group theory [jnl article] - j. milne

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1、GROUPTHEORYJ.S.MILNEAugust21,1996;v2.01Abstract.ThesarethenotesforthefirstpartofMath594,UniversityofMichigan,Winter1994,exactlyastheywerehandedoutduringthecourseexceptforsomeminorcorrections.Pleasesendcommentsandcorrectionstomeatjmilne@umich.eduusing“Math594”as

2、thesubject.Contents1.BasicDefinitions11.1.Definitions11.2.Subgroups31.3.Groupsoforder<1641.4.Multiplicationtables51.5.Homomorphisms51.6.Cosets61.7.Normalsubgroups71.8.Quotients82.FreeGroupsandPresentations102.1.Freesemigroups102.2.Freegroups102.3.Generatorsandre

3、lations132.4.Finitelypresentedgroups14ThewordproblemTheBurnsideproblemTodd-CoxeteralgorithmMaple3.IsomorphismTheorems;Extensions.163.1.Theoremsconcerninghomomorphisms16FactorizationofhomomorphismsTheisomorphismtheoremThecorrespondencetheoremCopyright1996J.S.Mi

4、lne.Youmaymakeonecopyofthesenotesforyourownpersonaluse.iiiJ.S.MILNE3.2.Products173.3.Automorphismsofgroups183.4.Semidirectproducts213.5.Extensionsofgroups233.6.TheH¨olderprogram.244.GroupsActingonSets254.1.Generaldefinitionsandresults25OrbitsStabilizersTransiti

5、veactionsTheclassequationp-groupsActionontheleftcosets4.2.Permutationgroups314.3.TheTodd-Coxeteralgorithm.354.4.Primitiveactions.375.TheSylowTheorems;Applications395.1.TheSylowtheorems395.2.Classification426.NormalSeries;SolvableandNilpotentGroups466.1.NormalSe

6、ries.466.2.Solvablegroups486.3.Nilpotentgroups516.4.Groupswithoperators536.5.Krull-Schmidttheorem55References:DummitandFoote,AbstractAlgebra.Rotman,AnIntroductiontotheTheoryofGroupsGROUPTHEORY11.BasicDefinitions1.1.Definitions.Definition1.1.AisanonemptysetGtoget

7、herwithalawofcomposition(a,b)
→a∗b:G×G→Gsatisfyingthefollowingaxioms:(a)(associativelaw)foralla,b,c∈G,(a∗b)∗c=a∗(b∗c);(b)(existenceofanidentityelement)thereexistsanelemente∈Gsuchthata∗e=a=e∗aforalla∈G;(c)(existenceofinverses)foreacha∈G,thereexistsana
∈Gsuchtha

8、t

a∗a=e=a∗a.If(a)and(b)hold,butnotnecessarily(c),thenGiscalledasemigroup.(Someauthorsdon’trequireasemigrouptocontainanidentityelement.)Weusuallywritea∗bandeasaband1,orasa+band0.Tw