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研究生： 劉任浩
論文名稱： 整係數群環裡的有限乘法群
Finite Subgroups of Units in Integral Group Rings
指導教授： 劉家新
學位類別： 碩士
Master
系所名稱： 數學系
Department of Mathematics
論文出版年： 2007
畢業學年度： 95
語文別： 英文
論文頁數： 34
中文關鍵詞： 群環 、表現
英文關鍵詞： group ring, representation
論文種類： 學術論文
相關次數： 點閱：147  下載：30 
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    • 在1960年代中期， 關於 integral group rings 中的 torsion units 及 finite subgroups， H. Zassenhaus 提出了三個猜想。
    其中最強的一個猜想（ZC-3）如此敘述：
    如果 H 是 integral group ring ZG 裡係數和為 1 的 unit group 的有限子群， 則 H 會和 G 裡的一個子群在 QG 裡共軛。
    這篇論文裡， 我們要證明的是 ZC-3 對個數為 p^2q 的群皆成立， 其中 p, q 為相異質數。

    In the 1960's, H. Zassenhaus made three conjectures about torsion units and finite subgroups of the units in integral group rings.
    The strongest one (ZC-3) states:
    If H is a finite subgroup of the unit group of augmentation 1 in the integral group ring ZG, then H is conjugate to a subgroup of G in QG.
    In this thesis, we prove that ZC-3 holds for groups of order p^2q, where p, q are distinct primes.

    Contents 1 Introduction 1 2 Preliminary 4 2.1 Universal Property . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Representations and Characters . . . . . . . . . . . . . . . . . 5 2.3 Torsion Units and Finite Subgroups . . . . . . . . . . . . . . . 9 3 Some Observations 12 3.1 Groups of Order p2q . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Some Known Results and Simple Cases 20 5 Representations and Some Reductions 23 6 Main result 27

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