(34.239.176.198) 您好!臺灣時間:2021/04/23 20:48
字體大小: 字級放大   字級縮小   預設字形  
回查詢結果

詳目顯示:::

我願授權國圖
: 
twitterline
研究生:劉任浩
論文名稱:整係數群環裡的有限乘法群
論文名稱(外文):Finite Subgroups of Units in Integral Group Rings
指導教授:劉家新劉家新引用關係
學位類別:碩士
校院名稱:國立臺灣師範大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
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
外文關鍵詞:group ringrepresentation
相關次數:
  • 被引用被引用:0
  • 點閱點閱:425
  • 評分評分:系統版面圖檔系統版面圖檔系統版面圖檔系統版面圖檔系統版面圖檔
  • 下載下載:0
  • 收藏至我的研究室書目清單書目收藏:0
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
[DJ96] Michael A. Dokuchaev and Stanley O. Juriaans, Finite subgroups
in integral group rings, Canad. J. Math. 48 (1996), no. 6, 1170–
1179. MR MR1426898 (97j:20001)
[DJPM97] Michael A. Dokuchaev, Stanley O. Juriaans, and C´esar Polcino
Milies, Integral group rings of Frobenius groups and the conjectures
of H. J. Zassenhaus, Comm. Algebra 25 (1997), no. 7,
2311–2325. MR MR1451697 (98e:20010)
[GP06] J. Z. Gon¸calves and D. S. Passman, Linear groups and group
rings, J. Algebra 295 (2006), no. 1, 94–118. MR MR2188853
(2006g:16060)
[Her02] M. Hertweck, Another counterexample to a conjecture of Zassenhaus,
Beitr¨age Algebra Geom. 43 (2002), no. 2, 513–520. MR
MR1957755 (2004b:20012)
[Her06] Martin Hertweck, On the torsion units of some integral
group rings, Algebra Colloq. 13 (2006), no. 2, 329–348. MR
MR2208368 (2006k:16049)
[Hig40] Graham Higman, The units of group-rings, Proc. London Math.
Soc. (2) 46 (1940), 231–248. MR MR0002137 (2,5b)
[HK02] Martin Hertweck and Wolfgang Kimmerle, On principal blocks
of p-constrained groups, Proc. London Math. Soc. (3) 84 (2002),
no. 1, 179–193. MR MR1863399 (2002j:16030)
[JL93] Gordon James and Martin Liebeck, Representations and characters
of groups, Cambridge Mathematical Textbooks, Cambridge
University Press, Cambridge, 1993. MR MR1237401 (94h:20007)
[JPM00] Stanley O. Juriaans and C´esar Polcino Milies, Units of integral
group rings of Frobenius groups, J. Group Theory 3 (2000), no. 3,
277–284. MR MR1772015 (2001e:16054)
[Kim02] W. Kimmerle, Group rings of finite simple groups, Resenhas 5
(2002), no. 4, 261–278, Around group rings (Jasper, AB, 2001).
MR MR2015338 (2004k:20006)
[LS98] I. S. Luthar and Poonam Sehgal, Torsion units in integral group
rings of some metacyclic groups, Res. Bull. Panjab Univ. Sci. 48
(1998), no. 1-4, 137–153 (1999). MR MR1773990 (2001f:16065)
[LT90] I. S. Luthar and Poonam Trama, Zassenhaus conjecture for certain
integral group rings, J. Indian Math. Soc. (N.S.) 55 (1990),
no. 1-4, 199–212. MR MR1088139 (92b:20008)
[MRSW87] Z. Marciniak, J. Ritter, S. K. Sehgal, and A.Weiss, Torsion units
in integral group rings of some metabelian groups. II, J. Number
Theory 25 (1987), no. 3, 340–352. MR MR880467 (88k:20019)
[PMS84] C´esar Polcino Milies and Sudarshan K. Sehgal, Torsion units in
integral group rings of metacyclic groups, J. Number Theory 19
(1984), no. 1, 103–114. MR MR751167 (86i:16009)
[PMS02] , An introduction to group rings, Algebras and Applications,
vol. 1, Kluwer Academic Publishers, Dordrecht, 2002. MR
MR1896125 (2003b:16026)
[Rog91] Klaus W. Roggenkamp, Observations on a conjecture of Hans
Zassenhaus, Groups—St. Andrews 1989, Vol. 2, London Math.
Soc. Lecture Note Ser., vol. 160, Cambridge Univ. Press, Cambridge,
1991, pp. 427–444. MR MR1123997 (92g:20004)
[RS83] J¨urgen Ritter and Sudarshan K. Sehgal, On a conjecture of
Zassenhaus on torsion units in integral group rings, Math. Ann.
264 (1983), no. 2, 257–270. MR MR711882 (85e:16014)
[SSZ84] Sudarshan K. Sehgal, Surinder K. Sehgal, and Hans J. Zassenhaus,
Isomorphism of integral group rings of abelian by nilpotent
class two groups, Comm. Algebra 12 (1984), no. 19-20, 2401–
2407. MR MR755921 (85m:20008)
[Val94] Angela Valenti, Torsion units in integral group rings, Proc.
Amer. Math. Soc. 120 (1994), no. 1, 1–4. MR MR1186996
(94b:20008)
[Wei88] Alfred Weiss, Rigidity of p-adic p-torsion, Ann. of Math. (2) 127
(1988), no. 2, 317–332. MR MR932300 (89g:20010)
[Wei91] , Torsion units in integral group rings, J. Reine Angew.
Math. 415 (1991), 175–187. MR MR1096905 (92c:20009)
QRCODE
 
 
 
 
 
                                                                                                                                                                                                                                                                                                                                                                                                               
第一頁 上一頁 下一頁 最後一頁 top
系統版面圖檔 系統版面圖檔