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Orbits of the Actions of Finite Solvable Groups

Permanent Link: http://ufdc.ufl.edu/UFE0024783/00001

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Title: Orbits of the Actions of Finite Solvable Groups
Physical Description: 1 online resource (60 p.)
Language: english
Creator: Yang, Yong
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: actions, group, linear, solvable
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Suppose V is a completely reducible faithful G-module for a finite solvable group G, we show G has a 'large' orbit on V. Specifically, there exists v in V such that the centralizer of G on v is contained in a normal subgroup of derived length 9 contained in the seventh ascending Fitting subgroup of G. This is applied to generate many theorems showing that a solvable group must have characters of large degree. Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on a finite vector space V. Then G has a uniquely determined normal subgroup E which is a direct product of extraspecial p-groups for various p and we denote e to be the square root of |E/Z(E)|. We prove that when e > = 10 and e not equal to 16, G will have at least 5 regular orbits on V. We also construct groups with no regular orbits on V when e=8, 9 and 16.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Yong Yang.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Turull, Alexandre.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024783:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024783/00001

Material Information

Title: Orbits of the Actions of Finite Solvable Groups
Physical Description: 1 online resource (60 p.)
Language: english
Creator: Yang, Yong
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: actions, group, linear, solvable
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Suppose V is a completely reducible faithful G-module for a finite solvable group G, we show G has a 'large' orbit on V. Specifically, there exists v in V such that the centralizer of G on v is contained in a normal subgroup of derived length 9 contained in the seventh ascending Fitting subgroup of G. This is applied to generate many theorems showing that a solvable group must have characters of large degree. Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on a finite vector space V. Then G has a uniquely determined normal subgroup E which is a direct product of extraspecial p-groups for various p and we denote e to be the square root of |E/Z(E)|. We prove that when e > = 10 and e not equal to 16, G will have at least 5 regular orbits on V. We also construct groups with no regular orbits on V when e=8, 9 and 16.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Yong Yang.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Turull, Alexandre.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024783:00001


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Atthebeginningofmydissertation,Iwouldliketoexpressmygratitudetoallthosewhohavehelpedmeduringtheseyearsofstudy.Thedeepestgratitudebelongs,nodoubt,tomyadvisorProfessorAlexandreTurull.Withouthim,Iwouldnothaveevenenteredintothisfantasticresearcharea,norcompletedthisdissertation.Iamdeeplytouchedbyhispatienceinguidanceandhisenthusiasminresearch,andammuchimpressedbyhisintelligenceandpersonality.IwishtoalsothankmycommitteemembersProfessorJamesKeesling,ProfessorJosePrincipe,ProfessorPeterSinandProfessorPhamHuuTiepfortheirmanyhelpfulcommentsandsuggestions.IwouldespeciallyliketoacknowledgeProfessorJamesKeesling,oneofmycommitteemembers,whointroducedmetotheDepartmentofMathematics,elsethestoryofmysuccesswouldbedierent.Inaddition,asinceregratitudeisduetotheDepartmentofMathematicsoftheUniversityofFloridawhohassupportedmyPhD.studiesandprovidedanexcellentresearchenvironment.Fromapersonalperspective,Iamalsogratefultomyfriendsandfamily.IwouldliketothankmyfriendsJoelandLauraSutton,QingguoZeng,HuaWang,NingGuo,AnqiShaoandXiaojingYefortheirhelpduringmystudies.Lastbutnotleast,Iamdeeplyindebtedtomyfamily:myparents,whosupplythestrongestsupportandencouragementfromShanghai,China;andmydearwifeJunXiaowhotakescareofeverythingwhileIamwritingthisdissertation.Itistheirlovethatsmoothesthewayofaccomplishment.Mylifeoftheveyears'studiesintheUniversityofFloridaissuchacolorfulexperiencethatIwillneverforget.Bestwishestoallthosewhohavebeenwithmeduringthatspecialperiod. 4

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page ACKNOWLEDGMENTS ................................. 4 ABSTRACT ........................................ 6 CHAPTER 1INTRODUCTIONANDNOTATION ........................ 7 1.1Introduction ................................... 7 1.2Notation ..................................... 9 2NORMALCLOSURESOFCENTRALIZERS ................... 11 2.1PreliminaryLemmas .............................. 11 2.2NormalClosuresofCentralizersofSolvableLinearGroups ......... 17 2.3Counterexample ................................. 23 2.4Applications ................................... 24 3REGULARORBITSOFPRIMITIVESOLVABLEGROUPS .......... 26 3.1PreliminaryLemmas .............................. 26 3.2RegularOrbitsofPrimitiveSolvableGroups ................. 45 3.3PrimitiveSolvableGroupsWithNoRegularOrbits ............. 58 REFERENCES ....................................... 59 BIOGRAPHICALSKETCH ................................ 60 5

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13 ]publishedatthe"JournalofAlgebra".Thesecondresultisbasedonanotherpaperinpreparation.ByGaschutz'sTheorem,theorbitstructureofsolvablelineargroupsisausefultooltostudythestructureofsolvablegroups.Manyorbittheoremshavebeenprovedintheliterature.In2004,MoretoandWolfprovedonesuchorbittheorem[ 10 ].Byapplyingthistheoremtheyobtainanumberofresultsonseveralconjecturesonclasssizes,characterdegreesandzerosofcharacters.Inparticular,theyprovethatsolvablegroupshave"large"characterdegreesandconjugacyclasses.InChapter2weproveastrengthenedversionoftheirorbittheorem.Thecorollarieswhichfollowfromitarealsoimproved.SupposeVisafaithfulcompletelyreducibleGmodulewhereGisanitesolvablegroup.In[ 10 ],MoretoandWolfconsiderthecentralizersofGonVandtheyprovethatthereexistsv2VsuchthatCG(v)F9(G).HerewedenotebyFn(G)thenthascendingFittingsubgroupofG.Thisresulthasmanyapplications.Forexample,itimpliesthatthereexists2Irr(F10(G))suchthatG=2Irr(G)andinparticular,jG:F10(G)j

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8 ]conjecturedthatallthenon-vanishingelementsareinF(G).Thepreviousresultshowsthatallthenon-vanishingelementsareinF8(G).OtherapplicationsarelistedattheendofChapter2.LetGbeapermutationgrouponaniteset.Theorbitf!gjg2Ggiscalledregular,ifCG(!)=1holds.Intheproofoftherstmainresult,Iusedaregularorbittheoremofniteprimitivesolvablegroups,i.e.Proposition4.10of[ 9 ].SincethisresultisnotthebestpossibleIstartedtondawaytoimproveitandittookmeaboutoneyeartodrawaconcreteconclusion.InChapter3,wegivealistofalle8(aninvariantmeasuringthecomplexityofthegroup)suchthattheregularorbitexists.Theproofofthisresultislongandtechnical,andrequiresaverydetailedanalysisofniteprimitivesolvablelineargroups.Whenonestudiestheorbitstructureofsolvablelineargroups,thecommonstrategyistousereduction.Aswecanseeinmanyplaces(forexample[ 1 ],[ 9 ],[ 10 ]and[ 13 ]),manyproblemsarereducedtothesituationwheretheactionisirreducibleandquasi-primitiveandthenpropertiesarecheckedforallpossiblevaluesofe.Ourresultisquiteusefulsincetheexistenceofregularorbitisanicepropertywhichwillimplymanyotherproperties.Thisresultcanbeappliedtosimplifytheproofoftherstmainresultofthisdissertationandmanyotherresultsintheliterature([ 1 ],[ 9 ],[ 10 ]).SupposethatanitesolvablegroupGactsfaithfully,irreduciblyandquasi-primitivelyonanitevectorspaceV.Byatheoremin[ 9 ],Ghasauniquelydeterminednormal 8

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9 ]thatwhene>118,Gwillhaveatleast5regularorbitsonV.InChapter3,weextendthisresult,weprovethatwhene10ande6=16,Gwillhaveatleast5regularorbitsonV.Clearlywhene=1,itiseasytondprimitivesolvablegroupwhichhasnoregularorbitsonthevectorspace.Onemightwanttoknowwhatisthebiggestethatwillprovidesuchanexample.InChapter3,weconstructprimitivesolvablegroupswithnoregularorbitsonVwhene=8,9and16. (1) LetGbeanitegroup,letSbeasubsetofGandletbeasetofdierentprimes.Foreachprimep,wedenotebySPp(S)=fhxijo(x)=p;x2SgandEPp(S)=fxjo(x)=p;x2Sg.WedenotebySP(S)=[pprimesSPp(S),SP(S)=[p2SPp(S),EP(S)=[pprimesEPp(S)andEP(S)=[p2EPp(S).WedenotebyNEP(S)=jEP(S)j,NEPp(S)=jEPp(S)jandNEP(S)=jEP(S)j. (2) Letnbeaneveninteger,qapowerofaprime.LetVbeastandardsymplecticvectorspaceofdimensionnofFq.WeuseSCRSp(n;q)orSCRSp(V)todenotethesetofallsolvablesubgroupsofSp(V)whichactscompletelyreduciblyonV.WeuseSIRSp(n;q)orSIRSp(V)todenotethesetofallsolvablesubgroupsofSp(V)whichactsirreduciblyonV.DeneSCRSp(n1;q1)SCRSp(n2;q2)=fHIjH2SCRSp(n1;q1)andI2SCRSp(n2;q2)g. (3) LetVbeanitevectorspaceandletGGL(V).WedenePC(G;V;p;i)=fxjx2EPp(G)anddim(CV(x))=igandNPC(G;V;p;i)=jPC(G;V;p;i)j.WewilldropVinthenotationwhenitisclearinthecontext. (4) Wedenesemi-lineargroup(qn)=fx7!axja2GF(qn);2Gal(GF(qn)=GF(q))gandwedene0(qn)=fx7!axja2GF(qn)g WeuseHoStodenotethewreathproductofHwithSwhereHisagroupandSisapermutationgroup. (6) WeuseF(G)todenotetheFittingsubgroupofG.(G)meanstheFittingheightofagroupGanddl(G)meansthederivedlengthofagroupG. 9

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LetF1(G)F2(G)Fn(G)=GdenotetheascendingFittingseries,i.e.F1(G)=F(G)andFi+1(G)=Fi(G)=F(G=Fi(G)).Fi(G)istheithascendingFittingsubgroupofG,wealsocallG=Fn1(G)thetopFittingfactorofGandF1(G)=F(G)thebottomFittingfactorofG. (8) 10

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Lemma1. (1) (2) (3) (4) (5) (6) (7) (8) Ife=1,thenG.(qn)and(G)2anddl(G)2.Proof.ThisfollowsfromCorollary1.10and2.6of[ 9 ]. 11

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(1) Ifnisaprime,then(G)4anddl(G)6.Ifalson=q,then(G)3anddl(G)3. (2) Ifnisaprimeandq=2,then(G)2anddl(G)2. (3) Ifn=2sforanoddprimes,then(G)5. (4) Ifn=2,then(G)3anddl(G)4. (5) Ifn=4,then(G)4anddl(G)5.Ifalsoq=2,then(G)3anddl(G)3. (6) Ifn=6,q=2,then(G)4anddl(G)6.AlsojGj64. (7) Ifn=6,q=3,thendl(G)6.IfalsoGactsprimitively,then(G)4. (8) Ifn=8,q=2,then(G)5anddl(G)5. (9) Ifn=8,q=3,then(G)5.IfalsoGSp(8;3),theneitherdl(G)6orjGj245. (10) Ifn=10,q=2,thendl(G)6. (11) Ifn=12,q=2,then(G)5anddl(G)7.Proof.LetVbeanirreducibleG-modulewithjVj=qn.Weuseapair(n;q)torepresentthedierentcases.(1),(2)and(4)areconsequencesofTheorems2.12,2.13,2.11of[ 9 ]and(3)isprovedinLemma4.4of[ 10 ].Let(n;q)=(4;2),(G)3anddl(G)3byCorollary2.15of[ 9 ].Let(n;q)=(6;2),byCorollary2.15of[ 9 ],Gisasubgroupof(23)oZ2,S3oS3,(26)orGhasanextraspecialF(G)oforder33andF(G)=Z(F(G))isafaithfulirreducibleG=F(G)module.IfGisasubgroupof(23)oZ2,S3oS3or(26),then(G)4anddl(G)4.Intheremaincase,G=F(G)isasubgroupofGL(2;3),by(3)(G=F(G))3anddl(G=F(G))4,thus(G)4anddl(G)6.Sinceallofthesegroupshaveorderlessthanorequalto64,jGj64.Hence(6)holds. 12

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3 andusethenotationinit.Ife=1then(G)2anddl(G)2.Thuswemayassumee>1. (1) Let(n;q)=(4;q).Sincen=4andejn,ecanonlybe2or4,ThusjWij=22or24.IfjWij=22forsomei,thendl(A=F)max(dl(G=CG(Wi)))2by(2)anddl(G)2+3=5,also(A=F)max((G=CG(Wi)))2(2)and(G)2+2=4.IfjWij=24forsomei,thene=e1=n=4andG=A.dl(A=F)dl(G=CG(W1))3anddl(G)2+3=5,also(A=F)(G=CG(W1))3and(G)1+3=4. (2) Let(n;q)=(6;3).Since3-eiandeijn=6,ei=e1=2andjWij=jW1j=22.Thusdl(A=F)max(dl(G=CG(Wi)))2by(2)anddl(G)3+2=5.Also(A=F)max((G=CG(Wi)))2by(2)and(G)2+2=4. (3) Let(n;q)=(8;2).Sincen=8,ejnandgcd(q;e)=1,e=1andthisisacontradiction. (4) Let(n;q)=(8;3).(A=F)(G=CG(F=T))max((G=CG(Wi))).jWij=22;24;26sinceeijn=8.IfjWij=26forsomeithene=e1=n=8,thusG=Aand(G=CG(Wi))4by(6),thuswehave(G)=(A=F)+14+1=5.OtherwisejWij=22;24,(G=CG(Wi))isatmost2;3by(2),(5)respectivelyand(G)(A=F)+(F)+(G=A)3+2=5.Supposeei4foralli,thusei=2or4andjWij=22or24.dl(A=F)max(dl(G=CG(Wi)))3by(2),(5)respectivelyanddl(G)dl(F)+dl(A=F)+dl(G=A)2+3+1=6.Nowwecanassumeei=8forsomeiandweknowthate=ei=e1=8=n.G=AandTZ(GL(V))byLemma2.10of[ 9 ],thusjTj=jUj=jZj=2andjFj=27.NowA=FisasolvableirreduciblesubgroupofGL(6;2)andthusjA=Fj64by(6).SowehavejGj=jFjjA=FjjG=Aj2764245. (5) Let(n;q)=(10;2).Sinceq=2andgcd(q;e)=1,wehave2-ei,thuseicanonlybe5sinceeijn=10andthereisonlyoneei,thistellsusthate=ei=e1.NowjW1j=52,A=FisafaithfulsolvablecompletelyreduciblesubgroupofGL(2;5)andA=FSp(2;5),thusdl(A=F)3anddl(G)3+dl(A=F)6. (6) Let(n;q)=(12;2).Sinceq=2andgcd(q;e)=1,wehave2-ei,thuseicanonlybe3sinceeijn=12andthereisonlyoneei,thistellsusthate=ei=e1.NowjWij=32,(A=F)(G=CG(F=T))max((G=CG(Wi)))3byapplying(4),thus(G)2+(A=F)=5.Alsodl(A=F)=dl(A=CA(Ei=Z))dl(Sp(2;3))3,thusdl(G)3+dl(A=F)6.NowwecanassumeVisnotprimitive,thusGisisomorphictoasubgroupofHoS.HereSisaprimitivepermutationgroupofdegreemforanintegerm>1withmt=nandHisasolvableirreduciblesubgroupofGL(t;q).ByLemma 1 ,misaprimepower. 13

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Let(n;q)=(4;q).Sincen=4,mcanbe2or4andGwillbeisomorphictoasubgroupofoneofthefollowinggroups, (a) (b) (2) Let(n;q)=(6;3).Sincen=6,mcanbe2or3andGwillbeisomorphictoasubgroupofoneofthefollowinggroups, (a) (b) (3) Let(n;q)=(8;2).Sincen=8,mcanbe2,4or8andGwillbeisomorphictoasubgroupofoneofthefollowinggroups, (a) (b) (c) 1 and(2),thusdl(HoS)dl(H)+dl(S)0+3=3and(HoS)(H)+(S)0+3=3. (4) Let(n;q)=(8;3).Sincen=8,mcanbe2,4or8andGwillbeisomorphictoasubgroupofoneofthefollowinggroups, (a) (b) 14

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1 and(2),thus(HoS)(H)+(S)1+3=4.SincetheFittinglengthofallofthesegroups5,(G)5.AssumealsoGSp(8;3).Sincen=8,mcanbe2,4or8andGwillbeisomorphictoasubgroupofoneofthefollowinggroups, (a) (b) (c) 1 and(2).Thusdl(A=F)4. (5) Let(n;q)=(10;2).Sincen=10,mcanbe2or5andGwillbeisomorphictoasubgroupofoneofthefollowinggroups, (a) (b) 1 andthusby(2)dl(HoS)dl(H)+dl(S)2+2=4. (6) Let(n;q)=(12;2).Sincen=12,mcanbe2,3or4andGwillbeisomorphictoasubgroupofoneofthefollowinggroups, (a) (b) (c) 15

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9 ])andinalltheseexceptionalcases(G)3anddl(G)4.Nowweassumer(G)=3orr(G)=4.IfVisimprimitive,thenGisisomorphictoasubgroupofHoS,assumes=r(H)andSisapermutationgrouponmletters,thenr(G)isatleastm+s1mbyLemma2.6of[ 2 ].Ifs3orifm4,thenGhasatleast5orbitsonV.Wethusassumes=2andm=2or3.Sinces=2,then(H)3anddl(H)4bythepreviousparagraph.Ifm=2,then(G)(H)+(Z2)4anddl(G)dl(H)+dl(Z2)=5.Ifm=3,then(G)(H)+(S3)=5anddl(G)dl(H)+dl(S3)=6.ClearlyS3hasa2groupasthetopFittingfactor.WethusassumeVisprimitiveandGisnotsemi-linear.Whenr(G)=3,Theorem1.1of[ 2 ]showsthat2n4andso(G)4anddl(G)6byLemma 4 (4),(1)and(5)respectively.Whenr(G)=4,Theorem1.1of[ 2 ]showsthat2n4orGisasolvablesubgroupofGL(6;3)orGL(10;3).Suppose2n4then(G)4anddl(G)6byLemma 4 (4),(1)and(5)respectively.SupposethatGisasolvablesubgroupofGL(6;3)andVisairreducibleG-module,(G)4anddl(G)6byLemma 4 (7). 16

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2 ]weknowthatG=HTHX,wherejHTj=24andHThasQ8asanormalsubgroup,jHXj=5112.Thus(HT)2and(HX)2and(G)2.dl(HT)3anddl(HX)2anddl(G)3. 5 wecanassume(H)=5andr(H)=4.ByLemma 5 againHhasa2groupasthetopFittingfactorandclearlyS4hasa2groupasthebottomFittingfactor,thus(G)7byLemma 2 3 .IfjGjjVj3=4=jVj1=4=jVj1=2.Sincewehavee4andejdim(V)andthusjVj1=224=2=4,thencertainlythereareatleast5regularG-orbitsonV. 3 ,ife4andtheinequalitye2624jWje8issatised,thenGwillhaveatleast5regularorbitsonV.Proof.ItisshowninProposition4.10of[ 9 ]ifjGjjVj1=4,thenwewillhavethate26>24jWje8,thusifthisconditionisnotsatisedthenjGj
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3 andweadoptthenotationinit.Assumethate>118.InthelastparagraphofProposition4.10of[ 9 ],itisshownthatjGj118.Thencertainlythereareatleast5regularG-orbitsonVbyLemma 7 ,acontradiction.Thuse118andeachei118.Setei=pnii,wherepiisaprime.Wenowarguethat(G)7.Assumeni3,thistellsusdim(Wi)=2;4;6,thenbyLemma 4 (4),(5)and(3)respectively,(G=CG(Wi))5.Nowforni4,theonlycasesleftareei=24,25,26or34,thusjWij=28,210,212or38and(G=CG(Wi))5byLemma 4 (8),(3),(11)and(9)respectively.SinceA=FactsfaithfullyonF=T,(A=F)(G=CG(F=T))max((G=CG(Wi)))5anditfollowsthat(G)5+(F)+(G=A)7.Wenowarguethatdl(G)9.Assumeitisfalseandwehavethatdl(G)10.Assumeni2,thistellsusdim(Wi)=2;4,thenbyLemma 4 (4),(5)respectively,dl(G=CG(Wi))5.Nowforni3,theonlycasesleftareei=23,24,25,26,33or34,thusjWij=26,28,210,212,36or38.IfjWij=26,28,210or36,thendl(G=CG(Wi))6byLemma 4 (6),(8),(10)and(7)respectively.Assumealltheeiarefromthepreviouscases,thendl(A=F)dl(G=CG(F=T))max(dl(G=CG(Wi)))6anddl(G)dl(F)+dl(A=F)+dl(G=A)2+6+1=9,acontradiction.Thusei=26orei=34forsomei.Ifei=26forsomei,i.e.jWij=212forsomei,thene=ei=e1sinceotherwisee>118.SupposepisanoddprimeandpjjTj.ThenjUj6andjWj7.Examinetheinequalitye2624jWje8.Itissatisedforall(e;jWj)suchthate>62andjWj7bydirectcalculation,andthusGwillhaveatleast5regularorbitsbyLemma 8 18

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4 (11),acontradiction.Ifei=34forsomei,i.e.jWij=38forsomei,thene=ei=e1sinceotherwisee>118.ThusjE=Zj=38.AlsowehavethatjTj=jUj=3sinceotherwisejUj6andjWj7.Examinetheinequalitye2624jWje8.Itissatisedforall(e;jWj)suchthate>62andjWj7bydirectcalculation,andthusGwillhaveatleast5regularorbitsbyLemma 8 ,acontradiction.ThusjTj=jUj=jZj=3,jFj=jEj=39andjG=AjjAut(Z3)j=2.Sincedl(G=A)+dl(A=F)+dl(F)dl(G)10anddl(G=A)1anddl(F)2wehavedl(A=F)7.A=FisafaithfulcompletelyreduciblesolvablegrouponVwherejVj=38andalsoA=FSp(8;3).AssumeA=FisreduciblethenbyLemma 4 (4),(5),(7),dl(A=F)6,acontradiction.ThusweknowA=FisirreducibleandbyLemma 4 (9)wehavejA=Fj245andjGj=jFjjA=FjjG=Aj392452.jWj4sincejUj=3andthusjVjjWj81481.NowjGj392452<4209thenGhasatleast5orbitsonVsuchthatCG(v)K=(LLL)\G. (2) SupposeHhasatleast3orbitsonV1suchthatCH(v1)L,L/H.Ifalsom>4,thenGhasatleast5orbitsonVsuchthatCG(v)K=(LLL)\G. (3) SupposeHhasatleast5orbitsonV1suchthatCH(v1)L,L/H.ThenGhasatleast5orbitsonVsuchthatCG(v)K=(LLL)\G.Proof.GisisomorphictoasubgroupofHoSandSisasolvableprimitivepermutationgroupon.LetHhaveatleastsorbitsofelementsinV1whosecentralizerinHisinLwhereL/H. 19

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12 ].Weconstructthevweneedinthefollowingway.Ifv=(v1;;vm)inVissuchthatvaandvbarenotH-conjugatewhenaandbbelongtodistincti's,thenCG(v)(HHH)\G.Converselyv=(v1;;vm)inVdeterminesapartitionof:vaandvbareinthesamesubsetofifandonlyifvaandvbareH-conjugate.ThenweknowCG(v)(HHH)\GifandonlyifthepartitiondeterminedbyvistriviallycentralizedbyS.Themethoddevelopedby[ 12 ]tocountregularorbitsofGonValsoappliestocountingtheorbitsofVwhosecentralizerisin(LLL)\GintermsofthenumberofH-orbitsofxinV1suchthatCH(x)LandthenumberoftriviallycentralizedpartitionsofSon.Inparticular,ifHhassorbitsxwithCH(x)L,thenumberofG-orbitsvwithCG(v)in(LLL)\Gis1 12 ]appliesandnotethatpartbasstatedthereisnotasstrongastheproofindicates,ratherifm>9,thenGhasatleast4s(s-1)regularorbits.Foroursituation,weseefromparts(i),(ii)and(iii)thatCorollaryshows (1) Ifs>4,thenGhasatleastsorbitsin(LLL)\G. (2) Ifm>9,thenGhasatleast4s(s1)orbitsin(LLL)\G. (3) Ifm>4ands3,thenGhasatleast2sorbitsin(LLL)\G.Theresultfollows. 20

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1 wehavedl(G)9and(G)7andwemaychooseK=G.NowweassumethatVisnotquasi-primitive,andthereexistsNnormalinGsuchthatVN=V1Vmform>1homogeneouscomponentsViofVN.BychoosingNmaximalsuch,thenS=G=NprimitivelypermutestheVi.AlsoV=VG1,inducedfromNG(V1).IfH=NG(V1)=CG(V1),thenHactsfaithfullyandirreduciblyonV1andGisisomorphictoasubgroupofHoS.Ifx12V1andC1istheHconjugacyclassesofx1,thentheonlyGconjugatesofx1inV1aretheelementsofC1.ThesetofGconjugatesofx1isC1[[CmwhereCiViisaGconjugateofC1.Choosey12V1inanHconjugateclassdierentthanthatofx1.AlsochoosexiandyiinViconjugatetox1andy1(respectively)foralli.ThennoxiiseverGconjugatetoayj.Inparticular,ifg2Gcentralizesv=x1++xj+yj+1++ym,thengandNgmuststabilizethesetsfV1;:::;VjgandfVj+1;:::;Vmg.NowSisasolvableprimitivepermutationgroupon=fV1;:::;Vmg.GisisomorphictoasubgroupofHoS.ByinductioneitherthereisL/Handdl(L)9,(L)7suchthatHwillhaveatleast5orbitsofelementsinV1whosecentralizersareinLorHhaslessthan5orbitsonV1.IfHhasatleast5orbitsofelementsinV1whosecentralizersareinL,byProposition 1 (3),Ghasatleast5orbitsonVsuchthatCG(x)(LLL)\G.InthiscasewesetK=(LLL)\G.Thus,wecanassumethatHhaslessthan5orbitsonV1.Thus(H)5,dl(H)6byLemma 5 .Clearlyr(H)2isalwaystrue.Ifm>9,byProposition 1 (1),Ghasatleast5orbitsonVsuchthatCG(x)(HHH)\G.InthiscasewesetK=(HHH)\G. 21

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5 .Sincem9,(S)4anddl(S)5byLemma 1 andLemma 4 (2),(4).Thus(G)7anddl(G)9.InthiscasewemaysetK=G.Nowwehaver(H)3andifm>5,byProposition 1 (2),Ghasatleast5orbitsonVsuchthatCG(x)(HHH)\G.InthiscasewemaysetK=(HHH)\G.Nowthecasesleftare,r(H)=3,m=2;3;4;5.(H)4,dl(H)6byLemma 5 .Sincem=2;3;4;5,(S)3,dl(S)3.Thus(G)7anddl(G)9.InthiscasewemaysetK=G.r(H)=4,m=2;3;4;5.(H)5anddl(H)6byLemma 5 .Whilem=2;3;5,misaprime,(S)2anddl(S)2.Thus(G)7anddl(G)9andinthesecaseswemaysetK=G.Whilem=4,(G)7byLemma 6 ,dl(G)9byLemma 5 andinthiscasewemayalsosetK=G. 2 ,wemayassumeVisnotirreducible,i.e.,wecanwriteV=V1V2forcompletelyreduciblemodulesV1andV2.LetCi=CG(Vi).ThenGisisomorphictoasubgroupofG=C1G=C2.Byinductivehypothesis,wemaychoosexiandKi/(G=Ci)suchthatCG=Ci(xi)Ki,(Ki)7anddl(Ki)9.Setx=x1+x2,thenCG(x)(K1K2)\G.DenoteK=(K1K2)\G,clearlyK/G,(K)7anddl(K)9. AsconsequencesofTheorem 3 ,wehavethefollowingtheorems. 22

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4 orTheorem 5 canbeimproved.Hereweprovideacounterexampleshowingthatboththeoremsarethebestpossible.WeprovethatthereexistsanitesolvablegroupG,andVafaithfulirreducibleGmodulesuchthatforeveryv2VwehaveCG(x)6F6(G)anddl()9.Inordertoestimate()anddl()weneedthefollowinglemma. Theconstructionofthecounterexampleisasfollows.First,wecanndanirreduciblesolvablelineargrouponVwithr(H)=2and(H)=3.OneexampleforthisiswhenVisavectorspaceofdimension4overF3,F(H)isextraspecialoforder25,jF2(H)=F(H)j=5andH=F2(H)=Z4.ThisconcreteexampledoesexistduetoBucht(XII7.4of[ 6 ]).NowweconsiderK=HoS3actsonU=V+V+V,clearlyr(K)=4and(K)=5. 23

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9 ].Theonlyorbitsthathavenotdiscussedarex1+x2+x3+x4andx4+x4+x4+x4.Wecallthemexceptionalorbits.NowconsidertheactionofLoS4onZ.AssumesomeoftheorbitsofLappeartwicethenS4andthendl()9+2=11and()7byLemma 9 .AssumenoneoftheorbitsofLappeartwicethensomeofthenon-exceptionalorbitwillappearandthusdl()9and()7. 3 .ThenexttheoremstrengthensTheoremAof[ 10 ]byreplacingF10(G)byF8(G). 24

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10 ]butuseourTheorem 4 ThenexttheoremstrengthensTheoremCof[ 10 ]byreplacingF10(G)byF8(G). 10 ]butuseourTheorem 4 ThenexttheoremstrengthensTheoremC'of[ 10 ]byreplacingF10(G)byF8(G). 10 ]butuseourTheorem 4 ThenextcorollarystrengthensCorollary2.1of[ 10 ]byreplacingF10(G)byF8(G). 10 ]butuseourTheorem 4 ThenextcorollarystrengthensCorollary2.3of[ 10 ]byreplacing9by7. 10 ]butuseourTheorem 7 ThenextcorollarystrengthensCorollary5.1of[ 10 ]byreplacingp10byp8. 10 ]butuseourTheorem 7 25

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Denition1. 1 .TheneverynormalabeliansubgroupofGiscyclicandGhasnormalsubgroupsZUFAGsuchthat, (1) (2) (3) (4) (5) (6) (7) 9 ,Theorem1.9]thereexists~Ei;Ti/Gandallthefollowinghold, 26

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(ii) ~Eiisextraspecialor~Ei=Zi; (iii) exp(~Ei)=piorpi=2; (iv) (v) IfTiisnotcyclic,thenthereexistsUi/GwithUicyclic,UiTi,jTi:Uij=2andCTi(Ui)=Ui; (vi) If~Ei>Zi,thenEi=Zi=Ei1=ZiEid=ZiforchieffactorsEik=ZiofGandwithZi=Z(Eik)foreachkandEikCG(Eil)fork6=l.WedeneUi=Tiifpi6=2.WedeneU=Qmi=1Ui,T=Qmi=1Ti,F=EUandA=CG(U).Ifpi6=2,thenby(i),(ii),(iii)~Ei=1(Pi)andtherefore~Ei=Ei.Ifpi=2andassume~Ei>Zi,~Ei=Zi=QkEik=ZiforchieffactorsEik=ZiofGby(vi)andthusEik=[Eik;G]and~Ei=[~Ei;G].By(v),Aut(Ti)isa2-groupand[Ti;G;;G]=1.Thus[Pi;G;G;;G]=~Eiandtherefore~Ei=Ei.TheotherresultsmainlyfollowfromCorollary1.10,2.6andLemma2.10of[ 9 ].SinceCG(F)=CG(EU)CG(E)=TandCT(U)=U,wehaveCG(F)F.SinceA=CG(U),F(G)\A=CF(G)(U)=FandthusA=FactsfaithfullyonE=Z. 9 ,wehavejGjdim(W)jA=Fje2(jWj1).Proof.ByTheorem 9 ,jGj=jG=AjjA=FjjFjandjFj=jE=ZjjUj.SincejG=Ajjdim(W),jE=Zj=e2andjUjj(jWj1),jGjdim(W)jA=Fje2(jWj1). 9 (1) Ifg2FthenjCV(g)jjWj1 2eb. (2) Ifg2AnFthenjCV(g)jjWjb3 4ecb. 27

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Ifg2AnF,s3ands-jEj,thenjCV(g)jjWjb1 2ecb. (4) Ifg2AnF,s=2ands-jEj,thenjCV(g)jjWjb2 3ecb. (5) Ifg2GnAthenjCV(S)j=jCV(g)jjWj1 2eb.Itisprovedin[ 9 ,Proposition4.10]thatforg2F(G),jCV(g)jjWj1 2eb=jVj1=2.SinceFF(G),(1)follows.SinceCG(F)Fandg62F,[g;F]6=1.Sinceg2A=CG(U)andF=EU,[g;E]6=1.Sinces-jEj,thereexistsag-invariantq-subgroupQE,forsomeprimeq6=ssuchthatQisextraspecial,[Q;g]=Q,[Z(Q);g]=1,Z(Q)/GandtheactionofgonQ=Z(Q)isxed-pointfree.LetKbeasplittingeldforhgiQwhichisaniteextensionofFandsetVK=VFK.SincedimK(CVK(g))=dimF(CV(g)),wemayconsiderVKinsteadofV.Let0=V0V1Vl=VKbeahgiQ-compositionseriesforVKwithquotient 4 ,TheoremV.17.13]orHall-HigmanTheorem[ 5 ,TheoremIX.2.6],dimK( 2whens3and2 3whens=2.Assumesjqk+1andletqk+1=stwheret1isaninteger,then=t st1.Thus1 2whens3.Whens=2,t2sinceqisanoddprimeandwehave2=3. 28

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4ec.SincedimK(CVK(g))jdimK(C 4ecb.Thus(2)holdsbythisparagraphand(3),(4).Letg2AnF,thenthereexistsu2Usuchthat[g;u]2U#.SinceCV([g;u])=1,jCV(g)jjVj1=2,thus(5)holds. 7 ]Isaacsdenedgoodelement.LetC=Z=CE=Z(x),inoursituation,xisagoodelementif[x;C]=1.Wecallanelementbadifitisnotgood.Wehavethefollowing, (1) Ifxisagoodelement,thenwehavethattheBrauercharacterofxonV,say(x)isthensuchthatj(x)j2=jCE=Z(x)j. (2) Ifxisabadelement,then(x)=0. 29

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7 ,Theorem3.5]. 9 ,letx2EPs(AnF)and(s;charF)=1.LetC=Z=CE=Z(x),wecallxagoodelementif[x;C]=1,wecallxabadelementifitisnotgood. (1) Assumexisabadelementandlet=e=s,thenjCV(x)jjWjb. (2) AssumexisagoodelementandjCE=Z(x)ja,let=b1 (3) Assumeo(x)=2andxisagoodelement,thenjCE=Z(x)jisthesquareofaninteger.Assumefurther2je,thenjCE=Z(x)jisthesquareofaneveninteger.Proof.Notethatchar(F)-jhxiEj.LetKbeasplittingeldforhxiEwhichisaniteextensionofFandsetVK=VFK.SincedimK(CVK(x))=dimF(CV(x)),wemayconsiderVKinsteadofV.Let0=V0V1Vl=VKbeanhxiE-compositionseriesforVKwithquotient 12 canbeappliedonhxiEand s.Ifxisagoodelement,thenallthenontrivialelementsy2hxiaregoodelementsandjj(y)j2=jCE=Z(y)jaandthusjj(y)ja1=2.dimK(C

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2(dimK( 2(e+j(x))andthusjCE=Z(x)j=jj(x)j2isthesquareofaninteger.Since2je,2jj(x)andthusweknowthatjCE=Z(x)j=jj(x)j2isthesquareofaneveninteger. 9 andweadoptthenotationinit.Letpbeaprimeandx2EPp(AnF)andassumejCE=Z(x)j=Qipimi.DeneUp=gcd(jUj;p2).Wehavethefollowing, (1) NEPp(AnF)NEPp(A=F)jFj. (2) NEPp(AnF)NEPp(A=F)jFj (3) NEPp(xF)QiMiUpwhereMi=8<:pi2niifp=pi6=2;pi2nimiifp6=pi;2miifp=pi=2: Assumefurtherp=2andxisagoodelement.DeneS=fyjy2EP2(xF)andyisagoodelementg,thenjSjQiMiU2whereMi=8<:pi2nimiifpi6=2;2miifpi=2andnimi;22nimiifpi=2andni
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ti)2=1tellsthat 32

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(1) (2) (3) Ifp=2,thenNPC(G;2;n)2n+1,NPC(G;2;i)=0forall0i2n1andi6=nandNEP2(G)2n+1.Proof.By[ 11 ,Proposition3.1(1)],GmaybeidentiedwithasubgroupofthesemidirectproductofGF(p2n)byGal(GF(p2n):GF(p))actinginanaturalmannerofGF(p2n)+.AlsoG\GF(p2n)=F(G)andjG\GF(p2n)jjpn+1.ClearlyG=F(G)iscyclicoforderdividing2n.Now(1)and(2)hold.(3)Considerx2EP2(G)andthusx2GnF(G),x=awhere2Gal(GF(p2n):GF(p))isoforder2anda2GF(p2n).Sincex2=aa=1,a2n+1=1andthusNEP2(G)2n+1.Since2-22n1,xcannotactxedpointfreelyonVandthusxisconjugatewith,clearlyjCV(x)j=2n. 33

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16 (2) 17 (3) (1) GL(2;2)=SL(2;2)=S3,NEP2(S3)=3andNEP3(S3)=2. (2) (3) (1) LetV1=(2;F2),H=S3,n=2,S=S2andG=S3oS2,thenjGj=622,NEP2(G)=21,NEP3(G)=8,NPC(G;2;3)=6,NPC(G;2;2)=15,NPC(G;2;1)=0,NPC(G;3;2)=4. 34

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LetV1=(2;F2),H=S3,n=3,S=S3andG=S3oS3,thenjGj=64,NEP2(G)=135,NEP3(G)=98,NPC(G;2;5)=9,NPC(G;2;4)=45,NPC(G;2;3)=81,NPC(G;2;2)=0,NPC(G;2;1)=0. (3) LetV1=(2;F2),H=S3,n=4,S=S4andG=S3oS4,thenjGj=6424,NEP(G)=1883,NPC(G;2;7)=12,NPC(G;2;6)=90,NPC(G;2;5)=324,NPC(G;2;4)=513,NPC(G;2;3)=0,NPC(G;2;2)=0,NPC(G;2;1)=0. (4) LetV1=(2;F2),H=S3,n=5,S=F20andG=S3oF20,thenjGj=6520,NEP(G)=7169,NPC(G;2;8)=90,NPC(G;2;6)=585,NPC(G;2;4)=0,NPC(G;2;2)=0. (5) LetV1=(2;F3),H=SL(2;3),n=2,S=S2andG=SL(2;3)oS2,thenjGj=2422,NEP2(G)=27,NEP3(G)=80,NEP5(G)=0. (6) LetV1=(2;F3),H=SL(2;3),n=3,S=S3andG=SL(2;3)oS3,thenjGj=2436,NEP2(G)=151,NEP3(G)=1880,NEP(G)=2031,NPC(G;2;4)=75,NPC(G;2;2)=75,NPC(G;2;0)=1andNPC(G;3;5)=24.Proof.HoS=LoSwhereL=HHH.Letx2HoSando(x)=p(paprime).x=lwhere2Sandl2L,l=h1h2hn.Sinceo(x)=p,eithero()=poro()=1.Ifo()=1theno(l)=pandjCV(x)j=jCV1(h1)jjCVn(hn)j.ByLemma 20 ,NEPp(L)=(NEP(H)+1)n1.Ifo()=pthenisadisjointproductoftp-cycles,say=(1;;p)(p+1;2p)((t1)p+1;tp)(tp+1)(n),thenjCV(x)j=jV1jtjCVtp+1(htp+1)jjCVn(hn)jbyLemma 19 andthusNEPp(L)=jHj(p1)t(NEP(H)+1)ntpbyLemma 19 .ThefollowingcalculationsarebasedonthepreviousparagraphandLemma 21 oralternativelytheycanbecheckedbyGAP[ 3 ]. (1) (2) (3) 35

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(4) (5) (6) (1) Let(n;F)=(4;F2),thenjGj622,NEP2(G)21,NEP3(G)8,NEPf2;3g0(G)4andNEP(G)29.NPC(G;2;3)6,NPC(G;2;2)15,NPC(G;2;1)=0,NPC(G;3;2)4. (2) Let(n;F)=(6;F2),thenjGj64. (3) Let(n;F)=(4;F3),thenjGj4822.Proof.Weprovethesedierentcasesonebyone. (1) (a) 22 (1). (b) (2) Thisfollowsfrom[ 9 ,Corollary2.15]. (3) AssumetheactionofGonVisimprimitive,thenGwillbeasubgroupofoneofthefollowinggroups, (a) GL(2;3)oS2andthusjGj4822=4608. (b) GL(1;3)oS4andthusjGj2424=384. 36

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(a) (b) 10 (c) 24 (2).jWj31anddim(W)1,byLemma 10 (1) Let(n;F)=(2;F2),thenG.S3,jGj6,NEP2(G)3,NEP3(G)2,NPC(G;2;1)3. (2) Let(n;F)=(4;F2),thenjGj622,NEP2(G)21,NEP3(G)8,NEPf2;3g0(G)4,NEP(G)29,NPC(G;2;3)6,NPC(G;2;2)15,NPC(G;2;1)=0,NPC(G;3;2)4. (3) Let(n;F)=(6;F2),thenjGj64,NEP2(G)171,NEP3(G)242,NEPf2;3g0(G)6,NPC(G;2;5)9,NPC(G;2;4)45,NPC(G;2;3)162,NPC(G;2;2)=0,NPC(G;2;1)=0. (4) Let(n;F)=(8;F2),thenjGj6424,NEP(G)1883,NPC(G;2;7)12,NPC(G;2;6)90,NPC(G;2;5)324,NPC(G;2;4)513,NPC(G;2;3)=0,NPC(G;2;2)=0,NPC(G;2;1)=0. (5) Let(n;F)=(10;F2),thenjGj6524,NEP(G)7536,NPC(G;2;8)126,NPC(G;2;6)1647,NPC(G;2;4)=0,NPC(G;2;2)=0. (6) Let(n;F)=(12;F2),thenjGj(64)22. (7) Let(n;F)=(2;F3),thenG.SL(2;3)andjGj24. (8) Let(n;F)=(4;F3),thenjGj2422andNEP(G)107,NEP2(G)95,NEP3(G)95,NEP5(G)64andGwillhavenoelementswithotherprimeorder.AssumeG6.SL(2;3)oS2,thenNPC(G;3;3)=0. (9) Let(n;F)=(6;F3),thenjGj2436andNEP(G)2031.AssumeG6.SL(2;3)oS3,thenNEP2(G)192andNPC(G;3;5)8. (10) Let(n;F)=(8;F3),thenjGj245. (11) Let(n;F)=(2;F5),thenjGj24andNEPf2;3g0(G)=0. 37

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Let(n;F)=(4;F5),thenjGj622244andNEP(G)1920. (13) Let(n;F)=(2;F7),thenjGj48. (14) Let(n;F)=(4;F7),thenjGj622246. (15) Let(n;F)=(2;F11),thenjGj120andNEP(G)15. (16) Let(n;F)=(2;F13),thenjGj28. (17) Let(n;F)=(2;F17),thenjGj96. (18) Let(n;F)=(2;F19),thenjGj72. (19) Let(n;F)=(2;F23),thenjGj2224.Proof.Weprovethesedierentcasesonebyone. (1) Let(n;F)=(2;F2).ThisfollowsfromLemma 21 (1). (2) Let(n;F)=(4;F2).AssumeGisirreduciblethentheresultfollowsfromLemma 23 (1).AssumeGisreducible,thenG.S3S3andtheresultfollowsfromLemma 22 (1). (3) Let(n;F)=(6;F2).AssumeGisirreducible,thenGsatisesoneofthefollowing, (a) 16 ,NEP2(G);NEP3(G)jGj212=42,NEPf2;3g0(G)6,NPC(G;2;5)=0,NPC(G;2;4)=0,NPC(G;2;3)21,NPC(G;2;2)=0,NPC(G;2;1)=0. (b) 22 (2). (c) 17 (d) 10 .NEP3(G)NEP3(AnF)+NEP3(F)827+271=242byLemma 14 (2).NEP2(GnA)189=162byLemma 15 andNEP2(AnF)9byLemma 14 (2),thusNEP2(G)NEP2(GnA)+NEP2(AnF)171.Forallx2EP2(GnA),jCV(x)j=23.Forallx2EP2(AnF),sincehxiE/G,jCV(x)j=24byCliord'sTheoremandHall-HigmanTheorem[ 5 ,TheoremIX.2.6].Thustheresultfollows.AssumeGisreducible,thenGsatisesoneofthefollowing, (a) 38

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(4) Let(n;F)=(8;F2).AssumeGisirreducible.AssumefurthertheactionofGonVisprimitive.Since2-ej8,e=1andthusF(G)isabelian.ByLemma 17 ,NEP(G)jGj(24+1)8=136,NPC(G;2;4)17andNPC(G;2;i)=0forall1i7andi6=4.AssumefurthertheactionofGonVisimprimitive,thenGsatisesoneofthefollowing, (a) 23 (1),jGj(622)22,NEP(G)3030+622=972,NPC(G;2;7)6+6=12,NPC(G;2;6)15+15+66=66,NPC(G;2;5)615+156=180,NPC(G;2;4)622+1515=297,NPC(G;2;3)=0,NPC(G;2;2)=0,NPC(G;2;1)=0. (b) 22 (3).AssumeGisreducible,thenGsatisesoneofthefollowing, (a) (b) (c) (5) Let(n;F)=(10;F2).AssumeGisirreducible.AssumefurthertheactionofGonVisimprimitive,thenGsatisesoneofthefollowing, (a) 22 (4). (b) 16 andjCV(x)j=25forallx2EP2(G).AssumefurthertheactionofGonVisprimitive.Since5-221,e6=5ande=1,F(G)isabelian.ByLemma 17 ,NEP(G)jGj330.NPC(G;2;8)=0,NPC(G;2;6)=0,NPC(G;2;4)=0,NPC(G;2;2)=0.AssumeGisreducible,thenGsatisesoneofthefollowing, 39

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(b) (c) (6) Let(n;F)=(12;F2).AssumeGisirreducible.AssumefurthertheactionofGonVisimprimitive,thenGsatisesoneofthefollowing, (a) 23 (2)jGj(64)22. (b) 23 (1)jGj(622)36. (c) 16 ,jGj(73)224.AssumefurthertheactionofGonVisprimitive,thenGisisomorphictoasubgroupofoneofthefollowinggroups, (a) Ife=1,thenF(G)isabelianandjGj(26+1)12byLemma 17 (b) Ife=3,thenbyTheorem 9 ,jE=Zj=32,A=F.SL(2;3)andjA=Fj24,jWj24anddim(W)4.ByLemma 10 (a) 23 (2). (b) (c) (d) (7) Let(n;F)=(2;F3),thenG.Sp(2;3)=SL(2;3)andjGj24byLemma 21 (2). (8) Let(n;F)=(4;F3).AssumeGactsirreduciblyonV.AssumefurthertheactionofGonVisimprimitive,thenGsatisesoneofthefollowing, (a) 16 ,ifitispairedthenjGjj96andforx2EP3(G)wehavejCV(x)j32andthus 40

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22 (5). (b) 16 andthusjGjj822=32.NEP3(G)=0andNPC(G;3;3)=0.AssumefurthertheactionofGonVisprimitive,thene=1;2or4andforxwitho(x)=3,jCV(x)j32byLemma 11 (1),(3),(5)andthusNPC(G;3;3)=0.ItiswellknownthatamaximalsubgroupofSp(4;3)isisomorphictooneofthevegroupsM1,M2,M3,M4andM5,whereM1=SL(2;3)oS2andjM2j=jM3j=2434,jO3(M2)j=jO3(M3)j=33,M4=2:S6,M5=(D8Q8):A5.WecanhenceassumeGismaximalamongthesolvablesubgroupsofM2;M3;M4orM5.AssumeGisasubgroupofM2orM3,thenclearlyjGjj48.AssumeGisasubgroupofM4,itisnothardtoshowthatjGjj96orjGjj40.AssumeGisasubgroupofM5,thenG=(D8Q8):LwhereLisS3,A4orF10andthusweknowjGj384.ItischeckedbyGAP[ 3 ]thatforallGprimitiveandjGj384,NEP(G)107,NEP2(G)75,NEP3(G)80,NEP5(G)64andGwillhavenoelementswithotherprimeorder.AssumeGisreducible,thenGsatisesoneofthefollowing, (a) 21 (3)andforx2EP3(G)wehavejCV(x)j32andthusNPC(G;3;3)=0. (b) (9) Let(n;F)=(6;F3).AssumeGisirreducible.AssumefurthertheactionofGonVisimprimitive.ThenGsatisesoneofthefollowing, (a) 22 (6). (b) 16 andjGj782=156.NEP2(G)NEP(G)jGj156.Forallelementsxwitho(x)=3wehavejCV(x)j34sincetheactionisapairandthusNPC(G;3;5)=0.AssumefurthertheactionofGonVisprimitive,thenjCV(x)j3b3 46c=34forallx2GbyLemma 11 (1),(2),(5)andthusNPC(G;3;5)=0.Since3-ej6,ecouldbe1or2. (a) Assumee=1,thenF(G)isabelianandNEP2(G)NEP(G)jGj(33+1)6=168byLemma 17 (b) Assumee=2,thenjWj33,jG=Ajjdim(W)3,A=FS3andjA=Fj6.NEP(G)jGj362226=1872byLemma 10 .NEP2(G)=NEP2(AnF)+NEP2(F)38+8=32byLemma 14 (2).AssumeGisreducible,thenGsatisesoneofthefollowing, (a) 41

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21 (2).IfG6.SL(2;3)oS3,thenH26.SL(2;3)oS2andNPC(H2;3;3)=0by(8),thusNPC(G;3;5)8byLemma 21 (2).Hencetheresultholdsinallcases. (10) Let(n;F)=(8;F3).AssumeGisirreducible.AssumefurthertheactionofGonVisimprimitive,thenGsatisesoneofthefollowing, (a) 16 ,ifitisnotpairedthenjGj(2422)22by(8),ifitispairedthenjGj48222byLemma 23 (3). (b) 16 ,ifitisnotpairedthenG.SL(2;3)oS4andjGj245,ifitispairedthenjGj48224byLemma 21 (3). (c) (a) Assumee=1,thenF(G)isabelianandjGj(34+1)8byLemma 17 (b) Assumee=2,thenA=F.S3andthusjA=Fj6.jWj34anddim(W)4,jGj462280byLemma 10 (c) Assumee=4,thenA=F2SCRSp(4;2)andthusjA=Fj622by(2).jWj32anddim(W)2,jGj2622248byLemma 10 (d) Assumee=8,thenA=F2SCRSp(6;2)andthusjA=Fj64by(3).jWj31anddim(W)1,jGj6427byLemma 10 .AssumeGisreducible,thenGsatisesoneofthefollowing, (a) 23 (3). (b) (c) (11) Let(n;F)=(2;F5).BytheproofofLemma 18 ,jGjj8;12or24,thusjGj24andNEPf2;3g0(G)=0. (12) Let(n;F)=(4;F5).AssumeGisirreducible.AssumefurthertheactionofGonVisimprimitive,thenGsatisesoneofthefollowing, 42

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16 andNEP(G)jGj4424=384. (b) 16 ,ifitisnotapairthenH2SCRSp(2;5)andjGj2422=1152by(11),ifitisapairthenjGj24202=960.InanycasewehaveNEP(G)jGj1152.AssumefurthertheactionofGonVisprimitive,thene=1;2or4andGsatisesoneofthefollowing, (a) 17 (b) 10 (c) 14 (1).AssumeGisreducible,thenGsatisesoneofthefollowing, (a) (b) (13) Let(n;F)=(2;F7).Sincegcd(24;78)=8,jGjmax(16;48)=48byLemma 18 (14) Let(n;F)=(4;F7).AssumeGisirreducible.AssumefurthertheactionofGonVisimprimitive,thenGsatisesoneofthefollowing, (a) 16 andjGj6624=864. (b) 16 ,ifitisnotapairthenH2SCRSp(2;7)andjGj4822by(13),ifitisapairthenjGj48422.AssumefurthertheactionofGonVisprimitive,thenGsatisesoneofthefollowing, (a) 17 (b) primitivecasewithe=2,A=F.S3andjA=Fj6.jWj72anddim(W)2,jGj26448byLemma 10 (c) primitivecasewithe=4,A=F2SCRSp(4;2)andjA=Fj622by(2).jWj71anddim(W)1,jGj622246byLemma 10 .AssumeGisreducible,thenGsatisesoneofthefollowing, 43

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(b) (15) Let(n;F)=(2;F11).By[ 9 ,Theorem2.11]Gsatisesoneofthefollowing, (a) 16 .NEP(G)15. (b) 17 andNEP(G)15. (c) 14 (2). (16) Let(n;F)=(2;F13).Sincegcd(24;1314)=2,jGjmax(28;24)=28byLemma 18 (17) Let(n;F)=(2;F17).Sincegcd(24;1718)=6,jGjmax(36;96)=96byLemma 18 (18) Let(n;F)=(2;F19).Sincegcd(24;1920)=4,jGjmax(40;72)=72byLemma 18 (19) Let(n;F)=(2;F23).Sincegcd(24;2324)=24,jGjmax(48;2224)=2224byLemma 18 9 ,ife6andjGjjWjb3 4ecb=jWj(eb3 4ec)b=jWj(2b3 4ece)b.Sincewehavee6,jWj(2b3 4ece)b22=4,thencertainlythereareatleast5regularG-orbitsonV. 9 ,ife6andtheinequalitye2624jWje8issatised,thenGwillhaveatleast5regularorbitsonV. 44

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9 ,Proposition4.10]thatifjGjjVj1 4,thenwewillhavethate26>24jWje8.Thusbyourcondition,wehavethatjGj24jWje8wherejWjisaprimepower.p1;;psaredierentprimesdividingeandp1psjjZjjjUjjjWj1.Theexceptionalcasesare, (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) 45

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25 ,jGjjWj(d1 4ee)bjWjd1 4ee.Lete=81andjWj=4.A=F2SCRSp(8;3)andjA=Fj245byLemma 24 (10).ByLemma 10 ,jGj224539
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24 (1),(9).ByLemma 10 ,jGj624365426
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13 (3)and 24 (5).DeneA2=fhxijx2EP2(AnF),xisagoodelementandjCE(x)j=28g.Forallx2P2A2wehavejCV(P)j=jCV(x)jjWj24bbyLemma 13 (2)andweset2=24.jA2j12622(jWj1)=a2byLemma 24 (5)andLemma 14 (4).DeneA3=fhxijx2EP2(AnF),xisgoodelementandjCE(x)j=26g.Forallx2P2A2wehavejCV(P)j=jCV(x)jjWj20bbyLemma 13 (2)andweset3=20.jA3j164724(jWj1)=a3byLemma 24 (5)andLemma 14 (4).DeneA4=fhxijx2EP(GnA)g.Thusforallx2P2A4,jCV(P)j=jCV(x)jjWj16bbyLemma 11 (5)andweset4=16.jA4jjGjdim(W)6524210(jWj1).SinceA4isemptyifdim(W)=1weseta4=0ifdim(W)=1anda4=dim(W)6524210(jWj1)ifdim(W)6=1.Itisroutinetocheckthat?issatised,acontradiction.Lete=31andjWj=32.A=F2SCRSp(2;31)andjA=Fj313.ByLemma 10 ,jGj5313313
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10 .DeneA1=fhxijx2EP(F);x2EP3(AnF)andxisabadelement;x2EPs(AnF)forallprimess6=3g.Assumex2EP(F),thenjCV(x)j713:5bbyLemma 11 (1).Assumex2EP3(AnF)andxisabadelement,thenjCV(x)j79bbyLemma 13 (1).Assumex2EPs(AnF),sisaprimeands6=3,sinces-jEj,thenjCV(x)j718bbyLemma 11 (3),(4).ThusjCV(P)j718bforallP2A1andweset1=18.jA1j2032372=a1byLemma 24 (9)and 14 (1).DeneA2=fhxijx2EP3(AnF)andxisagoodelementg.Thenforallx2P2A2wehavejCE(x)j35andjCV(P)j=jCV(x)j7b1 3(27+32:5+32:5)cb=719bbyLemma 13 (2)andweset2=19.jA2j203237=2=a2byLemma 24 (9)and 14 (2).Itisroutinetocheckthat?issatised,acontradiction.NowweassumejWj=4andsincejG=Ajjdim(W)j2,jGj2243637=BbyLemma 10 .A=Fwillsatisfyoneofthefollowing, (1) 11 (1).Assumex2EP3(AnF)andxisabadelement,thenjCV(x)j49bbyLemma 13 (1).Assumex2EP3(AnF),xisagoodelementandjCE(x)j34,thenjCV(x)j4b1 3(27+9+9)cb=415bbyLemma 13 (2).ThusjCV(P)j415bforallP2A1andweset1=15.jA1j203237=a1byLemma 14 (1)and 22 (6).DeneA2=fhxijx2EP2(AnF)gandjCV(P)j418bforallP2A2byLemma 11 (4)andweset2=18.Forallx2A2,jCE(x)j=1;32or34byLemma 13 (3).jA2j7532+7534+36=a2byLemma 14 (3)and 22 (6).DeneA3=fhxijx2EP3(AnF),xisagoodelementandjCE(x)j=35g.Thusforallx2P2A3,jCV(P)j=jCV(x)j4b1 3(27+232:5)cb=419bbyLemma 13 (2)andweset3=19.jA3j2437=2=a3byLemma 14 (2)and 22 (6).DeneA4=fhxijx2EP(GnA)g.ForallP2A0wehavejCV(P)j413:5bbyLemma 11 (5)andweset4=13:5.SincejG=Ajj2,o(x)=2forallx2P2A4.ByLemma 15 ,jA4jjAj=27624337=27=a4.Itisroutinetocheckthat?issatised,acontradiction. (2) 49

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11 (1).Assumex2EP3(AnF)andxisabadelement,thenjCV(x)j49bbyLemma 13 (1).Assumex2EP3(AnF),xisagoodelementandjCE(x)j34,thenjCV(x)j4b1 3(27+9+9)cb=415bbyLemma 13 (2).Assumex2EPs(AnF),sisaprimeands5,sinces-jEj,jCV(x)j414bbyLemma 11 (3).ThusjCV(P)j415bforallP2A1andweset1=15.jA1j203237=a1.DeneA2=fhxijx2EP2(AnF)g.jCV(P)j418bforallP2A2byLemma 11 (4)andweset2=18.jA2j19236=a2byLemma 24 (9)and 14 (2).DeneA3=fhxijx2EP3(AnF),xisagoodelementandjCE(x)j=35g.Foranyx2P2A3,sincejCE(x)j=35,jCV(P)j=jCV(x)j4b1 3(27+232:5)cb=419bbyLemma 13 (2)andweset3=19.jA3j837=2=a3byLemma 24 (9)and 14 (1).DeneA4=fhxijx2EP(GnA)g.ForallP2A0wehavejCV(P)j=413:5bbyLemma 11 (5)andweset4=13:5.SincejG=Ajj2,o(x)=2forallx2P2A4.ByLemma 15 ,jA4jjAj=27624337=27=a4.Itisroutinetocheckthat?issatised,acontradiction.Lete=26.26jjWj1andthusjWj27.A=F2SCRSp(2;2)SCRSp(2;13)andjA=Fj628byLemma 24 (1),(16).ByLemma 10 ,jGjdim(W)628262(jWj1)
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14 (1).DeneA2=fhxijx2EP(GnA)gthenjCV(P)jjWj12bforallP2A2byLemma 11 (5)andweset2=12.jA2jjGjdim(W)6424242(jWj1)=a2.Itisroutinetocheckthat?issatised,acontradiction.ThusweknowjWj=19,13or7.AssumejWj=7;13;19,thenjG=Ajdim(W)=1.jGj6424242(jWj1)=BbyLemma 10 .NowA=F.HIwhereH2SCRSp(6;2)andI.SL(2;3).DeneA1=fhxijx2EP(F);x2EP(AnF)andxisabadelement;x2EP2(AnF),xisagoodelementandjCE(x)j26;x2EP3(AnF),xisagoodelementandjCE(x)j2432;x2EPs(AnF)forallprimess5g.Assumex2EP(F),thenjCV(x)jjWj12bbyLemma 11 (1).Assumex2EP(AnF)andxisabadelement,thenjCV(x)jjWj12bbyLemma 13 (1).Assumex2EP2(AnF),xisagoodelementandjCE(x)j26,thenjCV(x)jjWjb1 2(24+8)cb=jWj16bbyLemma 13 (2).Assumex2EP3(AnF)andxisagoodelementandjCE(x)j2432,thenjCV(x)jjWjb1 3(24+12+12)cb=jWj16bbyLemma 13 (2).Assumex2EPs(AnF),sisaprimeands5,sinces-jEj,jCV(x)jjWj12bbyLemma 11 (3).ThusjCV(P)jjWj16bforallP2A1andweset1=16.Since2;3jjUj,jA1j25362632(jWj1)=2=a1byLemma 14 (2).DeneA2=fhxijx2EP2(AnF),xisagoodelementandjCE(x)j=2432g.Thusforallx2P2A2,jCV(P)j=jCV(x)jjWj18bbyLemma 13 (2)andweset2=18.jA2j4524(jWj1)=3=a2byLemma 24 (3)and 14 (3).DeneA3=fhxijx2EP3(AnF),xisagoodelementandjCE(x)j=263g.Thusforallx2P2A3,jCV(P)j=jCV(x)jjWjb1 3(24+2830:5)cb=jWj17bbyLemma 13 (2)andweset3=17.jA3j832(jWj1)=2=a3byLemma 21 and 14 (3).Itisroutinetocheckthat?issatised,acontradiction.Lete=23.Then23jjWj1andthusjWj47.SinceA=F2SCRSp(2;23),jA=Fj2224byLemma 24 (19).ByLemma 10 ,jGjdimW2224232(jWj1)
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24 (1),(15).ByLemma 10 ,jGjdim(W)1206222(jWj1)24g.SincejCE(x)jisthesquareofanevennumberbyLemma 13 (3),jCE(x)j=2252.Thusforallx2P2A2,jCV(P)j=jCV(x)j11b1 2(20+10)cb=1115bbyLemma 13 (2)andweset2=15.jA2jjHj2362223=a2byLemma 14 (3).Itisroutinetocheckthat?issatised,acontradiction. 52

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24 (18).ByLemma 10 ,jGjdim(W)72192(jWj1)
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3(18+2630:5)cb=jWj12bbyLemma 13 (2)andweset3=12.jA3j9534(jWj1)=2=a3byLemma 24 (8)and 14 (3).DeneA4=fhxijx2EP3(AnF),xisagoodelementandjCE(x)j=34g.Thenforallx2P2A4wehavejCV(P)j=jCV(x)jjWjb1 3(18+232)cb=jWj12bbyLemma 13 (2)andweset4=12.jA4j22234(jWj1)=2=a4byLemma 14 (3).Itisroutinetocheckthat?issatised,acontradiction.Lete=17.Then17jjWj1andthusjWj103.SinceA=F2SCRSp(2;17),jA=Fj323byLemma 24 (17).ByLemma 10 417e=jWj5,acontradiction.Lete=15.SinceA=F2SCRSp(2;3)SCRSp(2;5),jA=Fj2424byLemma 24 (7),(11).AssumejWj31.jGjdim(W)2424152(jWj1)=BbyLemma 10 .DeneA1=fhxijx2EP(AnF)gthenjCV(P)jjWjb3 415cb=jWj11bforallP2A1byLemma 11 (2)andweset1=11.SinceNEP5(A=F)=0byLemma 18 ,jA1j2424152(jWj1)=5=a1byLemma 14 (2).DeneA2=fhxijx2EP(F)orx2EP(GnA)gthenjCV(x)jjWj7:5bforallx2A2Lemma 11 (1),(5)andweset2=7:5.jA2jjGjdim(W)2424152(jWj1)=a2.Itisroutinetocheckthat?issatised,acontradiction.AssumejWj=16thenjUj15,jFj=jE=ZjjUj=153andjG=Ajjdim(W)=4.jGj42424153=BByLemma 10 .A=F=HIwhereH.SL(2;3)andI2SCRSp(2;5).2,3aretheonlyprimesthatmightdividejA=Fj.DeneA1=fhxijx2EP(F);x2EP2(AnF);x2EP3(AnF)g.Assumex2EP(F),thenjCV(x)j167:5bbyLemma 11 (1).Assumex2EP2(AnF),thenjCV(x)j1610bbyLemma 11 (4).Assumex2EP3(AnF),thenjCE(x)j523andjCV(x)j16b1 3(15+2530:5)cb=1610bbyLemma 13 (1),(2).Thus,jCV(P)j1610bforallP2A1and 54

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14 (2).DeneA2=fhxijx2EP(GnA)g.ThenforallP2A2wehavejCV(P)j167:5bbyLemma 11 (5)andweset2=7:5.jA2jjGj42424153=a2.Itisroutinetocheckthat?issatised,acontradiction.Lete=14.SinceA=F2SCRSp(2;2)SCRSp(2;7),jA=Fj648byLemma 24 (1),(13).AssumejWj43.ByLemma 10 ,jGjdim(W)648142(jWj1)
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11 (1).Assumex2EP(AnF)andxisabadelement,thenjCV(x)jjWj6bbyLemma 13 (1).Assumex2EP3(AnF),xisagoodelementandjCE(x)j223,thenjCV(x)jjWjb1 3(12+2230:5)cb=jWj6bbyLemma 13 (2).Assumex2EPs(AnF),sisaprimeands5,sinces-jEj,jCV(x)jjWj6bbyLemma 11 (3).ThusjCV(P)jjWj6bforallP2A1andweset1=6.jA1j(NEP2(H)+1)(NEP2(I)+1)2432(jWj1)=3+(NEP3(H)+1)(NEP3(I)+1)2432(jWj1)=2+(NEPf2;3g0(H)+1)(NEPf2;3g0(I)+1)2432(jWj1)=62222432(jWj1)=3+992432(jWj1)=2+52432(jWj1)=6=a1byLemma 24 (2), 21 (2)and 14 (2).DeneA2=fhxijx2EP2(AnF),xisagoodelementandjCE(x)j=2232g.Thenforallx2P2A2wehavejCV(P)j=jCV(x)j=jWjb1 2(12+6)cb=jWj9bbyLemma 13 (2)andweset2=9.jA2j1522(jWj1)=3=a2byLemma 24 (2)and 14 (4).DeneA3=fhxijx2EP2(AnF),xisagoodelementandjCE(x)j=24g.Thenforallx2P2A3wehavejCV(P)j=jCV(x)j=jWjb1 2(12+4)cb=jWj8bbyLemma 13 (2)andweset3=8.jA3j132(jWj1)=3=a3byLemma 21 (2)and 14 (4).DeneA4=fhxijx2EP2(AnF),xisagoodelementandjCE(x)j=22g.Thenforallx2P2A4wehavejCV(P)j=jCV(x)j=jWjb1 2(12+2)cb=jWj7bbyLemma 13 (2)andweset4=7.jA4j152232(jWj1)=3=a4byLemma 24 (2)and 14 (4).DeneA5=fhxijx2EP3(AnF),xisagoodelementandjCE(x)j>223g.Thenforallx2P2A5wehavejCE(x)j243andjCV(P)j=jCV(x)jjWjb1 3(12+22230:5)cb=jWj8bbyLemma 13 (2)andweset5=8.SincejCE(x)j>223,jCE(x)j=2232or243andjA5j(42232+832)(jWj1)=2=2=a5byLemma 24 (2), 21 (2)and 14 (3).DeneA6=fhxijx2EP(GnA)g.Thusforallx2P2A6,jCV(P)j=jCV(x)jjWj6bbyLemma 11 (5)andweset6=6.jA6jjGjdim(W)62224122(jWj1). 56

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24 (15)andjGjdim(W)120112(jWj1)=BbyLemma 10 .DeneA1=fhxijx2EP(AnF)g,thenjCV(P)jjWjb3 411cb=jWj8bforallP2A1byLemma 11 (2)andweset1=8.SinceNEP11(A=F)=0byLemma 18 ,jA1j15112(jWj1)=11=a1byLemma 24 (15)and 14 (2).DeneA2=fhxijx2EP(F)orx2EP(GnA)gthenjCV(P)jjWj5:5bforallP2A2byLemma 11 (1),(5)andweset2=5:5.jA2jjGjdim(W)120112(jWj1)=a2.Itisroutinetocheckthat?issatised,acontradiction.Lete=10andthus10jjWj1.A=F=HIwhereH.S3andI2SCRSp(2;5).2,3aretheonlyprimesthatmightdividejA=Fj,jA=Fj246andjGjdim(W)246102(jWj1)=BbyLemma 24 (1),(11)andLemma 10 .DeneA1=fhxijx2EP(F);x2EP2(AnF)andxisabadelement;x2EP3(AnF)g.Assumex2F,thenjCV(x)jjWj5b.Assumex2EP2(AnF)andxisabadelement,thenjCV(x)jjWj5bbyLemma 13 (1).Assumex2EP3(AnF)thenjCV(x)jjWj5bbyLemma 11 (3).ThusjCV(P)jjWj5bforallP2A1andweset1=5.Since2;5jjUj,byLemma 14 (2),jA1j246102(jWj1)=2=a1.DeneA2=fhxijx2EP2(AnF)andxisagoodelementg.Thusforallx2P2A2,jCE(x)j=22byLemma 13 (3),jCV(P)j=jCV(x)jjWj6bbyLemma 13 (2)andweset2=6.jA2jjIj5222(jWj1)245222(jWj1)=5=a2byLemma 14 (3).DeneA3=fhxijx2EP(GnA)g.Thusforallx2P2A3,jCV(P)j=jCV(x)jjWj5bbyLemma 11 (5)andweset3=5.jA3jjGjdim(W)246102(jWj1).SinceA3isemptyifdim(W)=1weseta3=0ifdim(W)=1anda3=dim(W)246102(jWj1)ifdim(W)6=1. 57

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3 ].LetG=S2n(G1G1).SincejGj=2(2724)2=3>jVj=49,GwillhavenoregularorbitsonV.Lete=16,n=16andV=F163.WeconstructaprimitivesolvablegroupGonV.LetG=S4n(GL(2;3)GL(2;3)GL(2;3)GL(2;3)).ItischeckedbyGAP[ 3 ]thatGhasnoregularorbitsonV. 58

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[1] S.Dol.Largeorbitsincoprimeactionsofnitesolvablegroups,TransactionsoftheAMS,360(1)(2008),135-152. [2] D.A.Foulser,Solvableprimitivepermutationgroupsoflowrank,TransactionsoftheAMS,143(1969),1-54. [3] GAP.TheGAPGroup,GAP-Groups,Algorithms,andProgramming,Version4.3;2002. [4] B.Huppert,FiniteGroupsI,Springer-Verlag,Berlin,1967. [5] B.HuppertandN.Blackburn,FiniteGroupsII,Springer-Verlag,Berlin,1982. [6] B.HuppertandN.Blackburn,FiniteGroupsIII,Springer-Verlag,Berlin,1982. [7] I.M.Isaacs,Charactersofsolvableandsymplecticgroups,AmericanJournalofMathemtics,95(3),Autumn(1973),594-635. [8] I.M.Isaacs,G.Navarro,T.R.Wolf,FiniteGroupelementswherenoirreduciblecharactervanishes,J.Algebra,222(1999),413-423. [9] O.ManzandT.R.Wolf,RepresentationsofSolvableGroups,CambridgeUniversityPress,1993. [10] A.MoretoandT.R.Wolf,Orbitsizes,characterdegreesandSylowsubgroups,AdvancesinMathematics,184(2004),18-36. [11] A.Turull,Supersolvableautomorphismgroupsofsolvablegroups,Math.Z.,183(1983),47-73. [12] T.R.Wolf,Regularorbitsofinducedmodulesofnitegroups,Finitegroups2003,deGruyter,Berlin,2004,389-399. [13] Y.Yang,Orbitsoftheactionsofnitesolvablegroups,J.Algebra,toappear. 59

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YongYangwasbornin1976inShanghai,China.HegrewupmostlyinShanghai,graduatingfromWeiyuHighSchoolin1994.HeearnedhisB.S.inAutomaticControlandhisM.S.inImageProcessingandArticialIntelligencefromtheShanghaiJiaoTiongUniversity(SJTU)in1998and2001,respectively.UpongraduatinginJune2001,YongenteredtheShanghaiAcerLabCorpasanapplicationspecicintegratedcircuit(ASIC)designer.InAugust2003,YongcametotheUniversityofFloridaandstudiedintheDepartmentofElectricalEngineeringasaPh.D.student.Beinginspiredbythemathematicscourseshetook,hechangedhismajorandbecameaPh.D.studentintheDepartmentofMathematicsinAugust2004.UponcompletionofhisPh.D.program,YongwillbecomeaVisitingAssistantProfessorintheTexasStateUniversityatSanMarcos,TX.YonghasbeenmarriedtoJunXiaofor2years. 60