群表示论是近代数学的一个重要分支,而特征标的理论是研究有限群常表示的最主要工具之一.Isaacs在创立特征标理论时提出了特征标稳定限制的概念，引入了特征标三元组的诱导子，拟本原特征标三元组以及相对正规子群的特征标稳定限制的概念。本学位论文由这几个概念出发，讨论了相对于一个特殊的正规子群的特征标的稳定限制之间的关系，证明了对于一个有限群G,N正规于G,N是相对SBPC-相关幂零群,则相对于这样的正规子群N的G的不可约特征标的稳定限制有相同的次数. 本学位论文共分为三章. 第一章，我们对与论文有关的背景、研究方向以及发展动态进行介绍，并概述了本学位论文的主要工作. 第二章，我们给出一些与论文...The representation of finite groups is an important branch of mordern mathematics. Character theory is one of the main tools of the representation of finite groups over the complex field.Isaacs put forward character stabilizer limits conception when he studied character theory of finite goups.Using the concepts of inductor of a character triple,stabilizer limits,and quasi-primitive character tripl...学位：理学硕士院系专业：数学科学学院数学与应用数学系_基础数学学号：20042305

林静

Xiamen University Institutional Repository

^t 10384  )  200423058 UDC:X '   dJ H d P $ S℄	s SBPC- ℄Æ*";K wV\Character stabilizer limits of a SBPC-relatednilpotent subgroup uDb/6 K C ÆA9fRx  t Æ k / {G I  t&UL>5 2007 C 5 *&Uv>5 2007 C *fRIt>5 2007 C *~uOxW3 z <2007 C 5 *厦门大学博硕士论文摘要库Character stabilizer limits of a SBPC-relatednilpotent subgroupByJing LinSupervisor: Professor Jiwen ZengSpeciality: Foundation MathematicsInstitution: College of Mathematics ScienceXiamen UniversityXiamen,P.R.ChinaMay, 2007厦门大学博硕士论文摘要库e Q % T℄	s SBPC- ℄Æ*";K wV\! Y (  e 0 0 H C J *厦门大学博硕士论文摘要库Y(eeQ%Tw|aB,e`is?&BZ0ayPaMw:>ixv>)Z0&BanX4 iI#ZEM/ixv>)}0BX3T %3ZY3D\O&B7vi2[t3Z (L5):B ) ^厦门大学博硕士论文摘要库Y(eeQ%T:GrB,Z:ShP,Ys #~s?&Bi%sP,YsR#,[(W_"2,EIPsH`&BiRV3pfR^s?&B9iikT,-m&Bo5#6+R^s?&Bi=`" {og[R^s?&Bi%-37}B"F/is?&B0h/9	%ss?&B1 /	0 Bh/9	R2 //	P0iV/=W ”√” 	8. =4 2 y =C8. =4 2 y =厦门大学博硕士论文摘要库1 #	U|j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1qU|j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2m~ ph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4~ vsF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7~ j7- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9zU[ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 ` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17厦门大学博硕士论文摘要库ContentsChinese Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1English Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Chapter 2. Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Chapter 3. Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Acknowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17厦门大学博硕士论文摘要库26 SBPC- 2!>%*:-1< 1{ iW('r[tj\~F+A&j'ywUW:(jk`~|JÆ Isaacs 1M+A&'z,G+A&DtUUjDd+A&g%ijbg&+A&g%iOW}C&gWj+A&DtUUjDt'C=QDG*'W} +jC&gWj+A&jDtUUJZj!LD4} UW G, N ⊳ G, N W} SBPC- W!0 W (t3.2), 4W} ={jC&gW N j G j0(+A&jDtUUW4jQÆt'C>g9Æn9G.}#'C!jtyw	\O8u&phjn-t'Cj`~oÆn9G.G`#'C!jGD#\~g'>:9qt!~jGIÆng9G.RtdjWP SBPC- W!0 gW-z4=WT1jR(Cj*'jJQ+A&g%i(*+A&jDtUU#+A&g%ij&bgJZj!Lk:2+}+A&j&bgj*'4W} +jC&gWj+A&jDtUUJZj!LÆ} DtUU Clifford }+A& SBPC- W!0 W厦门大学博硕士论文摘要库26 SBPC- 2!>%*:-1< 2ABSTRACTThe representation of finite groups is an important branch of mordernmathematics. Character theory is one of the main tools of the represen-tation of finite groups over the complex field. Isaacs put forward charac-ter stabilizer limits conception when he studied character theory of finitegoups. Using the concepts of inductor of a character triple,stabilizer lim-its,and quasi-primitive character triple, Isaacs proved in his article[2],thatall N−relative stabilizer limits of irrecducible characters of G have equaldegrees, if N is a normal nilpotent subgroup of G.After giving these concepts,we discuss another special kind of goup,thatis SPBC-related nilpotent subgroup, and prove that the stabilizer limits ofirrecducible characters of G relative to a SPBC-related nilpotent subgroupalso have equal degree.This academic dissertation is divided into three chapters altogether.In chapter one, we introduce the groundwork of this text, also theresearch direction and the trends of the development.In chapter two,we give some fundamental conceptions and importantresults related to the thesis.This can provide necessary preparation for thenext chapter.In chapter three,first we difine a new kind group,that is SBPC-relatednilpotent subgroup,and give an example to show this kind of group isexistential.Thus the discussion of this paper is sensible;second,using con-cept of primitive character triple, we discuss the relationship between thecharacter stabilizer limits of a normal SBPC-related nilpotent subgroup厦门大学博硕士论文摘要库26 SBPC- 2!>%*:-1< 3and quasi-primitive character triple;third,by discussing the quasi-primitivecharacter triple,we show the relationship between the stabilizer limits ofa normal SBPC-related nilpotent subgroup.Keywords: Stabilizer limit, Clifford correspondence, Characters,SBPC-related nilpotent group厦门大学博硕士论文摘要库26 SBPC- 2!>%*:-1< 4l} o gq G >UWN > G jC&gW-O θ ∈ Irr(N) > N j0(+A&Æb* θ > G-0$jP}\ g ∈ G 4 n ∈ N , ~ θ(gng−1) = θ(n), 4< (G,N, θ) >+A&g%iÆ.q (H,M,ϕ)+A&g%i(h1℄ G = NH OM = N∩H . b* θ = ϕN ,4< (H,M,ϕ) > (G,N, θ) jbgÆbN6!(N, θ) → G↑ ↑(M,ϕ) → H+A&g%i+A&'Yk>Gjyw}℄J:MsQgZsj?*ÆJYNgbj Dade [4] 4 Isaacs [2] M+A&jDtUU'J1fRCj+A&g%iOJbg4DtUUlDÆ>nC!ywjE.4>aG.RjnIsaacs !dj&j+A&g%i4+A&jDtUUjtÆn 1.1[2] q (G,N, θ) >+A&g%ib*}* K⊳G OK ≤ N , ~ θK >KQ+A&4<+A&g%i>&jÆn 1.2[2] b* (H,M,ϕ)> (G,N, θ)jbg-O (H,M,ϕ)>&j+A&g%i4< (H,M,ϕ) > (G,N, θ) j&jbgÆn 1.3[2] q G >UW M ⊳ G,θ ∈ Irr(M),χ ∈ Irr(G), θ χM j0(?S T = IG(θ), 4 η ∈ Irr(T |θ), g ηG = χ, 4<厦门大学博硕士论文摘要库26 SBPC- 2!>%*:-1< 5η > χ ! θ j Clifford }ÆC!ywj6 Clifford }} Clifford }j![7$$? fj Clifford }ÆN2xG6 Clifford }jtÆUpt 1.3 j1℄} η ∈ Irr(T ), qQ M1 ⊳ T , T1 θ1 ∈Irr(M1)O θ1  ηM1 j0(?S T1 = IT (θ1),4 η1 ∈ Irr(T1|θ1),g ηT1 = η, 0{\mphP1℄gNtn 1.4[2] 0{\jpht 1.3 Yj1℄R*DtgW Tigdj ηi ∈ Irr(Ti) el χ j6 Clifford }S> CCC(χ).n 1.5[2] S (N, θ) ≤ (G,χ), ( N ≤ G, θ ∈ Irr(χN), Eo!L CCC(χ) YjL_%<> χ jDtUUÆt 1.5 G CCC(χ) YjL_%P> CCC(χ) Yj&+A&PP+A&UUdC&gWjKQjÆn 1.6[2] q G >UWet G jC&gW N , gt 1.4jr[+BY M jq3(h M ≤ N , Rgdj6 Clifford }x<> χ W} N j6 Clifford }S> CCCN(χ).  CCCN(χ) YjL_%x<> χ W} N jDtUUÆ1CT [2] Y Isaacs 2+D4_ N  G j0 C&gW χ W} N jDtUU# (G,N, θ) j&jbg}jRD4 G j0(+A&W} N jDtUUW4jQÆC$!C&gWP>C& SBPC- W!0 gW N , =WiW^0t0 W (g\:). !	2}\0 W N , O\j厦门大学博硕士论文摘要库26 SBPC- 2!>%*:-1< 6θ ∈ Irr(N),N v θ- W!0 W (t 3.1), ^ N 0t SBP W (t 2.1). !Cywj04 0 Wj!8C&gW4{';XD4} =C&gW N , ℄X4{jg'?Æj~4jC!*'jW~>UW+A&~>+A&!j04$v&dj5CT [1].厦门大学博硕士论文摘要库26 SBPC- 2!>%*:-1< 7} urE1=9Æ>y<,y	#G.RG~dj`D4Hg*=`g*} 'C:`~tjD4\~jÆ}\UW G 4BH | G | j p j8	0 pa,G Slpa fjgWRGbNtÆn 2.1[5] b*UW G j?gWQ0fW4<=[W>SBP W (Sylow Best Possible).p 2.2[6] (Frattini '{) qW G o1N6 Ω j-O G ,gW N , $1 Ω jjoLoj4 G = GαN, ∀α ∈ Ω, JY Gα( α 1 G YjDtgWÆp 2.3[2] b* (H1,M, ϕ), (H2,M, ϕ) ~ (G,N, θ) jbg4 H1 = H2.p 2.4[3] " (H,M,ϕ)  (G,N, θ) jbg4 Irr(H|ϕ) #Irr(G|θ) JZT1}Æp 2.5[2] q (H,M,ϕ)&+A&g%iO M 0 4M/Z(ϕ)aÆp 2.6[6] UW G jL_C&gW N !>4℄WjLIÆp 2.7[1] (Clifford t) q G >W N ⊳G,χ ∈ Irr(G),θ  χNj0(O θ = θ1, θ2, · · · , θt  θ j;0W4j G p4χN = e∑ti=1 θi, JY e = [χN , θ].厦门大学博硕士论文摘要库26 SBPC- 2!>%*:-1< 8p 2.8[1] (Clifford }) q H ⊳ G, θ ∈ Irr(H), T = IG(θ). "A={ψ ∈ Irr(T ) | [ψH , θ] 6= 0}, B={χ ∈ Irr(G) | [χH , θ] 6= 0}, 4(a) e ψ ∈ A, 4 ψG 0((b) o ψ → ψG > A d B jo(c) e ϕG = χ, 4 ψ  χT 1 A Y<j0((d) e ψG = χ, O ψ ∈ A, 4 [ψH , θ] = [χH , θ].p 2.9[1] (Frobenius Reciprocity) q G >W H > G jgW.q θ 4 ϕ )> G 4 H jj-4 [ϕ, θH ] = [ϕG, θ].p 2.10[1] ( Going Down t) " K/L  G ja`ÆgÆXq θ ∈ Irr(K) O θ  G− 0$j4Ng'!?(a) θL ∈ Irr(L);(b) θL = eϕ, JY ϕ ∈ Irr(L),e2 = |K : L|.(c) θL =∑ti=1 ϕi, JY ϕi ∈ Irr(L) O;0W4 t = |K : L|.p 2.11[1] q H ≤ K ≤ G,ϕ ∈ Cha(H), 4 (ϕK)G = ϕG.p 2.12[1] q H,K ≤ G, O HK = G,ϕ ∈ Cha(H), 4 (ϕG)K =(ϕH∩K)K .厦门大学博硕士论文摘要库26 SBPC- 2!>%*:-1< 9?} i6+>`~g*jD4RdNDÆN >UW} θ ∈ Irr(N), " Hθ(N)={L ≤ N |∃ξ ∈ Irr(L), gξN = θ}.N ∈ Hθ(N), ÆP Hθ(N) Æn 3.1 q N >UW θ ∈ Irr(N), b*}\j L ∈ Hθ(N),~ L⊳ ⊳N , 4< N > θ- W!0 WÆn 3.2 q θ ∈ Irr(N),N > θ- W!0 WO N > SBP W4<N > SBPθ- W!0 Wb*}\j θ ∈ Irr(N), N ~> SBPθ- W!0 W4< N > SBPC- W!0 WÆ (Sylow Best Possible Character)SBPC-W!0 W~iW̂ 0t0 WÆbG = S34GM-W!}\j χ ∈ Irr(G),T1H ≤ G, λ ∈ Irr(H), λ(1) = 1,g λG = χ. +A&(Hcd(S3)={1, 2},!N2Q~$ χ(1) = 2, 2=χ(1)=| G : H |, R H=A3 ⊳ G. ÆP}\j χ ∈ Irr(G), G ~> χ- W!0 W G jf	H G j\?gW~>0fPG > SBPC- W!0 W^ G Q 2- 0 W00 WÆ0M 3.3 q θ ∈ Irr(N),N > θ- W!0 W4}\j L ∈Hθ(N), O L < N ~ LN < N ?-Æ QC&iOC& jtg'SXjÆI.M.Isaacs 1CT [2] Y94W} C&0 gWj+A&DtU厦门大学博硕士论文摘要库Degree papers are in the “Xiamen University Electronic Theses and Dissertations Database”. Fulltexts are available in the following ways: 1. If your library is a CALIS member libraries, please log on http://etd.calis.edu.cn/ and submitrequests online, or consult the interlibrary loan department in your library. 2. For users of non-CALIS member libraries, please mail to etd@xmu.edu.cn for delivery details.厦门大学博硕士论文摘要库

Character stabilizer limits of a SBPC-related nilpotent subgroup

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Character stabilizer limits of a SBPC-related nilpotent subgroup

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