Transformation group

Introduction

LetGbeanon-emptyset,anddefineanoperation"○"betweentheelementsofG.IfGsatisfiesthefollowingconditions:

1.(Operationalclosure)ForanytwoelementsaandbinG,therewillalwaysbea○b∈G;

2.(Associationlaw)Foranythreeelementsa,b,cinG,therewillalwaysbe(a○b)○c=a○(b○c);

3.(identityelement)existsTheidentityelemente∈G,sothatforanyelementainG,thereise○a=a;

4.(Inverseelement)ForanyelementainG,thereisaninverseelementbofa∈G,suchthatb○a=e.

ItissaidthatGregardingtheoperation"○"asagroup.Gforshortisagroup.

SupposeAisanon-emptyset,andthegroupformedbythemultiplicationoftransformationsbyseveralone-to-onetransformationsofAiscalledatransformationgroupofA.

Definition

Asetoftransformations,agroupformedbytheproductoftransformations.LetGbeasetoffiniteorinfinitetransformationsonM,ifthefollowingtwoconditionsaremet:①TheproductofanytwotransformationsinsetGstillbelongstoG;②EverytransformationinsetGmusthaveitsinversetransformation,andthisTheinversetransformationalsobelongstoG,soGiscalledatransformationgrouponM.

Forexample,translationaltransformationcanformagroup:theproductofanytwotranslationaltransformationsontheplaneisstillatranslationaltransformation;eachtranslationaltransformationhasaninversetransformation,andthisinversetransformationisatransformationintheoppositedirectionoftheoriginaltransformation.,Soitisstilltranslationaltransformation(see"TheNatureofTranslationalTransformation").

Theideaof​​usingtransformationgroupstostudythecorrespondinggeometrywasfirstproposedbytheGermanmathematicianKlein.In1872,Kleinpresentedapaperentitled"ComparisonofResearchonModernGeometry"inhisinaugurallectureattheUniversityofErlangen,discussingtheleadingroleoftransformationgroupsingeometry.HetookwhathehaddiscoveredsofarAllgeometriesareunifiedundertheviewpointoftransformationgroup,andanewdefinitionofgeometryisclearlygiven,thatis,geometryisdefinedasasciencethatstudiestheinvariantpropertiesandinvariantsoffiguresunderacertaintransformationgroup.Thispointofviewhighlightedthepositionoftransformationgroupsinthestudyofgeometryandopenedthewayfortheuseofmodernmathematicalmethodstostudygeometry,soitwaslaterreferredtoasthe"ErlangenProgram".

Accordingtotheviewpointoftransformationgroup,geometrycanbeclassifiedasfollows:studythegeometryofprojectivetransformationgroup,affinetransformationgroup,similartransformationgroup,invariantpropertiesandinvariantunderorthogonaltransformationgroup,respectively.Geometry,affinegeometry,parabolicgeometry,Euclideangeometry.Orthogonaltransformationgroupisalsocalledmotiongroup.ThemaincontentofEuclideangeometryistostudythegeometryofinvariantpropertiesandinvariantsundermotiongroup.Topology,asubjectthathasdevelopedrapidlyinmoderntimesandhasbecomemoreandmorewidelyused,istostudythegeometryofinvariantpropertiesandinvariantsundertopologicaltransformation.

Group

Groupisarelativelysimplealgebraicstructurewithonlyoneoperation;itisabasicstructurethatcanbeusedtobuildmanyotheralgebraicsystems.

SupposeGisanon-emptyset,anda,b,andcareitsarbitraryelements.Ifthealgebraicoperation"·"definedbyG(called"multiplication"andtheresultoftheoperationiscalled"product")satisfies:

(1)Closure,a·b∈G;

(2)Associativelaw,thatis(a·b)c=a·(b·c);

(3)Foranyelementa,binG,inGThereareuniqueelementsxandy,suchthata·x=bandy·a=b,thenitissaidthatGformsagroupforthedefinedoperation"·".Forexample,allrealnumbersthatarenotequaltozeroformagroupwithregardtousualmultiplication;theclockwiserotation(withrespecttoadditionofmodulo12)formsagroup.

Thegroupthatsatisfiesthecommutativelawiscalledthecommutativegroup.

Groupisoneofthemostimportantconceptsofmathematics,whichhaspenetratedintoallbranchesofmodernmathematicsandothersubjects.Wheneversymmetryisinvolved,thereisagroup.Forexample,itispossibletodefinevariousgeometriesbystudyingthepropertiesofgraphsthatremainunchangedunderthetransformationgroup,thatis,usingtransformationgroupstoclassifygeometries.Itcanbesaidthatwithoutunderstandinggroups,itisimpossibletounderstandmodernmathematics.

In1770,whenLagrangediscussedthepermutationbetweentherootsofalgebraicequations,hefirstintroducedtheconceptofgroup,anditsnamewasfirstproposedbyGaloisin1830.

Characterintroduction

KleinisafamousGermanmathematician,mathematicalhistorian,andphysicist.Hehasmademanycontributionsintopologyandgeometry.Herecognizedtheimportanceofgrouptheoryandappliedtheconceptofgrouptomanybranchesofmathematics.In1872,thefamous"ErlangenProgram"waspublished.Heputforwardtheviewpointofclassifyinggeometryaccordingtothepropertiesthatremainunchangedunderthetransformationgroup,andunifiedgeometrywithgrouptheory.Ithadaprofoundinfluenceonthedevelopmentofmoderngeometryandpreparedtheconditionsfortheestablishmentofthespecialtheoryofrelativity.After1886,hetaughtattheUniversityofGöttingenforalongtimeandwasanoutstandingrepresentativeoftheheydayoftheGöttingenSchool.HisviewsontheunityofmathematicshadagreatinfluenceonHilbert.Healsoproposedthatmathematicsshouldbecloselylinkedwithreality.Heorganizedmanyseminarsonmathematics,throughteachingactivities,sothatstudentsgetacomprehensiveunderstandingofthewholeofmathematics.Whenheteachestheorems,heonlytalksabouttheoutlineoftheproof,andleavestheprooftothestudentstocomplete.Hefirstadvocatedreformingthemathematicscontentofsecondaryeducation,whichhadanimportantinfluenceonmodernmathematicseducation.

Classification

Projectivetransformationgroup

referredtoasprojectivegroup.Itisabasictransformationgroup.Thatis,thetransformationgroupformedbyalltheprojectivetransformationsintheprojectivespace.Forexample,theentireprojectivetransformationontheplaneconstitutesaprojectivegroupontheplane.Allprojectivetransformationsinspaceconstituteprojectivegroupsinspace.Thegeometrythatstudiesinvariantpropertiesandinvariantsundertheprojectivegroupiscalledprojectivegeometry.

Affinetransformationgroup

referredtoasaffinegroup.Abasictypeoftransformationgroup.Thatis,thetransformationgroupformedbyallaffinetransformationsinaffinespace.Forexample,theentireaffinetransformationontheplaneconstitutestheaffinetransformationgroupontheplane,whichisanautomorphismgroupwiththeinfinitestraightlineastheabsoluteshapeintheplaneprojectivetransformation.Allaffinetransformationsinthespaceconstitutetheaffinetransformationgroupofthespace,whichisanautomorphismgroupwiththeinfiniteplaneastheabsoluteshapeinthespaceprojectivetransformation.Thegeometrythatstudiesinvariantpropertiesandinvariantsundertheaffinegroupiscalledaffinegeometry.

Similartransformationgroup

Alsoknownasparabolicmetricgroup.Referredtoassimilargroup.Akindofbasictransformationgroup.Thesetofallsimilartransformationsontheplaneconstitutesagroup,whichiscalledthesimilartransformationgroup.Itisafour-dimensionalgroup.Asubgroupoftheaffinetransformationgroup.Intheaffinetransformation,ifapairofpointsI(1,i,0)andJ(1,-i,0)remainunchanged,itisasimilartransformation.Thesimilaritytransformationmaintainshomogeneity,associativityandthesingleratioofthethreecollinearpoints,andtheangleformedbythetwostraightlinesremainsunchanged.Similaritytransformationchangesafigureintoafiguresimilartoit.Thegeometrycorrespondingtothesimilaritytransformationgroupiscalledsimilaritygeometryorparabolicgeometry.

OrthogonalTransformationGroup

Alsoknownasmotiongroupormetricgroup.Referredtoasorthogonalgroup.Abasictypeoftransformationgroup.Thatis,thetransformationgroupformedbytheentireorthogonaltransformation.Forexample,asetofallorthogonaltransformationsonaplaneconstitutesanorthogonalgroupontheplane,andallorthogonaltransformationsinaspaceconstituteanorthogonalgroupinaspace.Theorthogonalgroupontheplane(inspace)isasubgroupoftheaffinegroupontheplane(inspace).ThegeometrythatstudiestheinvariantpropertiesandinvariantsundertheorthogonalgroupiscalledEuclideangeometryormetricgeometry.