group theory(j s milne)

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1、GROUPTHEORYJ.S.MilneAugust29,2003∗AbstractThesenotes,whicharearevisionofthosehandedoutduringacoursetaughttofirst-yeargraduatestudents,giveaconciseintroductiontothetheoryofgroups.Pleasesendcommentsandcorrectionstomeatmath@jmilne.org.v2.01(August21,1996).Firstversionontheweb;57pages.v2.11.(August29,2

2、003).Fixedmanyminorerrors;numberingunchanged;85pages.ContentsNotations.......................................3References......................................3Prerequisites.....................................3Acknowledgements.................................31BasicDefinitions4Definitions...........

3、...........................4Subgroups......................................6Groupsofsmallorder................................8Multiplicationtables................................9Homomorphisms..................................9Cosets........................................10Normalsubgroups.......

4、...........................12Quotients......................................13Exercises1–4....................................142FreeGroupsandPresentations15Freesemigroups...................................15Freegroups.....................................16Generatorsandrelations...................

5、...........18Finitelypresentedgroups..............................20Exercises5–12...................................21∗Copyrightc1996,2002,2003.J.S.Milne.Youmaymakeonecopyofthesenotesforyourownpersonaluse.Availableatwww.jmilne.org/math/.1CONTENTS23IsomorphismTheorems.Extensions.23Theoremsconcernin

6、ghomomorphisms.......................23Directproducts...................................24Automorphismsofgroups.............................26Semidirectproducts.................................28Extensionsofgroups................................32TheHolderprogram.................................34

7、¨Exercises13–19...................................354GroupsActingonSets36Generaldefinitionsandresults...........................36Permutationgroups.................................42TheTodd-Coxeteralgorithm.......