Calegari - Classical Geometry.pdf

《Calegari - Classical Geometry.pdf》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、CLASSICALGEOMETRY—LECTURENOTESDANNYCALEGARI1.ACRASHCOURSEINGROUPTHEORYAgroupisanalgebraicobjectwhichformalizesthemathematicalnotionwhichex-pressestheintuitiveideaofsymmetry.Westartwithanabstractdefinition.Definition1.1.AgroupisasetGandanoperationm:GG!Gcalledmultiplic

2、ationwiththefollowingproperties:(1)misassociative.Thatis,foranya;b;c2G,m(a;m(b;c))=m(m(a;b);c)andtheproductcanbewrittenunambiguouslyasabc.(2)Thereisauniqueelemente2Gcalledtheidentitywiththepropertiesthat,foranya2G,ae=ea=a(3)Foranya2GthereisauniqueelementinGdenoteda

3、1calledtheinverseofasuchthataa1=a1a=eGivenanobjectwithsomestructuralqualities,wecanstudythesymmetriesofthatobject;namely,thesetoftransformationsoftheobjecttoitselfwhichpreservethestructureinquestion.Obviously,symmetriescanbecomposedassociatively,sincetheeffectofas

4、ymmetryontheobjectdoesn’tdependonwhatsequenceofsymmetriesweappliedtotheobjectinthepast.Moreover,thetransformationwhichdoesnothingpreservesthestructureoftheobject.Finally,symmetriesarereversible—performingtheoppositeofasymmetryisitselfasymmetry.Thus,thesymmetriesofan

5、object(alsocalledtheautomorphismsofanobject)areanexampleofagroup.Thepoweroftheabstractideaofagroupisthatthesymmetriescanbestudiedbythemselves,withoutrequiringthemtobetiedtotheobjecttheyaretransforming.Soforinstance,thesamegroupcanactbysymmetriesofmanydifferentobject

6、s,oronthesameobjectinmanydifferentways.Example1.2.Thegroupwithonlyoneelementeandmultiplicationee=eiscalledthetrivialgroup.Example1.3.TheintegersZwithm(a;b)=a+bisagroup,withidentity0.Example1.4.ThepositiverealnumbersR+withm(a;b)=abisagroup,withidentity1.Example1.5.T

7、hegroupwithtwoelementsevenandoddand“multiplication”givenbytheusualrulesofadditionofevenandoddnumbers;hereevenistheidentityelement.ThisgroupisdenotedZ=2Z.Example1.6.Thegroupofintegersmodnisagroupwithm(a;b)=a+bmodnandidentity0.ThisgroupisdenotedZ=nZandalsobyCn,thecycl

8、icgroupoflengthn.12DANNYCALEGARIDefinition1.7.IfGandHaregroups,onecanformtheCartesianproduct,denotedGH.Thisisagroupwhoseelementsaretheelem