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1、GroupTheoryandtheRubik’sCubeJanetChenANotetotheReaderThesenotesarebasedona2-weekcoursethatItaughtforhighschoolstudentsattheTexasStateHonorsSummerMathCamp.Allofthestudentsinmyclasshadtakenelementarynumbertheoryatthecamp,soIhaveassumedinthesenotesthatreadersarefamiliarwiththeintegersmodnaswe
2、llastheunitsmodn.BecauseonegoalofthisclasswasacompleteunderstandingoftheRubik’scube,IhavetriedtousenotationthatmakesdiscussingtheRubik’scubeaseasyaspossible.Forexample,Ihavechosentouserightgroupactionsratherthanleftgroupactions.2IntroductionHereissomenotationthatwillbeusedthroughout.Zthese
3、tofintegers...,−3,−2,−1,0,1,2,3,...Nthesetofpositiveintegers1,2,3,...Qthesetofrationalnumbers(fractions)RthesetofrealnumbersZ/nZthesetofintegersmodn(Z/nZ)×thesetofunitsmodnThegoalofthesenotesistogiveanintroductiontothesubjectofgrouptheory,whichisabranchofthemathematicalareacalledalgebra(or
4、sometimesabstractalgebra).Youprobablythinkofalgebraasaddition,multiplication,solvingquadraticequations,andsoon.Abstractalgebradealswithallofthisbut,asthenamesuggests,inamuchmoreabstractway!Ratherthanlookingataspecificoperation(likeaddition)onaspecificset(likethesetofrealnumbers,orthesetofint
5、egers),abstractalgebraisalgebradonewithoutreallyspecifyingwhattheoperationorsetis.Thismaybethefirstmathyou’veencounteredinwhichobjectsotherthannumbersarereallystudied!AsecondarygoalofthisclassistosolvetheRubik’scube.WewillbothdevelopmethodsforsolvingtheRubik’scubeandprove(usinggrouptheory!)
6、thatourmethodsalwaysenableustosolvethecube.ReferencesDouglasHofstadterwroteanexcellentintroductiontotheRubik’scubeintheMarch1981issueofScientificAmerican.ThereareseveralbooksabouttheRubik’scube;myfavoriteisInsideRubik’sCubeandBeyondbyChristophBandelow.DavidSingmaster,whodevelopedmuchoftheus
7、ualnotationfortheRubik’scube,alsohasabookcalledNotesonRubik’s’MagicCube,’whichIhavenotseen.Foranintroductiontogrouptheory,IrecommendAbstractAlgebrabyI.N.Herstein.Thisisawonderfulbookwithwonderfulexercises(andifyouarenewtogrouptheory,youshoulddolotsoftheexercis
