Student Ought to Know. Springer, Dordrecht, third edition, 2012. doi:10.1007/978-94-007-2636-9.

    [Gor80] D. Gorenstein. Finite Groups. Chelsea Publishing Co., New York, second edition, 1980.

    [Gre75] W. Greub. Linear Algebra, volume 23 of Graduate Texts in Mathematics. Springer, New York, 1975. doi:10.1007/978-1-4684-9446-4.

    [GS95] A. Giambruno and S.K. Sehgal. Generators of large subgroups of units of integral group rings of nilpotent groups. J. Algebra, 174(1):150–156, 1995. doi:10.1006/jabr.1995.1121.

    [Has66] H. Hasse. Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl a≠0 von gerader bzw. ungerader Ordnung mod. p ist. Math. Ann., 166:19–23, 1966. doi:10.1007/BF01361432.

    [HK71] K. Hoffman and R. Kunze. Linear Algebra. Prentice-Hall, New Jersey, second edition, 1971.

    [HLP90] A.W. Hales, I.S. Luthar, and I.B.S. Passi. Partial augmentations and Jordan decomposition in group rings. Comm. Algebra, 18(7):2327–2341, 1990. doi:10.1080/00927879008824023.

    [HP91] A.W. Hales and I.B.S. Passi. Integral group rings with Jordan decomposition. Arch. Math., 57:21–27, 1991. doi:10.1007/BF01200034.

    [HP99] A.W. Hales and I.B.S. Passi. Jordan decomposition. In Algebra: Some Recent Advances, Trends in Mathematics, pages 75–87. Birkh¨auser, Basel, 1999. doi:10.1007/978-3-0348-9996-3_5.

    [HP17] A.W. Hales and I.B.S. Passi. Group rings and Jordan decomposition. In Groups, Rings, Group Rings, and Hopf Algebras, volume 688 of Contemp. Math., pages 103–111, Providence, RI, 2017. Amer. Math. Soc. doi:10.1090/conm/688/13829.

    [HPW07] A.W. Hales, I.B.S. Passi, and L.E. Wilson. The multiplicative Jordan decomposition in group rings, II. J. Algebra, 316(1):109–132, 2007. doi:10.1016/j.jalgebra.2007.07.009.

    [HPW12] A.W. Hales, I.B.S. Passi, and L.E. Wilson. Corrigendum to “The multiplicative Jordan decomposition in group rings, II” [J. Algebra 316 (1) (2007) 109-132]. J. Algebra, 371:665–666, 2012. doi:10.1016/j.jalgebra.2012.08.003.

    [Hum72] J.E. Humphreys. Introduction to Lie Algebras and Representation Theory, volume 9 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1972. doi:10.1007/978-1-4612-6398-2.

    [IR90] K. Ireland and M. Rosen. A Classical Introduction to Modern Number Theory, volume 84 of Graduate Texts in Mathematics. Springer, New York, second edition, 1990. doi:10.1007/978-1-4757-2103-4.

    [JdR16] E. Jespers and Á. del Río. Group Ring Groups, volume 1. De Gruyter, Berlin, 2016. doi:10.1515/9783110372946.

    [JLP03] E. Jespers, G. Leal, and A. Paques. Central idempotents in the rational group algebra of a finite nilpotent group. J. Algebra Appl., 2(1):57–62, 2003. doi:10.1142/S0219498803000398.

    [JP93] E. Jespers and M.M. Parmenter. Units of group rings of groups of order 16. Glasgow Math. J., 35(3):367–379, 1993. doi:10.1017/S0017089500009952.

    [JS] E. Jespers and W.-L. Sun. Nilpotent decomposition in integral group rings. Preprint.

    [Kle87] E. Kleinert. A theorem on units of integral group rings. J. Pure Appl. Algebra, 49(1-2):161–171, 1987. doi:10.1016/0022-4049(87)90126-5.

    [KS19] W. Kuo and W.-L. Sun. The multiplicative Jordan decomposition in the integral group ring Z[Q8xCp]. J. Algebra, 534:16–33, 2019. doi:10.1016/j.jalgebra.2019.06.015.

    [Lam01a] T.Y. Lam. Finite groups embeddable in division rings. Proc. Amer. Math. Soc., 129(11):3161–3166, 2001. doi:10.1090/S0002-9939-01-05961-5.

    [Lam01b] T.Y. Lam. A First Course in Noncommutative Rings, volume 131 of Graduate Texts in Mathematics. Springer, New York, second edition, 2001. doi:10.1007/978-1-4419-8616-0.

    [Lim68] F.N. Liman. 2-groups with normal noncyclic subgroups. Mathematical Notes, 4:535–539, 1968. doi:10.1007/BF01429816.

    [Lin20] S.-K. Lin. Primitive central idempotents in the rational group algebras of some non-monomial groups. Master’s thesis, National Taiwan Normal University, 2020. doi:10.6345/NTNU202000642.

    [Liu12] C.-H. Liu. Multiplicative Jordan decomposition in group rings and p-groups with all noncyclic subgroups normal. J. Algebra, 371:300–313, 2012. doi:10.1016/j.jalgebra.2012.07.010.

    [LP09] C.-H. Liu and D.S. Passman. Multiplicative Jordan decomposition in group rings of 3-groups. J. Algebra Appl., 8(4):505–519, 2009. doi:10.1142/S0219498809003461.

    [LP10] C.-H. Liu and D.S. Passman. Multiplicative Jordan decomposition in group rings of 2; 3-groups. J. Algebra Appl., 9(3):483–492, 2010. doi:10.1142/S0219498810004026.

    [LP13] C.-H. Liu and D.S. Passman. Multiplicative Jordan decomposition in group rings with a Wedderburn component of degree 3. J. Algebra, 388:203–218, 2013. doi:10.1016/j.jalgebra.2013.04.015.

    [LP14] C.-H. Liu and D.S. Passman. Multiplicative Jordan decomposition in group rings of 3-groups, II. Comm. Algebra, 42(6):2633–2639, 2014. doi:10.1080/00927872.2013.766828.

    [LP16] C.-H. Liu and D.S. Passman. Groups with certain normality conditions. Comm. Algebra, 44(8):3308–3323, 2016. doi:10.1080/00927872.2015.1044104.

    [Odo81] R.W.K. Odoni. A conjecture of Krishnamurthy on decimal periods and some allied problems. J. Number Theory, 13(3):303–319, 1981. doi:10.1016/0022-314X(81)90016-0.

    [OdRS04] A. Olivieri, Á. del Río, and J.J. Simón. On monomial characters and central idempotents of rational group algebras. Comm. Algebra, 32(4):1531–1550, 2004. doi:10.1081/AGB-120028797.

    [Par02] M.M. Parmenter. Multiplicative Jordan decomposition in integral group rings of groups of order 16. Comm. Algebra, 30(10):4789–4797, 2002. doi:10.1081/AGB-120014667.

    [Pas70] D.S. Passman. Nonnormal subgroups of p-groups. J. Algebra, 15(3):352–370, 1970. doi:10.1016/0021-8693(70)90064-5.

    [Pas86] D.S. Passman. Group Rings, Crossed Products and Galois Theory, volume 64 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, 1986. doi:10.1090/cbms/064.

    [PS02] C. Polcino Milies and S.K. Sehgal. An Introduction to Group Rings, volume 1 of Algebras and Applications. Kluwer Academic Publishers, Dordrecht, 2002. doi:10.1007/978-94-010-0405-3.

    [PW50] S. Perlis and G.L. Walker. Abelian group algebras of finite order. Trans. Amer. Math. Soc., 68:420–426, 1950. doi:10.1090/S0002-9947-1950-0034758-3.

    [Rum07] W. Rump. Braces, radical rings, and the quantum Yang-Baxter equation. J. Algebra, 307(1):153–170, 2007. doi:10.1016/j.jalgebra.2006.03.040.

    [Seh78] S.K. Sehgal. Topics in Group Rings, volume 50 of Monographs and Textbooks in Pure and Applied Math. Marcel Dekker, Inc., New York, 1978.

    [Seh93] S.K. Sehgal. Units in Integral Group Rings, volume 69 of Pitman Monographs and Surveys in Pure and Applied Math. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. With an appendix by Al Weiss.

    [Sie43] C.L. Siegel. Discontinuous groups. Ann. Math., 44(4):674–689, 1943. doi:10.2307/1969104.

    [SW86] M. Shirvani and B.A.F. Wehrfritz. Skew Linear Groups, volume 118 of London Mathematical Society Lecture Note Series. Cambridge Univ. Press, Cambridge, 1986.

    [Wol] WolframjAlpha. Wolfram Alpha LLC. URL: https://www.wolframalpha.com/.

    [WZ17] X.-L. Wang and Q.-X. Zhou. Multiplicative Jordan decomposition in integral group ring of group K8xC5. Commun. Math. Res., 33(1):64–72, 2017. doi:10.13447/j.1674-5647.2017.01.07.

    [Yam74] T. Yamada. The Schur Subgroup of the Brauer Group, volume 397 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1974. doi:10.1007/BFb0061703.

    下載圖示
    QR CODE