某些对称群和交错群的新刻画

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1、425'Vol.42,No.5201310$ADVANCESINMATHEMATICSOct.,2013ANewCharacterizationofCertainSymmetricandAlternatingGroupsYANYanxiong1,2,3,∗,CHENGuiyun1,∗∗,XUHaijing1,∗∗∗(1.SchoolofMathematicsandStatistics,SouthwestUniversity,Chongqing,400715,P.R.Chi-na;2.Departm

2、entofMathematicsandInformationEngineering,ChongqingUniversityofEducation,Chongqing,400067,P.R.China;3.CollegeofMathematicsScience,ChongqingNormalUniversity,Chongqing,401331,P.R.China)Abstract:ThedegreepatternofafinitegroupGhasbeenintroducedin[AlgebraCol-loqu

3、ium,2005,12(3):431-442]anddenotedbyD(G).ThegroupGiscalledk-foldOD-characterizableifthereexistexactlyknon-isomorphicgroupsHsatisfyingconditions

4、H

5、=

6、G

7、andD(H)=D(G).Inaddition,a1-foldOD-characterizablegroupissimplycalledOD-characterizable.Thefollowingsimplegro

8、upsareuniquelydeterminedbytheirordersanddegreepatterns:allsporadicsimplegroups,thealternatinggroupsAp(p≥5isatwinprime)andsomesimplegroupsofLietype.Inthisproblem,thosegroupswithconnectedprimegraphsaresomewhatmuchdifficulttobesolved.Inthispaper,wecontinuethisin

9、vestigation.Inpartic-ular,weshowthatthesymmetricgroupsS81andS82are3-foldOD-characterizable.WealsoshowthatthealternatinggroupsA130andA140areOD-characterizable.Itisworthmentioningthatthelattergivesapositiveanswertoaconjecturein[FrontiersofMathematicsinChina,2

10、009,4(4):669-680].Keywords:primegraph;degreepattern;degreeofavertex;symmetricgroup;alternatinggroupMR(2000)SubjectClassification:20D05/CLCnumber:O152.1Documentcode:AArticleID:1000-0917(2013)05-0601-100IntroductionThroughoutthispaper,Gwillrepresentafinitegroup

11、.Weuseπe(G)todenotethesetofordersofitselementsandbyπ(G)thesetofprimedivisorsof

12、G

13、.Oneofthewell-knownsimplegraphsassociatedtoGistheprimegraph(orGruenberg-Kegelgraph)denotedbyΓ(G)(see[3]).Thisgraphisconstructedasfollows:Thevertexsetofthisgraphisπ(G)andtwodist

14、inctverticesp,qarejoinedbyanedgeifandonlyifpq∈πe(G).Inthiscase,wewritep∼q.ThenumberofconnectedcomponentsofΓ(G)isdenotedast(G)andtheconnectedcomponentsofΓ(G)asπi=πi(G)(i=1,2,···,t(G)).When

15、G

16、iseven,wesu