The Minimal Degree for a Class of Finite Complex Reflection Groups

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1、THEMINIMALDEGREEFORACLASSOFFINITECOMPLEXREFLECTIONGROUPSNEILSAUNDERSSchoolofMathematicsandStatisticsUniversityofSydney,NSW2006,AustraliaE-mailaddress:neils@maths.usyd.edu.auAbstract.WecalculatetheminimaldegreeforaclassoffinitecomplexreflectiongroupsG(p,p,q),forpandqprimesandestab-lishrela

2、tionshipsbetweenminimaldegreeswhenthesegroupsaretakeninadirectproduct.1.IntroductionTheminimalfaithfulpermutationdegreeµ(G)ofafinitegroupGistheleastnon-negativeintegernsuchthatGembedsinthesymmetricgroupSym(n).Itiswellknownthatµ(G)isthesmallestvalueofPnTi=1

3、G:Gi

4、foracollectionofsubgroupsT

5、{G1,...,Gn}satisfyingngi=1core(Gi)={1},wherecore(Gi)=g∈GGi.WewilloftendenotesuchacollectionofsubgroupsbyRandreferitastherepresentationofG.TheelementsofRarecalledtransitiveconstituentsandifRconsistsofjustonesubgroupG0say,thenwesaythatRistransitiveandthatG0iscore-free.Thestudyofthisaredat

6、esbacktoJohnson[3]whereheprovedthatonecanconstructaminimalfaithfulrepresentation{G1,...,Gn}consistingentirelyofsocalledprimitivegroups.Thesearegroupswhichcannotbeexpressedastheintersectionofgroupsthatproperlycontainit.HerewegiveatheoremduetoKarpilovsky[4],whichalsoservesasanintroductory

7、example.Wewillmakeuseofthistheoremlaterandtheproofofitcanbefoundin[3]or[8].Theorem1.1.LetAbeafiniteabeliangroupandletA∼=A1×...×Anbeitsdirectproductdecompositionintonon-trivialcyclicgroupsofprimepowerorder.Thenµ(A)=a1+...+an,AMSsubjectclassification(2000):20B35,51F15.Keywords:FaithfulPermu

8、tationRepresentations,ComplexReflectionGroups.12NEILSAUNDERSwhere

9、Ai

10、=aiforeachi.Theaimofthispaperiscalculateµ(G(p,p,q))wherepandqareprimenumbersandG(p,p,q)isthemonomialcomplexreflectiongroup(seeSection3forthedefinition).Whenq=2,thegroupG(m,m,2)isthedihedralgroupoforder2m(maninteger)andin[

11、2,Proposition2.8],EasdownandPraegercalculatedtheminimaldegreeforthedihedralgroups.Specifically,theyprovedthefollowing:QmαiProposition1.2.ForanyintegerPk=i=1pi>1,withthepidis-tinctprimes,defineψ(k)=mpαi,withψ(1)=0.Thenforthei=1idihedralgroupD2rnwithnodd,wehave2rifn=1,1≤r≤2r−12