PSL(3,q) as a totally irregular collineation group

80 Pages·1994·3.1 MB·English

by  Radas, Sonja, 1961-

PSL(3,q)ASATOTALLYIRREGULARCOLLINEATIONGROUP
By
SONJARADAS
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
1994
N
ACKNOWLEDGEMENTS
Gratefulacknowledgementsareduetopersonswhosehelpmadethisworkpossi-
ble. IwishtothankmyadvisorProf. ChatYinHoforgivingmesuchaninteresting
problemtoworkonandforintroducingmetotechniquesneededforsolvingit.
IamalsogratefultoProf. A.GongalvesandProf. E.Moorehousefortheirhelpful
suggestions.
IalsowishtothankProf. MirkoPolonijowhowasmyfirstteacherintheareaof
finitegeometryandwhodirectedmymagisterthesis.
AndfinallyIwishtoexpressmygratitudetomymotherMirjanaandhersisters
VjeraandDunjaforinspiringmebytheirexample,andtomyhusbandHrvojefor
hisunwaveringsupportandencouragementthroughallthestagesofmywork.
9
TABLEOFCONTENTS
ACKNOWLEDGEMENTS ii
ABSTRACT iv
CHAPTERS
1 PRELIMINARIES 1
1.1 Theproblemandtheresults 1
1.2 Backgroundoftheproblemandsomedefinitions 3
1.3 Reviewofallknownresults 11
1.4 AboutPSL{3,q) 15
1.5 Factsaboutperspectivities 20
2 ONELEMENTSOFG 23
22..12 IEnlveomleunttisonwihsoasepeorrsdpeercstidviivtiyde9, +1,orq—1aregeneralizedperspec- 23
tivities 34
3 THETYPEOFTHEPLANEn 42
3.1 OnfixedsetsofS-groups 42
33..23 WOhnatthecatnypbeeosfaHidiafbSo-ugtroHupwihsesnpeSc-igalroupsarenotspecial 4566
4 CONCLUSION 69
REFERENCES 71
BIOGRAPHICALSKETCH 73
111
AbstractofDissertationPresentedtotheGraduateSchool
oftheUniversityofFloridainPartialFulfillmentofthe
RequirementsfortheDegreeofDoctorofPhilosophy
PSL(3,q)ASATOTALLYIRREGULARCOLLINEATIONGROUP
By
SonjaRadas
April1994
Chairman: Dr. ChatYinHo
MajorDepartment: Mathematics
AcollineationgroupofaprojectiveplaneH,istotallyirregularifeverypointis
fixedbyanontrivialcollineation. Non-abeliansimplestronglyirreduciblecollineation
groups containingaperspectivityhavebeen characterizedby Reifart and Stroth.
TheyarePSL[2,q), qodd, PSL(3,q), PSU{3,q), A7or J2. Heringand Walker
provedthattheprojectiveplanewhichadmitsPSL{3,q),qodd,asastronglyirre-
duciblecollineationgroupwithaperspectivity,isisomorphictoPG{2,q).
Thefollowingaremajorresultsinthisdisertation. Let PSL{3,q) beatotally
irregular,notstronglyirreduciblecollineationgroupcontainingaperspectivity. By
anS-groupwedenoteacyclicsubgroupoforder(9^+9-|-1)/(3, —1)inG. Ifevery
elementofanS-groupSfixesthesamesetofpointsasSitself,thenHiseitherof
orderq(Desarguesianplane),orderq^(DesarguesianorgeneralizedHughesplane),
oroforderq^(Desarguesian,Figueroaplaneorperhapssomeotherplanes).
IfaS-groupdoesnothavetheaboveproperty,thenanupperboundfortheorder
ofnis9^/3. IfnisasubplaneofHwhichisleftinvariantbyalltheperspectivities
inAuf(H),thenthegroupAuf(H)antsonHwithkernelK. WehaveeitherH=
IV
n=PG{2,q')fori=1,2,3,orft=PG(2,q)^II. InthelattercaseAui(II)/A:<
PTH3,q).
V
CHAPTER
1
PRELIMINARIES
1.1 Theproblemandtheresults
Letnbeafiniteprojectiveplane,andletPSL{3,q)=G<Auf(H),q=p"*,pan
oddprime,beatotallyirregularcollineationgroupwithanontrivialperspectivity.
OurgoalistodeterminewhatisthestructureoftheplaneH.
ByS-groupwemeanthemaximalcyclicsubgroupoforder{q^+q+l)/(3,q—1)
inG. (ThisgroupisisomorphictoanintersectionofaSingercycleinPGL{3,q)
withPSL{3,q).)
Herearetheresults:
1)IfndenotestheorderoftheplaneH,thenn=qorn=Lq^{q—1)+ where
L>0.
2)IfanS-groupisspecial,i.e. Fix{s)=Fix{S)foreverys£S,thenwehave
oneofthefollowing:
(i)n=qandH=PG{2,q).
(ii)n=q^andH=PG{2,q^)orHisgeneralizedHughesplane.
(iii)n=q^(possibleplanesincludePG{2^q^)andFigueroaplane).
3)IfanS-groupisnotspecial,and|Fix(5)|=3,thenAu<(H)<PTL{3,q).
4)IfanS-groupisnotspecial,and |F’ia:(5)lgreaterthan3,thenAut{U)/K<
PTL(3,q), where K is kernelofAuf(H) on an invariant subplane isomorphicto
PG(2,q).
5)IfanS-groupisnotspecial,thenn<9^/3.
1
2
Herewewillgiveashortoverviewofthemainidea;sweusedinprovingtheabove
results.
IfPSL{3,q) ^ G < Aut(n), q = an odd prime, is atotallyirregular
collineationgroup withanontrivialperspectivity,then theexistenceofthat per-
spectivityimpliesthataninvolutioninGisaperspectivity,asprovedinsection2.1
(theorem2.1.2). SincealltheinvolutionsinGareconjugate,everyinvolutionisa
perspectivity.
TheexistenceofaninvolutoryperspectivityinGimpliestheexistenceofaDe-
sarguesiansubplane0oforderqintheplaneH(lemma2.1.5). Thisfacthelpsusto
controltheactionofGonthepointsofHoutsideoffl. Asaconsequenceofsections
2.1 and2.2itwasshownthatalltheelementsoforderdividingq,q—1 orq I
aregeneralizedperspectivitiesandtheirfixedpointsareonthelinesof (theorem
2.2.1).
SinceGistotallyirregular,pointsofHwhicharenotonthelinesofflmustbe
fixedbyelementsoforderdividing(?^+?+1)/(3,9—1),i.e.byelementsofS-groups.
AtthispointS-groupsstartplayinganimportantrole.
Insection3.2weassumedS-groupsarespecial,i.e. pointsfixedbyanyelement
ofagroupaxefixedbythewholegroup. UsingthepartitionoftheplaneHtogether
withsomeresultsoriginatingfromcodingtheory,weprovethattheorderofHis9,
q^orq^(theorem3.2.2).
IfS-groupsarenotspecial,asinsection3.3,itisveryhardtospecifytheplane
order,andevenhardertopindownthegeometricstructureoftheplaneH. Inlemma
3.3.1someboundsfortheorderofHaregiven.ThesetofpointsfixedbyaS-group
isverycomplicated. Itcouldbeaunionofplaneswhichdoordonotintersect. In
lemma3.3.1wealsogivesomedetailsaboutthatset.
3
TogathermoreinformationonII,weintroduceH,asubplaneofIIwhichisleft
invariantbythegroupofalltheperspectivitiesofII. Thetotalcollineationgroup
y4ut(n)actsonftwithacertainkernelK. Inlemma3.3.2itwasshownthateither
ft / nandthenft=0 ^PG{2,q)orU= U ^ PG{2,q^),fort = 1,2or3. If
ft n,thenAut{U)IK<PTL{^,q). ItwasprovedthatKiscontroledbyS-groups
andtheirfixedstructuresasshowninlemma3.3.4.
1.2 Backgroundoftheproblemandsomedefinitions
Themotivationforconsideringwhichtypesoffiniteprojectiveplanesadmitto-
tallyirregulaxcollineationgroups(i.e.suchthatstabilizerofanypointisnontrivial)
comesfromourwishtoclassifyallfiniteprojectiveplanes. Sinceallknownfinite
projectiveplanesexceptfortranslationplaneswithtrivialcomplementadmittotally
irregularcollineationgroups,itisreasonabletoconsidersuchgroupsandlookatthe
planestheyacton. Ourhopeisthat inthatwaywecoulddiscoveranewfinite
projectiveplane.
Wewillbeginwithdefinitionsofsomebasicgeometricobjects. Othernotation
andterminologyconcerninggroupsandprojectiveplanescanbefoundinGorenstein
[6],Hupert[16]orSuzuki[21](forgroups),andDembowski[5]andHughesandPiper
[15](forplanes).
Definition1.2.1 LetVbeasetofpoints,CasetofsubsetsofVcalledlines, andI
anincidencerelation. Wesay{V,C,X)isaprojectiveplaneifthefollowingistrue:
1)Foreverytwopointsthereisexactlyonelineincidentwiththem.
2)Foreverytwolinesthereisexactlyonepointincidentwiththem.
3) TherearefourpointsinVsuchthatnothreeofthemarecollinear.
WhenVisfinite, wesay{V,C,T) isafiniteprojectiveplane.
Wedenotepointswithcapitallettersandlineswithlowercaseletters.
.
4
Aneasyconsequenceofthedefinitionisthefollowingfact.
Assume('P,£,X)isafiniteprojectiveplane.Thenumberofpointsonthelineis
thesameforeveryline,sayn+1. Thenumberoflinesthroughapointisalsothe
sameforeverypoint,andisequalton 1. Thenumberofpointsintheplaneis
equaltothenumberoflinesintheplane,andthatnumberis +n+1
Wesaythat(P,£,J)isafiniteprojectiveplaneofordern.
Nowwewillintroducetheconceptofcollineation.
—
Definition1.2.2Ass—ume(P,£,J) isafiniteprojectiveplane. Leta:V >Vhea
bijection. Letfi:£ >Cbeabijectioninducedbya, i.e. {PQY=P°‘Q°‘ Ifotand
fipreserveincidence, i.e. PII-4==^P“/l^, wesaythataisacollineation.
Undertheusual definitionofproducts ofmappingsit isclearthat theset of
edlcollineationsofagivenplane IIformsagroup. This group is calledthefull
collineationgroupandisdenotedbyAut(II).Whenwerefertoacollineationgroup,
wemeanasubgroupofAut(II).
LetGbeagroupandflafiniteset. AnactionofGonfiisahomomorphism
—
a:G >Sym{Q),whereSym(fl)isthesymmetricgroupof17. WesaythatGacts
faithfullyon17ifker{a)=1. (WewillwriteP®insteadofP°‘^^\forgGG.)
ForapointX€ II,Gx = {gEG: =X} isthestabilizerofthepointX.
ThepointsetX^={X^:g£G}istheG-orbitcontainingX.
Thefollowingistrue: =|G|/|Gx|.
Letqbeanoddprimepowerthroughout,i.e. q=p*",m>1andpanoddprime.
Foraprimepowerq,wecanalwaysconstructafiniteprojectiveplaneasfollows.
T^lkeanyfieldGF{q). Wedefine{V,C,I)inthefollowingway:letPbetheset
oftillone-dimensionalsubspacesin{GF{q)Y,let£bethesetofalltwo-dimensional
5
subspcicesin{GF{q))^,andletIbetheinclusion. SuchaplaneiscalledDesarguesian
andisdenotedbyPG{2,q).
Uptonow,therehasbeennofiniteprojectiveplanewhoseorderisnotaprime
power,anditisthusconjecturedthataplaneofcompositeorderdoesnotexist.One
facttosupporttheconjecturewasprovedbyBruckandRyser[15]. Theyshowed
thatcompositenumberswhichsatisfycertainconditionscannotbeordersofplanes.
Theorem1.2.1 (Bruck-Ryser)
Letn= 1 or2 {mod4). Thenthereisnoplane ofordern, except ifn canbe
representedasasumoftwosquares.
Nothingisknownaboutnumbersn^10whichcanberepresentedassumoftwo
squares. Bruck-Rysertheoremdoesnotsayifsuchaplaneexists. Thefirstnumber
thatcouldnotberuledoutusingBruck-Rysertheoremis10. Foralongtimeexistence
ofaplaneoforder10wasanopenquestion,untilC.Lamcameupwithacomputer
prooffornon-existenceofsuchplanein1991. (Remark; forn=2,3,4,5,7,8there
isonlyDesarguesianplaneofsuchorder. Forn=9therearemoreplanes,andthey
areallclassifiedbyC.Lamin1992.)
Now we will definefixed substructure ofa collineation and related concepts.
Thesesubstructuresareimportanttodeterminegeometrywhichcorresponds toa
collineationgroup.
Definition 1.2.3Forg eG<Au<(II), wedefineV{g) asthesetofpointsfixedby
g, andC{g) asthesetoflinesfixedbyg. Also we defineV{G) = Dg^aV{g), and
C{G)=Hg^aCig).
Notethat{V{g),C{g),l)satisfiesthefirsttwoaxiomsofaprojectiveplane. The
incidencestructurewiththesetwopropertiesiscalledaclosedconfiguration ora