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1、群论第一次作业Problem1-1(30’)Thesymmetriesofaregularquadrilateral(i.e.asquare)isthesetofsymmetryoperationsthatkeepthesquaretobeinvariantinthespace.Provethatthesymmetriesofasquareformagroupcalled,givethemultiplicationtableofthegroup.SolutionLetusconsiderthegroupofallsymmetriesofth
2、esquare,D4.Itcontainseightelements:fourrotationsofC4andfourreflections.Denotingtheverticesofthesquarebythenumbers1,2,3,4,thoseeightelementsaree，d=(1234),f=(13)(24),g=(1432),h=(13),a=(24),b=(12)(34),c=(14)(23)Themultiplicationtableofitssymmetriesislistedbelow.edfghabceedfgh
3、abcddfgecbhaffgedahcbggedfbcahhhbacefdgaachbfegdbbachgdefcchbadgfeToprovethesymmetriesofasquareformagroupG,weshouldprovethefollowingfourfacets:(1)closure:theproductamongtheelementsareallbelongtoG;(2)associativity:theproductprovetheassociativity.Forexample,d(fa)=dh=c,(df)a=
4、ga=c,sod(fa)=(df)a;(3)Thereexistsanelement,e.Wecanknowitfromthefirstrowandthefirstcolumn；(4)ForeverypinG,thereisap-1inG,suchasdg=e,ff=e,gd=e,hh=e,aa=e,bb=e,cc=e.thatis,d(f)istheinverseelementoff(d),anda,(b,c,h)istheinverseelementofa,(b,c,h)itself.Sowecanknowthesymmetriesof
5、thesquareisagroup.Problem1-2(30’)Provethewholesetofpermutationoperationofthreesymbols,say1,2and3,formsagroupcalled，andproveitisisomorphictothegroup,thesetofsymmetriesofanequilateraltriangle.SolutionTheelementsofS3areE，D=(123),F=(132),A=(23),B=(13),C=(12)Themultiplicationta
6、bleislistedbelowEDFABCEEDFABCDDFECABFFEDBCAAABCEDFBBCAFEDCCABDFEToproveS3isagroupG,weshouldprovethefollowingfourfacets:(1)closure:theproductamongtheelementsareallbelongtoG;(2)associativity:theproductprovetheassociativity.Forexample,d(fa)=db=a,(df)a=ea=a,sod(fa)=(df)a;(3)Th
7、ereexistsanelement,e.Wecanknowitfromthefirstrowandthefirstcolumn；(4)ForeverypinG,thereisap-1inG,suchasdf=e,fd=e,aa=e,bb=e,cc=e.thatis,d(f)istheinverseelementoff(d),anda,(b,c)istheinverseelementofa,(b,c)itself.AsfortheD3,thesymmetriesoftrianglecontainsixelements,theyaree，d=
8、(123),f=(132),a=(23),b=(13),c=(12)Themultiplicationtableislistedbelowedfabceedfabcddfecab
