系統識別號 U0026-0507201922400600 論文名稱(中文) 共軛類對有限群的影響 論文名稱(英文) On the influence of conjugacy class in finite groups 校院名稱 成功大學 系所名稱(中) 數學系應用數學碩博士班 系所名稱(英) Department of Mathematics 學年度 107 學期 2 出版年 108 研究生(中文) 任靜宜 研究生(英文) Ching-Yi Jen 學號 L16064038 學位類別 碩士 語文別 英文 論文頁數 32頁 口試委員 指導教授-黃世昌口試委員-黃柏嶧口試委員-蕭仁傑 中文關鍵字 共軛類大小  質數次方指數元素  q-Baer 群  normal p-complement 英文關鍵字 conjugacy class size  element of prime-power index  q-Baer group  normal p-complement 學科別分類 中文摘要 在這篇論文中，我們列出一些與元素的指數(index)相關的結果。首先是和指數為質數次方的元素有關的結果，包含單一性(simplicity)，normal p-complement 的存在性，以及一個元素擁有質數次方的指數的充分條件。接著我們舉一些例子說明滿足 order 為質數次方的元素都有質數次方指數這種條件的群還有 q-Baer 群。最後列出由元素的指數所推導出的和群的可解性(solubility)有關的結果。 英文摘要 In this thesis, we show some results about index of an element. We first show results about elements of prime-power indices, which include simplicity, the existence of normal p-complements, and sufficient conditions for an element to have prime-power index. Next, we give some examples of groups such that elements of prime-power order have prime-power indices, and q-Baer groups. The last part, we show some results that the indices of elements in a group can imply the solubility of it. 論文目次 1 Introduction 1 2 De finitions and Basic Results 3 2.1 De finitions 3 2.2 Basic Results 4 3 Elements of Prime-power Indices 7 3.1 Nonsimplicity 7 3.2 Normal p-complement 8 3.3 Sufficient Conditions for Elements to Have Prime -Power Index 14 4 Elements of Prime-power Order Having Prime-power Indices and the q-Baer Group 16 4.1 Characterisation by Baer 16 4.2 q-Baer Group 17 5 Solubility 20 5.1 Index Not Divisible by p^2 20 5.2 Square-free Index 24 Notation 27 Index 29 Bibliography 30 參考文獻 [1] R. Baer(1953), Group elements of prime power index, Trans. Amer. Math. Soc., 75:20-47. [2] Y. Berkovich and L. Kazarin(2005), Indices of elements and normal structure of finite groups, J. Algebra, 283(2):564-583. [3] W. Burnside(1904), On groups of order paqb, Proc. Lond. Math. Soc., 2:388-392. [4] A. R. Camina(1972), Arithmetical conditions on the conjugacy class numbers of a finite group, J. London Math. Soc. (2), 5:127-132. [5] A. R. Camina and R. D. Camina(1998), Implications of conjugacy class size, J. Group Theory, 1(3):257-269. [6] A. R. Camina and R. D. Camina(2006), Recognising nilpotent groups, J. Algebra, 300(1):16-24. [7] A. R. Camina and R. D. Camina(2011), The influence of conjugacy class sizes on the structure of finite groups: a survey, Asian-European J. Math. 4, 559-588. [8] D. Chillag and M. Herzog(1990), On the length of the conjugacy classes of fi nite groups, J. Algebra, 131(1):110-125. [9] A. R. Camina, P. Shumyatsky, and C. Sica(2010), On elements of prime-power index in finite groups, J. Algebra, 323(2):522-525. [10] J. Cossey and Y. Wang(1999), Remarks on the length of conjugacy classes of fi nite groups, Comm. Algebra, 27(9):4347-4353. [11] S. Dol and M. S. Lucido(2001), Finite groups in which p0-classes have q0-length, Israel J. Math., 122:93-115. [12] B. Fein, W. M.Kantor, and M. Schacher(1981), Relative Brauer groups, II. J. Reine Angew. Math., 328:39-57. [13] D. Gorenstein(1968), Finite Groups, New York. [14] D. Gorenstein(1982), Finite simple groups, New York and London. [15] T. W. Hungerford(1974), Algebra, Springer-Verlag New York. [16] N. It^o(1953), On finite groups with given conjugate types. I, Nagoya Math. J., 6:17-28. [17] N. It^o(1970), On finite groups with given conjugate types. II, Osaka J. Math., 7:231-251. [18] L. S. Kazarin(1990), Burnside's p^a -lemma, Mat. Zametki, 48(2):45-48, 158. [19] S. Li(1996), Finite groups with exactly two class lengths of elements of prime power order, Arch. Math. (Basel), 67(2):100-105. [20] S. Li(1999), On the class length of elements of prime power order in fi nite groups, Guangxi Sci., 6(1):12-13. [21] X. Liu, Y. Wang, and H. Wei(2005), Notes on the length of conjugacy classes of fi nite groups, J. Pure Appl. Algebra, 196(1):111-117. [22] G. Qian and Y. Wang(2014), On conjugacy class sizes and character degrees of finite groups, J. Algebra Appl., 13. [23] B. Steinberg(2009), Representation theory of fi nite groups, School of Mathematics and Statistics, Carleton University. [24] M. L. Sylow(1872), Th eor emes sur les groupes de substitutions, Math. Ann., 5(4):584-594. [25] B. A. F. Wehrfritz(1999), Finite Groups: a second course on group theory, World Scienti c Publishing Co. Pte. Ltd. 論文全文使用權限 同意授權校內瀏覽/列印電子全文服務，於2019-08-27起公開。同意授權校外瀏覽/列印電子全文服務，於2019-08-27起公開。

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