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,它与有限群的结构有着密切的联系.事实上,有限群的非线性不可约特征标个数和次数如何影响群的结构已成为有限群常表示论的一个重要研究课题.本学位论文正是以这一热点问题展开工作的.本学位论文主要在复数域上探讨,对非线性不可约特征标个数|Irr_{1}(G)|为素数,非线性不可约特征标次数集合|cd(G)|=2的群G,我们给出了非2-群的幂零群和导群极小正规的非幂零群结构.本学位论文共分三章.第一章:我们对与论文有关的研究方向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有关的基本概念与重要...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,it has close relationship with the structure of finite groups. In fact, how the number and the degree of nonlinear irreducible characters of finite groups influnce the structure of groups is an important issue of ...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：20042300

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Xiamen University Institutional Repository

$A 10384 :RA ℄H$ A 200423004 UDC: [ * % } %  Y ÆM5 >9!"Q:Cl 1/5qthe finite groups with a prime number andsame degree of nonlinear irreduciblecharacters  |F,Kz#aZ J b ,xn$H ; #  `P + a %G K 6 XJsh 2007 O 5 MX(sh 2007 O M$~6sh 2007 O M)	8DLg < r2007 O 5 M厦门大学博硕士论文摘要库the finite groups with a prime number andsame degree of nonlinear irreduciblecharactersByLirongZhuSupervisor: 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&  Z M5 >9!"Q:Cl 1/5qJ U u \ + & 0 0 Z O j M厦门大学博硕士论文摘要库\+&&Z7&"y`Abia3d	(vU`t`pu"Yi{+vU3d}4i\FvFNRi{+	Ud}*Cn $Cv&|?1E\<Q3d~7iWxCv (`E):Q N y厦门大学博硕士论文摘要库\+&&ZNRo|3y`vYl)lr9Z?"/:a3di&rr9Z?k/+)W#09F\trH>b3divz1o	?k`a3d:D5Gii&x+P3dtV1$6O	?k`a3diL|!?"7~t\A	?k`a3di$Q1\ B!J=ia3dUl=9):&ra3d3D1 =	U Ql=9):.k12 /=hU*z4-LW ”√” QAjbÆ tiÆ d ; t-{jbÆ tiÆ d ; t厦门大学博硕士论文摘要库 WI=) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11=) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32,? /' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526? E;eBK)/S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7§2.1 p- o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7§2.2 Frobenius o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8§2.3 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102v? L4=8 !P9Bk0.4p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13§3.1 >,og . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13§3.2 ^oLh%g>,og . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14#N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20G  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22厦门大学博硕士论文摘要库ContentsChinese Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1English Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Chapter 2. Fundamental definitions and main lemmas . . . . . . . . . . 7§2.1 p-group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7§2.2 Frobenius group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8§2.3 Main Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Chapter 3. The finite groups with a prime number and samedegree of nonlinear irreducible characters . . . . . . . . . . . . . . . . . . . . . . . . . 13§3.1 The case when the group is nilpotent . . . . . . . . . . . . . . . . . . . . . . . 13§3.2 The case when the group is non-nilpotent and the commutatorsubgroup is minimal normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Acknowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22厦门大学博硕士论文摘要库Zha`YXegT[ibjS^`fVUhd_ 1> *o%!1&u[6g# j}Nf#g1&z=wo9%!gn#GG=wogj=	<eg#n%=wogx-	KNf#6/R6}26{ogj'>℄=wo9%!1g# zP`1bh&(#snfP℄g`1bSÆ6JKLxx-	KNf#6 |Irr1(G)|℄?6x-	KNf#R6M3 |cd(G)| = 2go G, g:I'2- og>,o/^oLh%g>,oj`1b_m#_g:xG1b= gz}O ℄sHr	m*5'`1bgm_g:I#G1b= gGRG j1N/p- o;N/ p- o;N/ p- o Frobenius oj(O gGq}Dg Clifford x3qjj℄m_pqg1GKm_Q8og>,G>,yk`1bKLgfP_>$im#ix=?6x-	KNf#f[R6t#g 2-og>,oIiB'o	k℄;N/ 2- oG?6hogpImi*siB'_=x-	KNf#f[R6#g=w>,o[^oLh%[Rg:0#ggx^oLh%=?6x-	KNf#f[R6t#g>,oIg:iB'go=m#o G ℄ F - omo G 	k℄ F - oGHjaogpImo G go G/Z(G) ℄ F- o厦门大学博硕士论文摘要库2 ?I' x-	KNf#Nf#R6 Frobenius o;N/ p- o;N/ p- o厦门大学博硕士论文摘要库Zha`YXegT[ibjS^`fVUhd_ 3ABSTRACTThe representation of finite groups is an important branch of mordernmathematics. Character theory is one of the main tools of the representa-tion of finite groups over the complex field, it has close relationship withthe structure of finite groups. In fact, how the number and the degree ofnonlinear irreducible characters of finite groups influnce the structure ofgroups is an important issue of category theory. This academic disserta-tion focuses on these problems.In this academic dissertation, we study the groups which have a primenumber and same degree of nonlinear irreducible characters over the com-plex field, in these conditions, we classify the finite groups when theyare nilpotent but not 2-group, also we classify the groups when they arenon-nilpotent and the commutator subgroup is minimal. The whole dis-sertation includes three chapters.In the first chapter, we introduce the groundwork of this text, alsothe research direction and the trends of the development.In the second chapter, we list some fundamental concepts and facts,including special p-group, extraspecial p-group, semi-extraspecial p-groupand Frobenius group, also some important lemmas such as the famousClifford’s correspondence theorem. They give a necessary preparation forthe third chapter.In the third chapter, we solve the problems which this academic dis-sertation focuses on by means of the properties of nilpotent and non-nilpotent groups.The Chapter 3 is divided in two sections:厦门大学博硕士论文摘要库4In the first section,we prove that the nilpotent group but not 2-groupsatisfying our conditions is a direct product of an extraspecial-2 groupwith a prime oreder group.In the second section, first we prove that the commutator subgroupof a non-nilpotent group which has three nonlinear irreducible charactersand their degrees are same is minimal normal. Then we consider moregeneral case. We prove that the groups which satisfying our conditionsand whose commutator subgroup is minimal normal are divided into threedistinct classes. The first are Frobenius Groups. The second are directproducts of a Frobenius group with an elementary abelian group. Thethird have their factor group G/Z(G) as a Frobenius group.Keywords: nonlinear irreducible character; the degree of irreduciblecharacter; Frobenius group; extraspecial p-group; semi-extraspecial p-group.厦门大学博硕士论文摘要库Zha`YXegT[ibjS^`fVUhd_ 53- 0 (=wo9%!1 gzKz&Nf#1Nf#1=5gQd=wo%!1S ℄gHX|&Nf#1Soj|g38 W.Burnside C 1904 O8Nf#1iBg C paqb hog	kqpa 1972 O1=No1giB.tw℄g& Frobenius  Cdh%oUSg#GS^wniNo1giBnNf#1Gojgz=	<eg#n%=wogx-	KNf#6/R6}26{ogj'>℄=wo9%!1g# zP`1bh&[r#P℄x_=?6x-	KNf#f_6#go G, g:KL' 2- og>,o/^oLh%g>,oj_g:xC= z}g ℄sH(Ob_gÆZ℄Bg85g:B0go℄=woNf#℄ÆNf# Irr(G) [%oGg-	KNf#M3Irr1(G)[%o Ggx-	KNf#M3cd(G)[%o Gg-	KNf#R6M3} cd(G)zo'x=5{j1} Passman,Isaacs / Garrisonx |cd(G)| ≤ 3gor	~KL/zean*j*} [7][8][9][11]. }x#qM3 S = {1, a, b},a, b℄q6℄ a, b4#qS^Thomas Noritzsch[10]I'4cd(G) = S gog#y}} Irr1(G) (O Irr1(G) G cd(G)g nzogjBerkovich ea'{ j* 1992 OD<IB=x-	KNf#R6t-# (P |Irr1(G)| = |cd(G)| − 1)g=wo [1].1994ODx |Irr1(G)| = 8g=woI' [3]. } 1996厦门大学博硕士论文摘要库6 ODBxx-	KNf#R6_=$# (P |Irr1(G)| = |cd(G)|)g=woI' [4][5].m_g:I#G1b= gGRG j1N/p- o;N/ p- o;N/ p- o Frobenius oj(O gGq}Dg Clifford x3qjj℄m_pqg1GKm_Q8og>,G>,yk`1bKLgfP_>$im#i8'lg_=#x-	KNf#g=wox=?6x-	KNf#f[R6t#g 2- og>,oIiB'o	k℄;N/ 2- oG?6hogpImi*s8'lg_=#/$x-	KNf#g=woiB'_=x-	KNf#f[R6#g=w>,o[^oLh%[Rg:0#ggx^oLh%=?6x-	KNf#f[R6t#g>,oIg:iB'go=m#o G ℄ F- omo G 	k℄ F-oGHjaogpImo G go G/Z(G) ℄ F- o:Bg&bB8g,/4It&#g	3bv[6].  N EG, θ ℄ N g#Nf#. IG(θ) ℄ θ S G |geqoP IG(θ) = {g ∈ G|θg = θ}.   IG(θ) = G, Y= θ & G- -"R.Irr(G|θ) %! G gHS θ g-	KÆNf#M3P Irr(G|θ) ={χ ∈ Irr(G)|[χH, θ] 6= 0}.Lin(G) %!o G gxNf#M3-Wg:8 G′ %!o G g^o Z(G) %!o G g| F (G)/ Φ(G) '%!o G g Fitting o/ Frattini o p [%#?6厦门大学博硕士论文摘要库Zha`YXegT[ibjS^`fVUhd_ 737 F<fCM*0TS#_g:*sm#= q//2j*xCb(7qgiB& g-W_|IiBg2bv [6] |glP§2.1 p- qgoI p- o= gR	3bv [15] / [16] |gq/4. 2.1.1[15,16] o G ℄ p- o=o G ℄N/ p- o}*G4o*$S^n#(i) o G ℄Hja p- o(ii) Z(G) = G′ = Φ(G) &Hjag4. 2.1.2[15,16] o G ℄agN/ p- of |Z(G)| = p, Y=oG ℄;N/ p- og:l G/Φ(G) &HjaoY;N/ p- o4o?2gS^28B';N/ p- ogV/S 2.1.1[15] . G ℄a p- oG/Z(G) ℄HjaoZ(G)=Y(i) |G′| = p(ii) |G/Z(G)| ℄#V6厦门大学博硕士论文摘要库8 4. 2.1.3[10] o G ℄a p- o x Z(G) gw#LXo N4 G/N ℄;N/ p- oY=o G ℄;N/ p- o C;N/ p- o={jYg<	3bv [10][14]. buXbv [10] |jYgNf#</S 2.1.2[10] G ℄a p- oY(o$j1jY(1) G ℄;N/ p- o(2) Z(G) = G′ f cd(G) = {1, |G : Z(G)| 12}§2.2 Frobenius qFrobenius oi1Sogj>&o%!1|t^} gl`o?sI Frobenius ogR	3bv [16] |gq/4. 2.2.1[16]  G &=wo 1 < H < G. }* H ⋂Hg = 1,∀g ∈G−H , Y= G ℄ Co H g Frobenius o (℄= F - o), *= H ℄G g Frobenius , (℄= F - ,). } N = G − ⋃g∈G(Hg − 1) f G gF - .O Frobenius =#gqP}* G ℄ Co H g F -oY ∃N EG, = G = NH,N ⋂H = 1.(Th%o N P℄ F -.&; H \#mqg)/S 2.2.1[6]  N E G,   CG(x) ≤ N x ∀1 6= x ∈ N , Y(o$j1> (i) ∀ϕ ∈ Irr(N), ϕ 6= IN , = IG(ϕ) = N / ϕG ∈ Irr(G).(ii) ∀χ ∈ Irr(G) f N 6⊆ Ker(χ), H6 ∃ϕ ∈ Irr(N) = ϕG = χ.厦门大学博硕士论文摘要库Zha`YXegT[ibjS^`fVUhd_ 9O 2 2.2.1 |j1 (i) T& G ℄ F - ogGS^P&F - ogjYNf#<	3bv [17].o?I F - ogjYNo1</S 2.2.2[6]  N EG,G = NH,N ∩H = 1, Yo G &( H ℄ F -,g F - ogGS^& CG(x) ≤ N x ∀1 6= x ∈ N .g:-%Nf#uQ8 F - ogGoyiB2E_ GuÆxw-g g ∈ G − H , iB H ⋂Hg = 1. g = yx, [| y ∈ H ,x ∈ N . ; g /∈ H = x 6= 1. Xq= 1 6= y1 ∈H⋂Hg = H⋂Hx, C&US 1 6= y2 ∈ H  y1 = yx2 = y2[y2, x]. 1N E G = [y2, x] ∈ N , C& y−12 y1 ∈ N , B y−12 y1 ∈ H , } N ∩H = 1, y−12 y1 = 1, P y2 = y1. ;O|= yx2 = y2, P y2 ∈ CG(x). G CG(x) ≤ N5zGeiiUS x ∈ N , = CG(x) 6≤ N . YUS y ∈ CG(x) 4y ∈ G−N . 1 G & F - o}q/ 2.2.1  y = hg, [| h ∈ H, g ∈ G.Y g−1hgx = xg−1hg, P h = gxg−1hgx−1g−1, Y Hgxg−1 ⋂H 6= 1, S}gxg−1 = 1, P x = 1 5zeiO(i) ;2 2.2.1O[S	l2 2.2.2|XaO G = NH, N∩H = 1 &{EgO(ii) F - ogjYNo1<T= 5 	3bv [6].o?g2xbiBw  	3bv [10] E [13]./S 2.2.3[10,13] o G >, cd(G) = {1, m}, Y=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

The finite groups with a prime number and same degree of nonlinear irreducible characters

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The finite groups with a prime number and same degree of nonlinear irreducible characters

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