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cmr10POLeYA-REDFIELDfTHEOREM Ja kVLADIMIRޟ;jfCEPULI7x;jC In!troMduction Our4aaimistopro!ve4aatheoremofG.POLeYA4Econcerning4aarelativ!elywideclassofcombina-
torialproblems.Hegotthisresultin1937.GEOR!GEPOLeYA(1887-1985)w!asanAmericanmathematicianofh!ungarianorigin.AnothermathematicianJ.H.REDFIELDprovedaspMecialcasefofthesametheoremteny!earsbMefore,in1927.Ja Theftheoremisdealingwithproblemsoffollo!wingkind: Ho!wTmanycolouringsofacubMethereexist,ifweapplytoitssurfacesblack,greenand
bluefcolour. Weeconsider,%ofcourse,t!wocolouringswhic!hturnbyrotationoneintoanotherasthesamecolouring.Feorfexample(B-forblue,G-forgreen): mQ|32 fd P e|32 fd P Q|32 fd P e|32 fd P QJ P feeJ P fe J P fe J P feQ| O
line10 [| Q| [| | | | | y| ':
3
cmti10(G)}| Gy| G | (G)U| (B) | B|32 fd P -|32 fd P |32 fd P -|32 fd P J P fe-J P feiJ P fe}J P fe| #| | #| i| s| i| s| A| (B)E| BA| GO| (G) | (G)q| G Japresen!tfthesamecoloring. RotationsLofacubMeconstituteagroup.Th!usforsolvingsuchproblemsthegrouptheoryisin!volvedandweshouldbMeginwithsomebasicresultsconcerningstructureandpropertiesoff nitegroups. 11 *iӍ:-iӍFN q cmbx12Chapter
17ǍN H cmbx12Finite T{Groups;Ja$"V
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cmbx10De nitionf1 b>
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cmmi10:1MjACgrpoupK(G;1!",
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cmsy10)isanonempt!ysetGwithabinaryopMeration
:G0G
7 !G
(thatfis:8a;1b
2Gf9!c2Ganb
=c)suc!hthat:Ja (
`)f8a;1b;c
2G;yan(bc)
=(anb)c(theassoMciativ!elaw). ( )f9e
2G;y8a2G;ena
=ane
=a;Lefiscalledneutrpalelemen!tofG. (
)f8a
2G9aK cmsy802G;ya0=%na=aa0=e;La0tisfcalledtheinverseelemen!tofa.Iffmoreo!ver:(j)8a;1b
2G;yanb
=bna,Giscalledabpelianorcommutativegroup.Examplef1:1Fɹ(!
3
msbm10Z;1+),fZbMeingthesetofin!tegers,isanabeliangroup:Z
=f:1::;1 3; 2; 1;0;11;2;3;:::g;fa;b
2Z)an+b
2Z.(
`)fan+(b+c)
=(an+b)+c;( )f90
2Z;y0n+a
=an+0
=a(
)f8a
2Z;y9( a)2Z;( a)n+a
=an+( a)
=0(j)f8a;1b;ybn+a
=an+b.Examplef1:2F(f1;1 1g;)9isanabMeliangroup:Qa;1bk2f1; 1g)axbk2f1; 1g.VIt9iseasytoc!heckftheconditions(
`)n (j).Examplef1:3FN(The9Wmainexample!)LetSbMeasetandɖu
3
cmex10PH(S )
=ff8c:S9!SjfmisfabijectionHgg7(Bijection:1
$1fmappingthatisu.(1)S f8c=
S
(2)x
6=yo:)xf8c6=ydf|K).Thenf(ɖP (S );1)isagroupforcompMositionofmappingsasbinaryoperationonit.Feorf ;1g23NɖP?(S );x3N2S`\w!ede ne:x(f/gd)=(xf-)g,xf솹bMeingtheimageofxunderactionoff-.Ja ThefgroupɖP5W(S )iscalledthesymmetricgrouponS.Itisindeedagroup:Vef ;1go:2
ɖP(S )
:u.A(1)fSf8c=S9=Sgo:)S(fngd)=(Sf-)go:=Sgo:=S
A(2)fx6=yo:)xf8c6=ydf)(xf-)go:6=(ydf)go:)x(fngd)6=y(fng)Th!usffngo:2
ɖP(S ). 12 iӍ:-iӍ-?t<
lcircle10't; ' fe?&? $ ; ' fe?% |? fe |? fe | q | q | x | y? خ' خ; ' fe?&?
$
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R fd fe Hᜄ fd fe ! fd fe fd fe zhv fd fe T@h fd fe F fd fe ! fd fe 1 fd fe xʄ fd fe Tޟz fd fe DSz fd fe !, fd fe ҄ fd fe v= fd fe Tܟ҄ fd fe B fd fe !lz fd fe F fd fe t ʄ fd fe Tڟ1 fd fe @ fd fe !} fd fe b fd fe rfq fd fe T؟A fd fe >
fd fe ! fd fe
Q fd fe p2 fd fe T֟= fd fe
j fd fe 2 fd fe !0 fd fe ҄ fd fe d fd fe Tʟ fd fe 0 fd fe !c fd fe Er fd fe b' fd fe Tȟ fd fe .ʄ fd fe ! fd fe , fd fe ` fd fe TƟt6 fd fe ,V fd fe !9 fd fe fd fe ^ fd fe Tğb fd fe * fd fe ! fd fe fd fe \r fd fe TV fd fe (;V fd fe ! fd fe fd fe Z釄 fd fe TΖ fd fe &τ fd fe !2 fd fe ~ fd fe Xdv fd fe TJW fd fe $0b fd fe ! fd fe fd fe V fd fe T2 fd fe " fd fe ! fd fe G fd fe Tf fd fe TN' fd fe 5ք fd fe ! fd fe 쟅 fd fe R߄ fd fe T6 fd fe fd fe ! fd fe Ꞔʄ fd fe PxȄ fd fe Ta fd fe KB fd fe !4 fd fe 蟄 d fd fe N4 fd fe T. fd fe R fd fe !Ơ fd fe 柃 fd fe L fd fe T fd fe q| fd fe !~\ fd fe 䟃G fd fe J3Z fd fe T fd fe
fd fe !| fd fe ⟂΄ fd fe H fd fe T fd fe » fd fe !z fd fe ÇЄ fd fe Fn fd fe T\* fd fe ĻI fd fe !x7, fd fe Ňޟ$ fd fe Dք fd fe T fd fe ƻ fd fe !v fd fe LJܟ˒ fd fe BM fd fe T2 fd fe ȻA fd fe !tz fd fe ɇڟv݄ fd fe @fj fd fe TV! fd fe ʻ F fd fe !r6
fd fe ˇ؟&B fd fe > fd fe T* fd fe ̻
݄ fd fe !p躄 fd fe ͇֟ fd fe < fd fe TM fd fe λ҄ fd fe !n fd fe χԟZ fd fe :] fd fe Tu fd fe лg fd fe !lZb fd fe чҟM
fd fe 8? fd fe T2 fd fe һ&
fd fe !j] fd fe ӇП ڄ fd fe 6҄ fd fe T fd fe Ի眄 fd fe !h fd fe ՇΠ fd fe 4Ć fd fe T( fd fe ֻ fd fe !f fd fe ׇ̠
fd fe 2T fd fe TȄ fd fe غxf fd fe !dn. fd fe هʠd fd fe 0Z< fd fe TP fd fe ںF fd fe !b= fd fe ۇȠ4P fd fe .+> fd fe T"V fd fe ܺ fd fe !` fd fe ݇Ơ fd fe , Z fd fe T~D fd fe ~X fd fe !^~薄 fd fe ߇Ġ~ fd fe *~ِ fd fe T~L fd fe ~2 fd fe !\~r fd fe ~ fd fe (~ fd fe T~ fd fe ~R fd fe !Z~3 fd fe ~> fd fe &~s fd fe T~҄ fd fe ~[ fd fe !X~ fd fe 凾~ fd fe $~| fd fe T~x# fd fe ~s~ fd fe !V~o fd fe 燼~j fd fe "~f fd fe T~b fd fe ~^ fd fe !T~[ fd fe 釺~W fd fe ~T> fd fe T~Q fd fe ~N fd fe !R~K; fd fe 뇸~H fd fe ~F fd fe T~C fd fe ~A fd fe !P~?~ fd fe 퇶~= fd fe ~; fd fe T~:k fd fe ~8 fd fe !N~6 fd fe ~5ބ fd fe ~4 fd fe T~4H fd fe ~3 fd fe !L~3Z fd fe ~3" fd fe ~3 fd fe T~~30 fd fe ~3v fd fe !J~3 fd fe ~4 fd fe ~5D fd fe T|~62 fd fe ~7J fd fe !H~8 fd fe ~9 fd fe ~; fd fe Tz~=N fd fe ~?8 fd fe !F~AL fd fe ~C fd fe ~E fd fe Tx~H fd fe ޠ~K@ fd fe !D~N& fd fe ~Q6 fd fe ~Tp fd fe Tv~WԄ fd fe ܠ~[b fd fe !B~_ fd fe ~b fd fe ~e fd fe Tt~j* fd fe ڠ~n fd fe !@~s fd fe ~wń fd fe ~| fd fe Tr~ fd fe ؠ~ڄ fd fe !>~5 fd fe ~ fd fe
~i fd fe Tp~B fd fe ֠~E fd fe!<~r fd fe~Ʉ fd fe~J fd feTn~ fd feԠ~ʄ fd fe!:~Ʉ fd fe~ fd fe~E fd feTl~ fd feҠ~i fd fe!8~: fd fe~5 fd fe Z fd feTj fd feР" fd fe!6ń fd fe" fd fe+ fd feTh4 fd feΠ= fd fe !4F6 fd fe OԄ fd fe Y fd fe
Tfc fd fe
̠m fd fe !2w fd fe ` fd fe fd fe Td fd fe ʠ fd fe
!0Ą fd fe
fd fe
r fd feTb fd feȠȄ fd fe!.粄 fd feƄ fd fe fd feT` l fd feƟ fd fe!,% fd fe2 fd fe? fd feT^L fd feğZN fd fe!*g܄ fd feu fd fev fd feT\ fd fe fd fe!( fd fe fd feV fd feTZ4 fd fe fd fe!& fd fer fd fe fd feTX% fd fe5 fd fe!$E fd feU fd fef fd feTVv fd fe2 fd fe!" fd fe fd fe/ fd fe TT˂ fd fe fd fe ! fd fe w fd fe 쟂r fd fe TR$ fd fe 6 fd fe ! I_ fd fe \ fd fe Ꟃnτ fd fe TPƄ fd fe fd fe!! 2 fd fe! fd fe!蟂F fd fe"TN fd fe" fd fe#! fd fe# f fd fe#柃2` fd fe$TLF fd fe$[ fd fe%!p fd fe%~Ą fd fe%䟃 fd fe&TJr fd fe& fd fe'!Ȅ fd fe'| fd fe'⟄Ƅ fd fe(TH fd fe(4l fd fe)!J fd fe)za fd fe)x fd fe*TF fd fe* fd fe+!N fd fe+x܄ fd fe+ޟ프 fd fe,TDv fd fe, fd fe-!5 fd fe-vN fd fe-ܟf fd fe.TBV fd fe.4 fd fe/!< fd fe/tn fd fe/ڟʄ fd fe0T@P fd fe0 fd fe1! /B fd fe1rIE fd fe1؟cr fd fe2T>}Ʉ fd fe2J fd fe3!
fd fe3pʄ fd fe3֟Ʉ fd fe4T< fd fe4 E fd fe5!: fd fe5nVi fd fe5ԟr: fd fe6T:5 fd fe6Z fd fe7!Ʃ fd fe7l" fd fe7ҟń fd fe8T8 fd fe89 fd fe9!V fd fe9js fd fe9Пj fd fe:T6 fd fe:҄ fd fe;!ń fd fe;h fd fe;Ο') fd feT2E fd fe> fd fe? fd fe?d=N fd fe?ʟ]8 fd fe@T0}L fd fe@ fd feA fd feAbބ fd feAȟ@ fd feBT. & fd feBA6 fd feC bp fd feC`Ԅ fd feCƟb fd feDT, fd feD fd feE fd feE^-> fd feEğO fd feFT*r( fd feF܄ fd feG fd feG\ fd feG fd feHT(!P fd feHDք fd feI h fd feIZ` fd feId fd feJT&Ԓ fd feJ fd feK l fd feKXB fd feKe fd feLT$ fd feL; fd feM Ԏ fd feMV fd feM fd feNT"E fd feNk~ fd feO fd feOT fd feOk fd fePT fd feP+ۄ fd feQ 쟐R҄ fd feQRy fd feQ> fd feRT ȳ fd feRR fd feS ꟑ fd feSP@ fd feSh+ fd feTT r fd feT fd feU 蟑~ fd feUN
C fd feU32 fd fe |~32 fd ά- |z fng |32 fd Z ά- |32 fd Z - |32 fd Z - |32 fd Z - | S | SR| S Ja (
`)fFeoran!ymappingsfn(gh)
=(fngd)h.Ja ( )fLet12 cmmi8S (:
S97!SGs.th.C8x;yx1S=x.Then
8f"x(1Snf-)
=(x1S )f8c=xf,fx(fn1S )=(xf)1S (=xf :Th!usf1Snf8c=
f1S (=f-;f1S Hiscalledtheiden!tityfmappingonS . (
)Eac!hbijectionff :8PS7!S0-hasitsinversemappingf-|{Y cmr8( 1)_ :8PS7!S ,de nedby:ydf-( 1)1W=
x,xf8c=yvǍ? ' ; ' fe?&? Į$ Į; ' fe?% |? fe |? fe | q | x? ' ; ' fe?&?P$P; ' fe?%2|? fe 2|? fe 7| q9| y |32 fd fe 4 fd fe yЄ fd fe _{ fd fe ?j fd fe J fd fe B fd fe 㟸& fd fe O fd fe a%5 fd fe Ɵ fd fe g@ fd fe 2c fd fe w)G fd fe J" fd fe 럶J fd fe H{ fd fe -BI fd fe Ο fd fe oD fd fe _ fd fe _& fd fe R'
fd fe / fd fe uz fd fe 5 fd fe ֟H fd fe Fw fd fe N fd fe ѹ҄ fd fe Zn fd fe \8 fd fe fd fe =͐ fd fe -ޟL fd fe sci fd fe . fd fe > fd fe Db fd fe fd fe Ϥ^B fd fe E*܄ fd fe Z柰 fd fe z fd fe (҄ fd fe +ɟ_= fd fe qj- fd fe fd fe R fd fe BM fd fe f fd fe ͏5n fd fe 0 fd fe Xџ! fd fe r fd fe sЄ fd fe )C fd fe oU{ fd fe fd fe " fd fe @86 fd fe ٟXń fd fe z*V fd fe d fd fe Vr fd fe ] fd fe s fd fe 'FS fd fe m@ fd fe fd fe fd fe >#q fd fe ğh fd fe e< fd fe fd fe T fd fe HƄ fd fe 韪 fd fe %fx fd fe k+<} fd fe ð̟r fd fe m fd fe <h fd fe āk fd fe PmZ fd fe D܄ fd fe R H fd fe Ř3I fd fe ԟ2 fd fe #uY fd fe i} fd fe ƮU fd fe X. fd fe 9 fd fe fd fe ;v fd fe
ܟ fd fe P}ö́ fd fe Ȗ J fd fe ۿ% fd fe !` fd fe go fd fe ɬ fd fe C fd fe 7䟦n fd fe }K fd fe &'N fd fe ǟD fd fe Nh fd fe ˔ fd fe ٪ʄ fd fe KyO fd fe d쟥W fd fe ̪5 fd fe .6 fd fe 5ϟ fd fe {p fd fe fd fe fd fe LSp fd fe ΑP: fd fe ו00 fd fe 6d fd fe bןQ fd fe Ϩx fd fe n fd fe 3 fd fe y[v fd fe оX2 fd fe : fd fe J> Ą fd fe яߟ fd fe ՀR fd fe !ź fd fe `܄ fd fe Ҧc< fd fe pb fd fe 1T? fd fe wF8 fd fe Ӽ矢 > fd fe b fd fe H)9 fd fe ԍʟ܄ fd fe k0 fd fe R fd fe ^~# fd fe դNd2 fd fe K fd fe /1 fd fe u1 fd fe ֺҟ fd fe s fd fe Fj fd fe DŽ fd fe Vʄ fd fe fd fe \q& fd fe آ9Y fd fe ڟC~ fd fe -{, fd fe s ҄ fd fe ٸ fd fe ^" fd fe CW fd fe ډn fd fe A fd fe ⟟ fd fe Z fd fe ۠$mR fd fe şY fd fe +fE fd fe q2c fd fe ܶ ʄ fd fe I fd fe A fd fe ݇ф fd fe ,2 fd fe ͟ fd fe Xn` fd fe ޞ| fd fe 㰟 fd fe )Q|# fd fe nk fd fe ߴZƄ fd fe 4J҄ fd fe ?՟:e fd fe v* fd fe fd fe
fd fe VY fd fe b fd fe ᛟ* fd fe '. fd fe0ߟ5 fd fevW fd fe!x fd fe fd feGc fd feኄ fd feҥ܄ fd feF' fd fe]矦Jń fd fem҄ fd fe) fd fe.ʟ4 fd fetkۋ fd fe fd fe%h fd feENI fd fepA fd feАB fd fe 1 fd fe [ҟᔄ fd fe sP fd fe . fd fe ,U fd fe rV}, fd fe fd fe r fd fe C9 fd fe ڟ fd fe {Dd fd fe m fd fe Y fd fe ^ fd fe ք fd fe * fd fe pA< fd fe ⟪e fd fe 8 fd fe!A$ fd fe!ş fd fe!fB fd fe"= fd fe"Whf fd fe"I fd fe" fd fe#( fd fe#n, fd fe#͟E fd fe#ns fd fe$?K fd fe$΄ fd fe$Q܄ fd fe%*ބ fd fe%UXi fd fe%42 fd fe%՟ fd fe&&v8 fd fe&lw fd fe&C: fd fe&Ys fd fe'<8 fd fe'u fd fe'<2 fd fe(
ݟ5 fd fe(S~f( fd fe( c fd fe( fd fe)$a fd fe)j- fd fe)^z fd fe)D fd fe*:埰 fd fe* fd fe*'*D fd fe+ ȟ^ fd fe+Qi fd fe+
Ƙ fd fe+ܫ fd fe,"L- fd fe,gퟲcU fd fe,v fd fe,/ͨ fd fe-8ПF fd fe-~q8 fd fe-n fd fe. B fd fe.OTڄ fd fe. fd fe.ږH fd fe/ 7~ fd fe/e؟^ fd fe/y. fd fe/' fd fe06^g fd fe0|\҄ fd fe0Ϝ fd fe1 fd fe1M?Ä́ fd fe1|6 fd fe1 fd fe2 " fd fe2cß)# fd fe2db fd fe2H fd fe34D fd fe3zGi fd fe3蟸N fd fe4 fd fe4K*| fd fe4˟ fd fe4l? fd fe5
zÄ fd fe5a fd fe5O҄ fd fe532 fd fe |32 fd ά- | f |32 fd fe 4 p fd fe y fd fe _h fd fe & fd fe JcQ fd fe Bd fd fe 㟻> fd fe ҄ fd fe a%R/ fd fe ƟN fd fe g$ fd fe 2Ƅ fd fe w= fd fe JwB fd fe 럽 fd fe H fd fe -$ fd fe Ο]F fd fe o fd fe _΄ fd fe > fd fe R?Z fd fe /wM fd fe u fd fe 5` fd fe ֟ ~ fd fe FwTw fd fe fd fe ѹ fd fe Z fd fe \- fd fe cr fd fe =Ԅ fd fe -ޟ fd fe s fd fe 7 fd fe l& fd fe Db p fd fe U fd fe Ϥ" fd fe E; fd fe Zn fd fe á fd fe (Ԓ fd fe +ɟ' fd fe qj9P fd fe kh fd fe ĝ fd fe BMέ fd fe fd fe ͏0 fd fe 0a fd fe XџŒC fd fe r¦ fd fe fd fe )"z fd fe oUQ fd fe ƁR fd fe ưB fd fe @8. fd fe ٟ
fd fe z< fd fe j fd fe VǗ fd fe ]Ŧ fd fe ڄ fd fe ' fd fe m@LƄ fd fe y fd fe ȥ fd fe ># fd fe ğ fd fe e)j fd fe T fd fe T fd fe Hɪ fd fe d fd fe % fd fe k+) fd fe ð̟S fd fe m}n fd fe <ʦ fd fe ā fd fe P
fd fe ! fd fe RJ fd fe Ř3r fd fe ԟ˚2 fd fe #u fd fe iL fd fe Ʈ fd fe X7j fd fe 9^I fd fe ̄ fd fe ;̪ fd fe
ܟв fd fe P} fd fe Ȗ ܄ fd fe ۿAD fd fe !`fn fd fe g͊ fd fe ɬͯ fd fe CӪ fd fe 7ڄ fd fe }c fd fe &? fd fe ǟb fd fe Nh΅V fd fe ˔ Χ fd fe ٪ʚ fd fe K fd fe d fd fe ̪0 fd fe .R. fd fe 5ϟs fd fe {pϔ~ fd fe ϵ fd fe ҄ fd fe LSY fd fe Α* fd fe ו64 fd fe 6V fd fe bןu fd fe ϨxДb fd fe в fd fe 3Ȅ fd fe y[݄ fd fe о2 fd fe +Ȅ fd fe J>I fd fe яߟf fd fe Հф fd fe !Ѡ fd fe `ѽ fd fe Ҧc( fd fe fd fe 1% fd fe wF- fd fe ӼI& fd fe d fd fe H)+ fd fe ԍʟҙ fd fe kҴ4 fd fe fd fe ^A fd fe դN2 fd fe R fd fe /4Ƅ fd fe u1Mg fd fe ֺҟf^ fd fe s~ fd fe FӖ fd fe Ӯ fd fe Vƚ fd fe ݾ fd fe \> fd fe آ9 fd fe ڟ" fd fe -{9 fd fe s O fd fe ٸeڄ fd fe ^{B fd fe Cԑ
fd fe ډԥ fd fe AԻD fd fe Ϯ fd fe Z fd fe ۠$ fd fe ş fd fe +f Ԅ fd fe q4 fd fe ܶG fd fe IZH fd fe Amd fd fe ݇ fd fe ,Ւ2 fd fe ͟գ fd fe Xnն fd fe ޞ fd fe 㰟ڄ fd fe )QA fd fe n fd fe ߴ fd fe 4 fd fe ?՟+ fd fe v;t fd fe Kd fd fe ZZ fd fe VYḯ fd fe y fd fe ᛟև: fd fe '<֕ fd fe lݟ֣ fd fe ~ֱ fd fe ֿ fd fe =ބ fd fe aٙ fd fe ڄ fd fe fd fe TDڄ fd fe fd fe ߆ބ fd fe %'# fd fe jȟ. fd fe i:r fd fe
D fd fe ;P fd fe LZ fd fe d fd fe n fd fe R/x+ fd fe Пׁ. fd fe qƄ fd fe #ה fd fe hלe fd fe TץB fd fe fd fe 9h fd fe 7 fd fe ؟Ē fd fe
yZ fd fe P fd fe ꕻ fd fe \ fd fe 杄 fd fe f( fd fe ?V fd fe b fd fe 7 fd fe }" fd fe ßԄ fd fe d
fd fe N fd fe 퓦( fd fe Gb fd fe ^ fd fe d / fd fe *! fd fe ˟$ fd fe 5l' fd fe {
(Մ fd fe +Z fd fe O, fd fe K. fd fe 𑑟/ fd fe 21 fd fe ӟ2c fd fe bt2z fd fe 3J fd fe 2 fd fe 3W35 fd fe x2N fd fe 2$ fd fe :0 fd fe I۟0 fd fe |.2 fd fe - fd fe * fd fe `_) fd fe '* fd fe 롟$ fd fe 1B! fd fe v fd fe 2 fd fe % fd fe GƟ fd fe g fd fe J fd fe , fd fe ^J܄ fd fe N fd fe 錟r fd fe /-g fd fe tΟ fd fe o愄 fd fe ߪ fd fe E٥ fd fe RL fd fe ʄ fd fe fd fe \5 fd fe ֟ fd fe w fd fe -ץf fd fe rלQ fd fe Zה fd fe fd fe Cׁ҄ fd fe =w fd fe ޟn fd fe c fd fe Z ZN fd fe Pb fd fe bE fd fe +: fd fe p.ڄ fd fe E# fd fe fd fe A C fd fe (v fd fe ɟ fd fe jJ fd fe X fd fe R fd fe MֿH fd fe(ֲ0 fd fen֣ fd fe0֖ fd feџև
fd fe?rx fd feiq fd feʴZ fd feUJ܄ fd feU;Є fd fe, fd fe8 fd fe&ٟ fd felz fd fet fd feز fd fe=] fd feյ fd feȟդr fd fe@Ց fd feS fd fel fd fe#Z fd fe$ğH
fd feje4
fd fe! fd fe fd fe;H. fd fe3 fd feƊF fd fe +Ժ̄ fd fe Q̟Ԧb fd fe mԐi fd fe { fd fe
"f] fd fe
hPO fd fe
: fd fe
"΄ fd fe 93 fd fe ~ԟ fd fe u^ fd fe
* fd fe Oӯ fd fe XӖ^ fd fe ~Ȅ fd fe
f fd fe
f;M fd fe
ܟ52 fd fe
}B fd fe7 t fd fe| fd fe`κ fd feҳ̄ fd feMҚ fd feC~ fd fedR fd fe Iτ fd fed&- fd feǟ fd feh fd fe5 } fd fezѽT fd feKѡZ fd feу fd feKg; fd fe.I fd feϟ, fd fe p fd feb fd feZ fd feSв fd fe2Дʄ fd fext fd fe6V> fd feן5܄ fd feIx fd feՄ fd feԺ2 fd fe[ϴ҄ fd fe_ϔ fd fetT fd fe>R6 fd fe0ߟ1[ fd fev fd fe!f fd feJ fd feGcΨu fd fe΄ڄ fd feҥb fd feF>n fd fe] fd fe fd fe)Ӻ fd fe.ʟͰ0 fd fetk͊ل fd fe f҄ fd fe@ fd feEN x fd fe# fd feА" fd fe 1̪N fd fe [ҟ̄Є fd fe s_ fd fe 7 fd fe ,I fd fe rV8 fd fe fd fe ˙ fd fe C9r fd fe ڟI fd fe {" fd fe r fd fe YE fd fe ^ʧڄ fd fe } fd fe *T fd fe pA)ۄ fd fe v fd fe , fd fe!A$ɫJ fd fe!ş fd fe!fU" fd fe"(ڄ fd fe"W fd fe"I fd fe"ȥބ fd fe#(zG fd fe#n,L fd fe#͟ fd fe#n fd fe$? fd fe$Ǘ fd fe$Qj fd fe%; fd fe%U
fd fe%42 fd fe%՟ưr fd fe&&vƂ, fd fe&lQ fd fe&#* fd fe&Yl fd fe'<, fd fe'ő fd fe'xfork>2.$Weepseethat;actsincycles:?Cx[!xd!xd2h!:1::~!xdt 1.!xdtR.=xofsomelengtht,Ewhic!hweshalldenoteNb!y(x;1xd;:::l;xt 1ٹ),-imageNofthelastelemen!tinparenthesesbMeingthe rstone.Infourexampleo:=
(175)(26)(389)(4). JaTheorem21.1K Each*sppermutation2MɖPt(
),zUj
jM=n*s-ofdegreenontheset
,zUcanberpepresentedinthecyclicform:,ro:=
( z1;1 z1d;:::l; z1dzk qAa cmr61* 1Xù)( z2; z2d;:::l; z2dzk q2* 1Xù):::lŹ( zt; ztd;:::l; ztdzk ; cmmi6t\p 1); z1;1 z2;:::l; ztP2a
,&asaprpoductofcyclesoflengthskz1;kz2;:::l;kztʹ;kz1Q+kz2+:1::+?kztP=an.QHerpe
the d-imageofthelastelementinepachparenthesesisthe rstone-( ziddk8:i,r 1Z)o:=
zi,lfor al Fli.ThecyclesofO}cponstituteapartitionof
.Prpoof: LӹLetv 2
.8Observ!ethesequence
`;1 d; d2$;:::
Sincev
is nitethereexistaminimalksuc!hLthat
`dk= dsϹ,FforLsomes0,Fthen
`dk%d 1= dsd 1) dk6 1|r= ds 1K,againstdtheminimalit!yofkX?.ThussG=0dand
`dk =G ds .Sodw!ecanwritetheabMovesequenceinthecyclicform:n(
`;1 d;:::l; dk6 1d).If 22.@
isnotincludedinthecycleof
`,qw!eproMceedinthesamewaye,qbuildingthecycleof :*( ;1 d; d2$;:::l; ds 1K).Q+Thecy-clesof Sand ^aredisjoin!t:&otherwise
`dt藹=: dr,forsomet4N 5]z"6l7
2 7/
T3>4N 1]z"6l5wmVu v=
(1275)(3)(4)(6)(1275)346(1275): C EspMeciallye,Cif}
#=1;12;:::ln,,w!e}write1
b=#(1). #Byourconvention,Callotherelementsarealsof xedb!yd. 2)fThem!ultiplicationofpMermutationsrepresentedincyclicformisquitesimple.Iffo:=
(:1:: Q5 ! g#:1::-)(:1:: c)1:::l;1%=(::: 95
O!M
n:::5O)(::: c):::l;fthenn%
=(:1:: Q5 !
:1::5O)(:1:: c)1:::FeorfexampleinSz7ʫ:
(16)(2375)4n(251)(3746)=(134627)5.Namely:1
!6!3,fth!us1
!3,3
!7!4,fth!us3
!4,4
!4!6,fth!us4
!6,6
!1!2,fth!us6
!2,2
!3!7,fth!us2
!7,7
!5!1,fth!us7
!1.Sofw!egetthecycle(134627).Feorelement5notbMelongingtothiscycle,wehave5
!1!5,fth!us5
!5. 15 iӍ:-iӍ|- 3)iThecyclesofapMerm!utationcanbewritteninan!yorderandeachcyclecanbMeginwith
an!yfofitselements:forex.(251)(3764)
=(3764)(251)=(7643)(125)=:1::
.Examplef1:6GWeejcanrepresen!ttherotationsofequilateraltriangleaspMermutationinthefollo!wingway:4eR=(1);1d=(123);d2=(132);a=(23);b=(13);c=(12).`No!wwecaneasilycompute:bǍ | | | | | | T | T | T | T | T |ff < | A | 1 | B | 2 | C | 3 | " | " | " | " | " | " | y | a | b | b | b | b | b | b | y | b ɻJ A fe | y | c Ja dna
=(123)n(23)
=(13)2=bJa and
=(23)n(123)
=(12)3=c anb
=(23)n(13)
=(132)=d2 Infthefollo!wingweshallwriteGfor(G;1).Theorem21.2K Lpeta;1x;yo:2
G. vThen 1)anx
=anyo:)
x=yd; 2)xna
=yna)x=yPrpoof: v:1)fanx
=anyo:)
a0=%(ax)=a0=%(ayd)-s 5)
ܹ(a0=%na)x
=(a0na)y5䍑|
o:)mex
=eny5䍑Q o:)x
=y: 2)",xfya
=y fya)(xa)a0=(y a)a0 5)ּx(aa09)=y (aa09)5䍑.
",) xe=y e5䍑$ )x=yTheorem21.3K Theneutrpalelemente,andtheinversea0#ofaareuniquelydetermined.!rPrpoof: S1)fane0=
ae=aT.:h:25b)e0=e.Th!usfeisunique.I- 2)fana00'=
e=ana0T.:h:25 1)a00=
a0)a0tisfunique.Theorem21.4K F)ora;1b
2Gtheepquationsanx
=bandyna=bhavetheuniquesolutions
inG. vx
=a0=%nbandyo:=bna09."BPrpoof:p1)'a%x5=b)a0^%(ax)5=a0%b 5)(a0%a)x5䍑
5=Hex5䍑: 5=x5=a0^%b.aNo!w'indeed:a(a0b)
=(aa09)b
=eb
=e.&2)h@yW2a=b)(ya)a0=
y(aa09)
=ye=yo:=ba09.Thesefsolutionsareuniqueb!yTh.2. 16 àiӍ:-iӍ|-Theorem21.5K Lpeta;1b
2G. vItis(a09)0=a;(anb)0=
b0=%a09. NPrpoof: S1)fa0=%n(a09)05䍑.
=ge5䍑
=a0a
T.:h:25b)ι(a09)0=
a o⍑ 2)(anb)x
=ev 5)a0=%n[a(bx)]
=a0ne=a0 5)ʍ"|ex6ּ͍z P` }| P` { 3ּ(az0na)6Z(bnx)
=a0
)
bnx=a0)b0=%(bx)=b0=%a0)(b0=%b)x=ex=x=b0=%a09:fNo!w,f(anb)(ab)0=
e=(anb)(b0=%a09)
T.:h:25b)ι(ab)0=
b0=%a09.!Remarks:Ja1)BecauseofassoMciativ!elawaS^(bc)
=(aS^b)c;theresultsofexpressionsinagroupdofnotdepMendonparen!theses,sowecandropthem.(\Feorgettheparentheses!"),e2),eInthefollo!wingweshallwritethegroupopMerationmultiplicativelye.oThatis,Mwede-note~thegroupopMerationb!y""orevendropthissign:a7bsca7bscab.~Inthiscasewedenotea0Ía 1 \|;"e=1G %1,andcall1G Ήtheunit!yofG.Usingthisnotation,w!ewrite:aa 1g#=
a 1 \|a=1;y(a 1) 1g#=a;y(ab) 1=b 1 \|a 1;yax
=b)x=a 1 \|b;yda=b)yo:=ba 1 \|:De nitionf1:3M Letfa
2G.Weede ne:a0ʫ:=
1;ya1:=a;yam k:=am 1@a=aa1laP`| P`
{z P`
}m{mftimes$Ԧ.$Y_Theorem21.6K Lpetaz1;az2;1lazm k2
G. vThen:ArR1) Q(az1az251,azmĹ) 1g#=
a 1Am
/1Aae 12 \|ae 11
rR2) Qa m:=
(amĹ) 1g#=(a 1 \|)mrR3) Qm;1n
2Z(!am+n3=amanP;(am)n=
amn(Prpoof:1)Weealreadykno!w:(az1az2) 1g#=
ae 12 \|ae 11;assume:(az1az251,azm 1@) 1=
ae 1m 1q1#Nae 12 \|ae 11֍)
(az1az251,azm 1@azmĹ) 1g#=((az1az251,azm 1@)azmĹ) 1g#=a 1Am \|(az1az251,azm 1@) 1g#=a 1Am \|ae 1m 1q1#Nae 12ae 11. 2)fa m=
(amĹ) 1g#=(aa1la)
,| P` L={z P` L=}ohm-XE 1 <;1);㍑;=GNaz 1 \|az 1
/1Aaz 1P`GN| P` {z P` }
Ke|Tm 9=(a 1 \|)m. 3)fOnepro!vesfitb!yinduction.De nitionf1:4M Letf
6=A;1BKG,thenAnBK:=
fabja2A;1b2B g.ItisABKG.By\A 1fw!edenotethesetA 1g#:=
fa 1 \|ja2Ag._Feor\A=fag,kwe\denoteABK=
fagBaB=fabjb
2B g.Similarlye,fforBK=
fbg,AB=
AfbgAb. Onefcaneasilyseethat: 1)fA(B C ȁ)
=fa(bc)ja2A;1b2B;1c2C ȁg=f(ab)cja2A;b2B ;c2C ȁg=(AB )C 2)fOb!viouslye,An1
=fan1ja
2Ag=f1naja
2Ag=1nA. De nitionf1:5N_ Let{fHQgmG;N 6=H .\H_'isasubpgroupofG,HQgmG,ifH_'isagroupwithrespMecttothesamegroupoperation(restrictedtoH ),Ⱦthatis,if:1)a;1b8q2H 2=,)a=b8q2H ;2) saһ2H|=,)a 1/72H .FTheconditions
`)and
)aresatis ed,0,nthenFalreadyan3a 1r=asH a 1)an 1ܹ=as 1J,nagainstFtheminimalit!yofn.6Thusan=
a0ʫ=1. No!w,lfor7anyt-Y2Z,lt=nq`+|rM,0r{7 ̎ ff 32 fd 2 32 fd 2 32 fd 2 32 fd 2 32 fd 2 HR\nK O HO HRnkO HRnK K9 GH Ki=fhkG'jh2HH;1k2K ȁg=[fH kjk2K ȁg.
XNo!wbH kz1
=Hkz2
(,)Hkz1ke X? 12=Hh(,)
1 kz1ke X? 12b2Hh(,)kz1ke X? 12b2HA\ K((,)kz1ʫ2(H\ K ȁ)kz2ʫ(,)(H\ K ȁ)kz1ʫ=(H\ K ȁ)kz2.Henceyfrom,>H kz1v6=>rHkz2(,)>r(Hgd\K ȁ)kz16=(Hgd\K ȁ)kz2.;Th!us,>thereyareasmanycosetsofHinfHK,ascosetsofHR\nKninK,thatis:jH K ȁj
:jHj=jK ȁj:jHR\nKj=,)jH Kj=jH jjK j=ڟ@ p p jH \K jDe nitionf1:9NM LetiygY2OG;1x2G,>w!ede nexg $Ĺ:=gd 1 xgd,>thebcponjugateofxunderg.'FeorStG,cw!e=de neS g }r=fsgjs2S g,ctheconjugateofS=underg.FeorHG,cw!esaythatX@<;1Yy
GfarpeconjugatedunderH,LXH #Yy(,)
9h2HH;ys:th:fX h L=Yn. BTheorem21.15QUkThecponjugation'zg U:G!G;?'zg:x7!xg;1g2G,isabijepctionandtherpelationHD,forHh
G,isanequivalencerelation. vIt'sequivalenceclassesarecal FledH-cponjugateclasses.Prpoof:A)f1)x'zgߥ=
yd'zg(,)gd 1 xgo:=gd 1ydgoT.:h:25o:=,) +xgo:=ygoT.:h:25o:=,) +x=yo:=,)'zg{disfinjectiv!e.2)yo:2
G=,)y=gd 1 gdyg 1go:=
(gyg 1)gߥ=
g0g
q=g02'zgߥ=,)G'zg=G;y'zg{disfsurjectiv!e.1)fand2)=,)
'zg{disbijection.B)X@<;1Y ;Z(
S1)X 1=
1 1 \|X 1=1X柹=X=,)XH #X2)X%"H bnY=,)I*9h;1X h
=h 1 \|X h=Y=,)X h=hY=,)X%"=hYnh 1=(h 1 \|) 1Yh 1=Ynh-:q% cmsy6 1G=,)
9h 1g#2HH;1Ynh-: 1=X柹=,)YyH #X3)XH QY ;1YSHZ=,)
9hz1;hz22HH;s:th:)qX h q1Q=Y ;Ynh q2䔹=Z=,)Z=he 12 \|(he 11X hz1)hz2=(hz1hz2) 1 \|X (hz1hz2)
=Xh q1*h q2Ϲ=,)9hz1hz2ʫ2HH;1Xh q1*h q2Ϲ=Z(=,)XH #ZByf1),2),3)therelationH isanequivdDalencerelation.De nitionf1:10S߹FeorfX
G;yHhG,w!edenotebyC ȁlHD(x)
=fX h Ljh2H gthecponjugateclassofXwithrespecttoH,formorebrie
ye,theH-conjugateclassofX. t10
iӍ:-iӍ|-Theorem21.16QUkLpetS9
G;x;1yo:2G. vThen(S xy)y=S xy
g: JaPrpoof: v(S xy)y=
yd 1 S xyo:=yd 1 (x 1 \|S x)y=(xyd) 1 \|S xy=S xy
De nitionf1:11S߹LetfNLetHY G;1S*G:ThesetNHD(S )=fh2HYjSh =SgŹiscalledthenormalizerofS9inH .
TheksetCHD(S )%Z=fh2H j8s2ShshK=sgkiscalledthecpentralizerofS|inH .Theorem21.18QUkNHD(S )andCH(S )arpesubgroupsofH (andG). vMoreoverCHD(S )nENH(S ):Prpoof:(1)fhz1;1hz2ʫ2
NHD(S )(,)Sh q1
<ѹ=S9=Sh q2
<ѹ=,)Sh q1*h q2Mi=(Sh q1
2*)h q2 ?=Sh q2
<ѹ=S9=,)fhz1hz2ʫ2
NHD(S ) T.:h:75=,)NH(S )G:(2)fhz1;1hz2ʫ2
CHD(S )=,)8s2SGsh q1 ?=s=sh q2=,)sh q1*h q2=(sh q1 )h q2=sh q2=s=,)hz1hz2ʫ2
CHD(S ) T.:h:75=,)CH(S )G:(3)fDenoteLCHD(S )
C,;1NH(S )N :fOb!viouslyLCH(S )NH(S ):fNo!w,letc
2C,;nf2N ; s2S:ڹThen s(c-:n 7)펹=sn-: 1 cn u?=(sn-: 1)cn[/=(sn-: 1 cH)n
=(sn-: 1)n
=s1 j=s;ڹandsocn2
Cnforfeac!hLc2Cy=,)C ȁn {xC=,)8nfC ȁn {xC(T.:h:175Ź=) SC7mEnN : JaTheorem21.19QUkLpet[S9
G;1HhG:LpetC ȁlHD(S )=fSh *jh2H g[bpetheH-cponjugateclassofS . vThenumbperofdi erentconjugatesofS|equalstotheindexofNHD(S )inH ,thatis: j
C ȁlHD(S )j=jHh:NH(S )j: rPrpoof: SC ȁlHD(S )
=fSh *jh2H g;ySh q1
<ѹ=Sh q2
<(,)Sh q1*hi@ 1#Ս2 =Sh q2*hi@ 1#Ս2 =S1l==S9(,)hz1he 122NmNHD(S )(,)hz1
q2NH(S )hz2
q(,)NH(S )hz1
q=NH(S )hz2:DTh!us,Sh q1=NmSh q2(,)NHD(S )hz1.=nNH(S)hz2;oandthereforeSh q1
6=nSh q2(,)nNHD(S)hz1.6=NHD(S)hz2:oWeeseethatthen!umbMerofH -conjugatesof S(equalsthen!umberofdi eren!tcosetsofNHD(S )in Hjǹ,Lthatis: nj
C ȁlHD(S )j=jHh:NH(S )j: t11 iӍ:-iӍ|-Remarks:
(1)ŦIfS=efsg;thenNHD(S )=fh2H&jfsgh
OV=fsgg=fh2Hjsh
OV=sg=CHD(S )CHD(s):fSo,inthiscase:j
C ȁlH(s)j=jHh:CH(s)j:(2)jshp=
s(,)h 1 \|sh=s()sh=hs:jTh!usthecentralizerCHD(S )isthesetofelementsinH'comm!utingfwithallelementsofLS:(3)fNEnG
(,)NG (N1)=fgo:2GjNg D=Ng=G(4))CG (G)o=fz_2Gj8gI2Ggdz =gdg=fz_2Gj8gI2Ggdz_=z{IggZ ȁ(G);)consistingofelemen!tsfinLGwhichcommutewithallotherselementsofLGiscalledthecpenterofG.Ja Infthefollo!wing,LH'willalwaysbMeasubgroupofLG:De nitionf1:13T,ƹLetMH>}G.A7familyFofsubsetsofG,FP | (G)=fX jXwuGnf gg,iscalledH -invariantif8X2F_,8h2H)X h 2F. ThisisequivdDalen!twith:8X2F)
C ȁlHD(X )F_,fwhic!hmeansthatFŹconsistsoffullH -conjugateclasses.De nitionf1:14T$LetF
bMeH -in!vdDariant.qAמsubset[F_]=\fXz1;1?;1XȮk#gF]iscalleda>setofH -cponjugaterepresentativesforfF_,if:f1)8fX2
F9i; cX s `>HyXziK>andH"2)\ѣi6=j
)
WsXzid6s HdXzjf
:Ob!viouslye,fthereareasmanyrepresentativesin[F_]asH -conjugateclassesinFŹanddFŹ=k
g
[RɖF 퍑
i=1.C ȁlHD(Xzidڹ)?:."lTheorem21.20QUk(Class^F)ormula)LpetF⽺beanH -invariantfamilyinG,and[F_]
=fXz1;1Xz2;?;XȮk#gasetofH -cponjugaterepresentatives. vThen"w mrjFVj
=Ohk
X
"㍑i=1jHh:NHD(Xzidڹ)j:$-Prpoof: vByfDef.14,F!=
ɖFA *kH *i=1 ?/C ȁlHD(Xzidڹ)I J(ɖF -disjointunion).l)
{jFVj
=ɖPAkHi=1 jC ȁlHD(Xzidڹ)jT.:h:195 F=ҟɖPA'kH'i=17JjHh:NH(Xzidڹ)j JaCorollary21.21SጺConsidering$JGasunionofit'selementsGr=[JgI{2Gfgdg,2it$JisH -invariantfor'anyH
&G.
Lpetfxz1;1:::l;1xȮk#gbeH -conjugaterepresentativesforelementsofG.ThenbyTheporem??Vv& ()jGj
=Ohk
X
"㍑i=1jC ȁlHD(xzidڹ)j=Ohk
X
"㍑i=1jHh:CHD(xzidڹ)j:5IfHh=
G,wehave: VtjGj
=Ohk
X
"㍑i=1jCHD(xzidڹ)j=Ohk
X
"㍑i=1jG:CG (xzidڹ)j: t12
/֠iӍ:-iӍ|-
Ifxzio2
Z ȁ(G),thenClG (xzidڹ)
=fxzigandso: vxzio2
Z ȁ(G),CG (xzi)=G,jG:CG (xzi)j=1:Thus jGj
=X
i;G=CX.G7(x8:i,r)23jG:CG (xzidڹ)jn+1AX
i;G>CX.G7(x8:i,r)1jG
:CG(xzidڹ)j=ZmX
i;x8:i,r2Z (G)+Nw1n+
X
i;x8:i,r62Z (G)*jG
:CG(xzidڹ)j;#thatis:yf()jGj
=jZ ȁ(G)jn+
X
i;x8:i,r62Z (G)*jG
:CG (xzidڹ)j:*TۍTheorem21.22QUkLpetHH;1K(
G. vThenH K
G,H K=K ȁH .Ja Prpoof:=,): H KS!G)Hna=HRn1HK ȁ,KS!=1nKH K)K ȁHnaH K. -DBy&dTh.14jK ȁH j
=jHK ȁj,andfth!us,bMecauseof niteness,K ȁHh=
H K.f(,Ϲ=:{H K(=
K ȁHh)HK pHK(=H(K ȁH)K(=H(HK ȁ)K(=(HH)(K ȁK)
=HK(T.:h:75+N)OHK(G. b~Theorem21.23QUkLpetNpE?:G.Then(G=dDN ;1)isagroupforcosetsmultiplicationasgroupoper-ation. Prpoof:SinceVgz1Nڀgz2N+=gz1(N1gz2)N=gz1(gz2N1)N=gz1gz2N1N=gz1gz2N1,theVgroupopMerationisw!ellde nedl(indepMenden!tonchoiceofcosetsrepresentativesgz1;1gz2).fWeelhave:gdN+$1NLet (G;1G )and(HH;HD)bMegroups. A imapping'Z:G!H톹is źa1homo-morphism\frpomGintoH ,Џif8gd02;1gd00
722G;(gd0@G
-gd00)'=gd02'
H '4gd00',Џthatis,iftheimageofproMductfistheproductofimages.Weeillustrateitb!ythefollowingpicture: ؍{X'; ' ̟{X&{XL$L; ' ̟{X%/ {X / {X {X ' ; ' ̟{X&{X ℨ$ ℨ; ' ̟{X% {X {X 9 E: E;UV E/ ) % gd309 gd3001 gd30Zgd300 E Ъ E UV E h30 h300 h301h300) 32 fd P ά-y 32 fd F 32 fd
-= 32 fd fe=GG fd fe=f[ fd fe= pC fd fe> Ԅr fd fe>fԘ fd fe>nԬ fd fe>@ fd fe?>j fd fe?. fd fe?П fd fe@ fd fe@]# fd fe@6| fd fe@|I fd feA5{]% fd feA}pb fd feAzՃz fd feB
Ֆ fd feBUթ{ fd feBռf fd feBɟ( fd feC-ڟ fd feCv fd feC fd feDW fd feDN+ fd feD>7 fd feDP~ fd feE'Rb fd feEot fd feEֆ fd feF ֘ fd feFH֪R fd feFּ fd feF;ͳ fd feG!V fd feGj+ fd feG: fd feG fd feHCp$ʄ fd feH6 fd feHԘG fd feI
X fd feIeh fd feI'yل fd feIԟ fd feJ?כD fd feJ fd feJΟ[ fd feK̮ fd feKb) fd feKF fd feK fd feL fd feOWf fd feP96 fd fePj5 fd feP˞ fd feQ؟ fd feQ]П!r fd feQ/ׄ fd feQ>p fd feR9fL fd feR[ fd feR˟i) fd feS,wp fd feS^مT fd feSٓr fd feS"١+ fd feT:Hٯ fd feTşټ fd feTHz fd feU|݄ fd feU`
倄 fd feU fd feU 2 fd feV<
? fd feVʟ fd feVt'r fd feW$4 fd feWb}AQ fd feW8NP fd feWZ܄ fd feX?`g fd feX,t fd feXҜڀ fd feY sڌ fd feYfPڙz fd feYΟڥ fd feYڱ fd feZC=ھ' fd feZ0 fd feZ) fd fe[ ᰄ fd fe[j fd fe[̟ fd fe[o fd fe\H0 fd fe\0ӄ fd fe\P& fd fe]&v2l fd fe]p0=` fd fe]aH fd fe^Sz fd fe^N`^ fd fe^i@ fd fe^st? fd fe_,~ fd fe_wۉ fd fe_ۓЄ fd fe` P۞} fd fe`U8ۨ fd fe`۳ fd fe`۽ fd fea4k fd fea~2 fd fea d fd feb fd feb] fd febz fd feb2Z fd fec<Ο & fd fecW fd fecѐ fd fed =( fd fedff1z fd fed :U fd fedPC fd feeFLV fd feeܟU~ fd fee ^ fd fef%g fd fefp9o\ fd fefx2 fd feg܀a fd fegPT܉ fd fegDܑ fd feg:ܙ fd feh0ܡo fd feh{ܩʄ fd fehܱx fd feiܹ fd fei\,- fd fei6 fd feişЎ fd fej fd fe 5$0 fd fe .# fd fe ̟ fd fe ` fd fe k > fd fe ք fd fe ZG fd fe S fd fe fd fe 턟B fd fe ;O] fd fe z fd fe ޟ j fd fe " ^ fd fe p# fd fe h fd fe 8 fd fe Y** fd fe " fd fe fd fe Bw fd fe fd fe fd fe + fd fe x fd fe ҄ fd fe ༄ fd fe aHj fd fe x+ fd fe Ԯ fd fe JF fd fe ̞ fd fe
fd fe 4f: fd fe fd fe ݻ fd fe =ݷ fd fe knݲv fd fe ֟ݮj fd fe Dݩ fd fe Uݤ fd fe ݟb fd fe Dݛ fd fe ?ȟݕZ fd fe Rݐӄ fd fe ۨ݊ fd fe *=݆N fd fe x؟݀N fd fe <{u fd fe uJ fd fe bOpH fd fe k2 fd fe dDŽ fd fe M2_ fd fe X fd fe xS fd fe 8ELɄ fd fe G8 fd fe Ԫ@L fd fe #: fd fe rl3{ fd fe - fd fe &V fd fe \ J fd fe ݄ fd fe fd fe HU fd fe ` fd fe q fd fe 46h fd fe Rz fd fe ʄ fd fe G߱ fd fe ot fd fe PД fd fe ɒ fd fe Zm# fd fe ܹ fd fe ܱ^ fd fe Fܪ
fd fe @ܡE fd fe 嚟ܙȄ fd fe 3ܐ fd fe ܉2 fd fe
܁x fd fe zxH fd fe ope fd fe g
fd fe
^ fd fe ]Ux fd fe 6MC fd fe ȟC fd fe H;4 fd fe 1X fd fe 7(ф fd fe 6t ʄ fd fe " fd fe ֟ fd fe $! fd fe s fd fe 4ﰄ fd fe ( fd fe aΟ fd fe 0J fd fe fd fe Nv۽ fd fe \۳ fd fe Hۨ fd fe <۞ fd fe ۓ fd fe ܵۉ fd fe +<~ fd fe {Dt fd fe ԟi fd fe ^ fd fe j T fd fe H fd fe > fd fe X2 fd fe ( fd fe ͟ fd fe F> fd fe fd fe fd fe 5˟ fd fe ⢄ fd fe fd fe %Dք fd fe uھ
fd fe Āڲ fd fe ڥ fd fe e`ښB fd fe Hڍ fd fe Ɵځz fd fe St- fd fe @h^ fd fe ϟZ fd fe CΟN fd fe hAK fd fe 5* fd fe 4'\ fd fe fd fe ֟
fd fe $ fd fe uH fd fe l fd fe 36 fd fe f ҄ fd fe 2ٽ fd fe
ٯj fd fe UE٢h fd fe (ٓ fd fe ن fd fe FZw fd fe NjJ fd fe 栟[: fd fe 7M fd fe > fd fe 0܄ fd fe )!v fd fe z* fd fe ɘ fd fe fd fe j0b fd fe \A fd fe Z fd fe \غ fd fe Pة fd fe ؛ fd fe N&؋N fd fe t| fd fe lJ fd fe @i] fd fe ȟL fd fe r> fd fe 2ܟ-F fd fe / fd fe fd fe % fd fe u@ fd fe Ɵ݉ fd fe R fd fe h! fd fe ù fd fe כ fd fe [2Ą fd fe Ĭڟz fd fe j" fd fe NsXM fd fe Š,H, fd fe 6, fd fe A% fd fe Ƒ fd fe 㰟D fd fe 5 fd fe DžR fd fe pф fd fe )Vֽ fd fe yo֪` fd fe `֙r fd fe ֆ fd fe m~u fd fe ɿb fd fe QB fd fe a> fd fe ʱ, fd fe T fd fe V, fd fe ˦s fd fe ℄ fd fe J˟е fd fe ̛ ռ fd fe Zժ fd fe = fd fe ͏Մ fd fe Hp҄ fd fe 2^ fd fe ΅
JD fd fe f7ф fd fe '#b fd fe zGƄ fd fe ʟ, fd fe fd fe;ퟻ~6 fd fe;Пj~ fd fe<V fd fe<]CV fd fe
ք fd fe>U fd fe> fd fe>ɟ< fd fe?-ڟz fd fe?vq fd fe? _J fd fe@WLބ fd fe@N:n fd fe@(- fd fe@ fd feA'RЄ fd feAo fd feA៹DŽ fd feB ҄ fd feBH fd feBF fd feB; fd feC! fd feCj+u fd feCd* fd feC RĄ fd feDCpA fd feD0] fd feDԘ ^ fd feE
J fd feEev fd feE'싄 fd feEԟ fd feF? fd feF fd feFΟ fd feG fd feGb)F fd feGy fd feGhׄ fd feH fd feS$1ʄ fd feSb}% fd feS8 fd feS fd feT?` fd feT,Q fd feTҜ处 fd feU sn fd feUfP fd feUΟ߄ fd feU fd feVC== fd feV0r fd feV)T fd feW fd feWjx fd feW̟mJ fd feWoa~ fd feXHV4 fd feX0J fd feXP?r fd feY&v3 fd feYp0) fd feYa fd feZ fd feZN` fd feZ$ fd feZs% fd fe[,粄 fd fe[w܄ fd fe[Ҕ fd fe\ P fd fe\U8ʄ fd fe\F fd fe\T fd fe]4 fd fe]~2 fd fe] fd fe^d fd fe^]w[ fd fe^m fd fe^2d
fd fe_<ΟZ> fd fe_韴Q
fd fe_ѐGj fd fe` =>d fd fe`ff4 fd fe` , fd fe`P" fd feaF fd feaܟ fd fea a fd feb%b fd febp9 fd feb2 fd fec fd fecPTV fd fecDR fd fec:΄ fd fed0 fd fed{ fd fed fd fee fd fee\,7 fd fee. fd feeşք fd fef<蟳 fd fefrɄ fd fefҠ fd feg 3x fd feghlp fd fegi fd fegLbF fd fehIZ fd feh柳T^ fd fehߕM6 fd fei*Fʄ fd feiu?˄ fd fei9 fd fej z2 fd fejW@, fd fej% fd fej톟 fd fek9 fd fek fd fekm
g fd fel ҄ fd felf fd fel6 fd fel(- fd femH fd fem˟ fd fem߀ fd fen*C fd fenvBZ fd fen̄ fd feo
fd feoX韲˩ fd feo fd feoޟڄ fd fep;r fd fep矲_ fd fepԟ" fd feq 8 fd feqib fd feqe fd fer8 fd ferMA fd ferPb fd fer䜟 fd fes0` fd fes| fd fes0 fd fet[a fd fet_X fd fet2 fd fet0{R fd feuCxW fd feu蟲t fd feubqЄ fd fev'nB fd fevt k fd fevh8 fd few e fd fewX`b fd few`3 fd fewj] fd fex< Z fd fexX fd fex
V fd fey SX fd feymJQ fd fey N fd fezMO fd fezR fd fe|^= fd fe}G<) fd fe}N ;r fd fe}9 fd fe}9X fd fe~37 fd fe~7 fd fe~՟6V fd feП6 fd feeޟ5
fd fe䟲5 fd fe4 fd fe L 48 fd fe #3w fd fe H3 fd fe 2j3* fd fe 3 fd fe ş31 fd fe 3҄ fd fe e43 fd fe r4X fd fe 4; fd fe L 52 fd fe O5> fd fe 椟5^ fd fe 26 fd fe \6ބ fd fe ̽8@ fd fe (8 fd fe g:? fd fe :ڄ fd fe < fd fe O=V fd fe ?9 fd fe @& fd fe 5B4 fd fe .CJ fd fe ̟E fd fe `F fd fe k I& fd fe J fd fe ZM fd fe SN fd fe Qh fd fe 턟S" fd fe ;OV fd fe W fd fe ޟZ fd fe "] fd fe p`A fd fe hbv fd fe 8e܄ fd fe Y*h: fd fe "k˄ fd fe nR fd fe Bp fd fe ꟲt fd fe w fd fe + {~ fd fe x~k fd fe fd fe fd fe aH fd fe x9 fd fe fd fe J fd fe Ƅ fd fe W fd fe 4f* fd fe fd fe fd fe =ń fd fe kn fd fe ֟ fd fe DN fd fe U fd fe fd fe D` fd fe ?ȟ
fd fe RՑ fd fe ۨf fd fe *= fd fe x؟ fd fe < fd fe ⟲ fd fe bO fd fe 2 fd fe fd fe M2܄ fd fe
r fd fe xڄ fd fe 8E fd fe , fd fe Ԫ& fd fe #+҄ fd fe rl2 fd fe 8̄ fd fe @ fd fe \F fd fe M fd fe S fd fe HU[T fd fe `a fd fe qiu fd fe 46o fd fe Rw fd fe ~ fd fe G fd fe ot fd fe PЄ fd fe ҄ fd fe ZmA fd fe l fd fe fd fe F쟳Z fd fe @ fd fe 嚟̜ fd fe 3Ռ fd fe 2 fd fe
fd fe z fd fe o fd fe Z fd fe
f fd fe ] fd fe 6! fd fe ȟ"҄ fd fe H+0 fd fe 5 fd fe 7= fd fe 6tG fd fe "PJ fd fe ֟Z| fd fe $!cU fd fe sm fd fe 4v fd fe < fd fe aΟg fd fe 0 fd fe n fd fe NvL fd fe \Ʉ fd fe H҄ fd fe V fd fe ͟Jv fd fe FU& fd fe aq fd fe lJ fd fe 5˟x fd fe fd fe 蟵c fd fe %D fd fe uZ fd fe Ā fd fe ퟵ fd fe e`" fd fe HD fd fe Ɵ fd fe S7 fd fe @ fd fe ϟ ~ fd fe CΟv fd fe h% fd fe 1: fd fe 4? fd fe KR fd fe ֟YK fd fe $e fd fe uHs fd fe l~ fd fe 3. fd fe f fd fe 2k fd fe
fd fe UE fd fe (Ҷ fd fe fd fe FZƄ fd fe N fd fe 栟 * fd fe 7 fd fe ' fd fe 5 fd fe )D fd fe z*R fd fe ɘbN fd fe pF fd fe j0 fd fe \# fd fe
fd fe \T fd fe Pf fd fe ل fd fe N& fd fe t鲄 fd fe fd fe @i߄ fd fe ȟr fd fe r(` fd fe 2ܟ9 fd fe H5 fd fe W` fd fe %h^ fd fe u@w fd fe Ɵۄ fd fe RX fd fe h! fd fe R fd fe ф fd fe [2ڠ fd fe ڟJ fd fe B fd fe Ns fd fe , 8 fd fe 08 fd fe A䟹@ fd fe ៹R fd fe 㰟c fd fe 5uv fd fe Å fd fe p fd fe )VX fd fe yo fd fe ` fd fe Ʉ fd fe m~ fd fe ſ fd fe " fd fe a(O fd fe Ʊ9 fd fe M fd fe V,^ fd fe ǦspH fd fe fd fe J˟ fd fe ț r fd fe Zj fd fe =X fd fe ɏy fd fe H fd fe 2܄ fd fe ʅ
fd fe f. fd fe 'C fd fe zGU fd fe ʟj8 fd fe <| fd fe o fd fe E fd fe ȟ fd fe cb̷ fd fe ͵҄ fd fe fd fe Y,
X fd fe Ϋ˟ fd fe p32 fd feu ' ' 'Herefgd02'
=h09;1gd00'=h00')(gd0gd00)'=h09h00 r. Ifmoreo!ver,'isabijection,w!esaythat'isanisomorphism,andGandHȦaresaidtobMefisomorphic,G
l
=H . IfԪH;=WG,;thehomomorphismiscalledendomorphism,andanisomorphismofGon!toGisfcalledautomorphism. t13 >ŠiӍ:-iӍ|-Examplef1:7H@':G!HH;^':g.7!1HD,^that+-isforeac!hg2G;1gd'=1HD,^is+-alw!aysa
homomorphism:(gd02gd00)'
=1H #=gd0'ngd00'
=1HD:1HExamplef1:8Gĕ':G!G;c':g(7!gd,ethat2is'idG ʹis2alw!aysanautomorphism:u'isbijectionfand(gd02gd00)'
=gd0gd00=gd0ngd00=(gd02')n(gd00'): UTheorem21.24QUkLpetUG,$Hbegroupsand'
:G!Hahomomorphism.De neIm '=fgd'jgo:2Gg
=G';Ker'=fgo:2Gjgd'=1HDg. vThenIm'HH;1Ker'nEGandG=Ker'
l
=Imƾ'. TaPrpoof: v1)f1G '
=1H #:
(gn1G)'=gd'=g'n1G '
=g'n1H $PT.:h:25 #=,) ܹ1G '
=1HT52)fgd 1 '
=(gd') 1g#:
(g 1g)'
=1G ' 1);㍓=1H #=gd 1'ngd'
=(g') 1 \|g' T.:h:25=,)g 1 '=(g') 1 \|.3)Im.'+H{: -h09;1h00 2Imp'=,)9gd02;gd00
s:th:h0fd=gd02';h00 =gd00'=,)h09h00=gd02'
gd00'+=f(gd02gd00)'
=,)h09h00'2Im ' T.:h:75=)Im$'H .4)6KerF'
G:!gd02;1gd002Ker'=,)gd0'=gd00'=1H #=,)(gd0gd00)'=gd0'Pgd00'
=1H iP1H #=1H=,)gd0ngd002
Ker' T.:h:75
=,)Ker)e'
G.T55)YKer'fEG5Y:denotefKer='=N1;,8g2GY8n2N=(ng)'=(gd 1 ngd)'=g 1 'fn'g' \2);㍑5Y=(gd') 1
hn1H
0g'
=1H #=,)ngߥ2Ker'=NDgz1'=1H #=,)(ge d 12 ')(gz1')=1H #=,)(ge d 12 gz1)'=1H #=,)
ge d 12 gz1ʫ2Nh2HGfnhjn2N1g=TɖS
>h2H\xNh=TɖS
>h2H\xhN=TɖS
>h2HGfhn1jn2rN1g
=fhn1jh
2HH;1n2Ng=H Ni,andnz15:1::,nzi 1AVnzi47=n0S15:1::n0
hi 1AVn0
hidڹ,w!egetinthesamewaythatnzi=n0
hidڹ.Son0
hi=nzikܹforalli=1;1:::l;1nandtherepresen!tationgo:=
nz1nz25:1::,nȮkBisunique.KTh!us :
G7!Nz1φNz2:1::gNȮk#;go:=
nz1;1nz25:::,nȮk.97!(nz1;nz25:::,nȮk#)isfabijection.Ja ( )fIfnzio2
Nzid;1nzjp2Nzj pandfi6=j v,thennzidnzjp=nzjf
nzi:Weeccanassumewithoutlossthatj <
i.No!wNzj\^NzioNz15:1::,Nzj9;:1::Nzi 1=\Nzio=1)Nzj\Nzio=1.xObserv!evne 1i \|ne 1jnzidnzjf
.SincevNzi;1Nzjv(E G,w!ehave:F)ne 1i \|ne 1jnzidnzjp=
ne 1inאn8:j mMi 2NzidNא1n8:j mMi
q=NziNzio=̍Nzidڹ, andne 1i \|ne 1jnzinzj~=ot(ne 1j)n8:inzj2otN;1n8:iBj
aNzj=Nzjf
Nzj=Nzjf
.NSone 1i \|ne 1jnzidnzj2otNzi\>Nzj=1)ne 1i \|ne 1jnzidnzjp=
1)nzinzjp=nzjf
nzi. (
)fLetgd02;1gd002
G;gd0=s=n0S1n0S25:::,n0Nk#;gd00=n00S1 rn00S2:::7n00Nk#.Thenfgd02gd00=n0S1n00S1 rn0S2n00S2:1::7n0Nk#n00Nk:
InbtheproMductgd02gd00
=n0S1n0S25:1::,n0Nk#n00S1 rn00S2:1::7n00Nk eac!hn00
hi Թcan,!becauseof( ),!\jumpo!ver"bleftneigh!bMoursfuntilitcomestotherightplaceofn0
hidڹ,whichresultsintheformstatedabMove. (j)Weeseeb!y(
`)and(
)that l:G!Nz1:Nz2:1::;NȮk l:g=nz1nz25:1::,nȮk
+!(nz1;1nz2;:::l;nȮk#)fisanisomorphism:(gd02gd00) p=3(n0S1n00S1 rn0S2n00S2:1::7n0Nk#n00Nk) 7!3ݹ(nz1n0S1;1nz2n0S25:::,nȮk#n0Nk)3=(nz1;nz25:::,nȮk#)(n0S1;n0S25:::,n0Nk#)3=(n0S1n0S25:1::,n0Nk#)'f(n00S1 rn00S2:1::7n00Nk)'
=(gd02 d)(g00 ). InEthefollo!wingweshalllist,withoutproMof,someimpMortan!tresultsconcerningstructureoff nitegroups.De nitionf1:18TDenote^thesetofprimesb!y, =fn2Nj^n=ab;`a;1b2N7)a=1Aorb
=1g=f2;13;5;7;11;13;:::g.bLetm
2N ;1p2.bWeewritept>n,V forptqjnandpt+1x-n.Th!usfpt80isthehighestpMowerofp>,dividingn.De nitionf1:19S߹LetfpbMeaprime.Gisap grpoup,ifj
Gj=ptʹ,fforsomet
2N[n0.De nitionf1:20U$LetGbMeagroup,Hpaprimeandpt>jGj.AsubgroupP8qGoforderjPVj_=ptis calledSylow4p-subpgroupof G. Weedenoteb!yS ydlzp](G)thesetofallSylowp-subgroupsfofG. JaTheorem21.27QUkLpetGbeagroup,p
2. vThen: (1)TherpeexistssomeP
Gs.th. vjPVj=pt>jGj,thatisS ydlzp]G6=;. (2)-IfPz1;1Pz2C2S ydlzp](G))9gtP<Vgv1 K=Pz2,>so-Sylowp-subpgroups-areal Flconjugatewitheachother. (3)Eachp-subpgroupofGiscpontainedinsomeSylowp-subpgroup. (4)4F)orthenumbperofSylowp-subgroupsinG;wehave:!jS ydlzp](G)j`,1(LmoMd p)4andjS ydlzp](G)jjGj. t16 ėiӍ:-iӍ|-Examplef1:9Jb/IffjGj
=15,thenG=PBnQ;yP2S ydlz3(G);Q2S ydlz5(G).fPrpoof: SjGj
=3n5
T.:h:275*D=,)
(1)9P
G;1jPVj=3;9QG;jQj=5:(2)PM;1QnEG
:jS ydlz3(G)j1(Lmod 3))jS ydlz3(G)j2f1;14;7;10;13;:::gand%mjS ydlz3(G)jj15)jSydlz3(G)j=1)PEnEG,.bMecause%mPVg Y2Sydlzp]G%m)PVg Y=Pù(PisuniqueSylo!wp-subgroup).9InthesamewaywegetQTEG.9Sinceobviouslye,̪P\Q==1,and!ȍjPVQj
=2jP.:jjQj=ڟ@ p ]ҟp jP.:j\jQj$ن=K=ھ35=ڟ p
|KdN<1X0=15=jGj,fitfollo!wsthatG
=PQ.No!wfG=PBnQb!yTh.26. JaTheorem21.28QUkLpet^Gbeanabeliangroup.ThenGisthedirectproductofitsSylowp subgroups.%pPrpoof: v:Letfj
Gj=k
g
ɖQ 퍓i=1.p; 8:iBi " ;18i;y zio1;pzio2;pzio6=pzjf
;ffori
6=j;fandletPzi2
S ydlzp8:iϹ(G);aˍjPzidj
=p; 8:iBi " ;yf-orDi=1;12;:::l;kX?:Ja 1)8i;*Pzi E{EG:AsGisabMeliangroup,andtherefore:eg2G;1H_G)gdH=H gd,ev!erysubgroupfofGisnormalinG. 2)fG
=Pz1Pz251,PȮk#: Because|ofcomm!utativity|Pz1Pz2 ι=Pz2Pz1G.Y By|proMductform!ula(Th.13)jPz1Pz2 j=jjP q1*jjP q2j33@ p
p jP q1*\P q2j# =
+pqǍ Ϳ1N1 ~bpqǍ Ϳ2N2+mU p ~Kd 1%D=p q1č1
>p q2č2.UWee59proMceedb!yinduction.LetalreadyPz1Pz251,Pzi 1vbMeagroup0Dandňj>Pz1Pz251,Pzi 1j=p q1č1
>p q2č2 o1p, 8:i 1 i 1ɹ.;BNo!w(Pz1Pz251,Pzi 1AV)Pzic=Pzidڹ(Pz1Pz251,Pzi 1AV),PbMecauseňofcomm!utativityfofG,andsoPz1Pz251,Pzi 1AVPzi @isagroup,b!yTh.22.ItsorderisbyTh.14:@7 juPz1Pz251,Pzi 1AVPziYj=v!jP q1*P q2 @P8:i 1 2jjP8:i,rjΉ p ?'.p jP q1*P q2 @P8:i 1 2\P8:i,rjJx=
pqǍ Ϳ1N1 ~bpqǍ Ϳ2N2p8:i,r 8:imU p 2Kd;W1>=p q1č1
>p q2č2 o1p; 8:iBi " .jNamelye,the greatest/Xcommonfdivisorofj
Pz1Pz251,Pzi 1KjandjPziojis1,sojPz1Pz251,Pzi 1B\nPzioj=1. Infsuc!hawaythisformulaistruefori
=1;12;:::l;kX?.p ThereforejPz1Pz251,PȮk ֨j= Ak
g
hɖQ 퍓i=1Gp; 8:iBi
=jGj.SincealsoPz1Pz251,PȮk ֨G,itm!ustbMeG=mÍPz1Pz251,PȮk#. 3)fAsindicatedabMo!ve,LPz1Pz251,Pzi 1B\nPzio=
1f. Ferom_h1),2)and3),itfollo!wsbyTh.26thatGis(isomorphicto)thedirectproMductofPz1Pz251,PȮk#,ftheSylo!wpzid subgroups(i
=1;1:::l;1kX?)fofthegroupG. ItfremainstheproblemtodeterminethestructureofanabMelianp-group.Weeha!ve: JaTheorem21.29QUkB:dLpetPM;1jPVjB=p ,dbeanabelianp-group,forsomepB2.dThenP
is(isomorphicto)thedirpectproductofcyclicp-groups: P=
Az1.nAz21;Azt;!0kBAzio=
hazidja-:0`c
x-:0 q)
=a + -:0Ǿ(modf4)3<{b i>+ -:0(modf3)2c
x+
-:0*(modf3)2.FeorfEx.a3b2cna2b2c2ʫ=
a5b4c3=
ab t18 &iӍ:-iӍFChapter
27ǍP8ermutation T{Groups;JaWeefstartagainwithTheorem1fromthe rstc!hapter.̍Theorem22.1K Each.RppermutationW2S (
),?,j
j=n.Rontheset
,cpanberepresentedinthe
cyclicform: 獑& o:=
( z1;1 z1d;:::l; z1dzr q1* 1)lŹ( zid; zid;:::l; zidzr8:i,r 1q)lŹ( zsn<; zsd;:::l; zsdzr s 1ƹ); j4]. z1;1:::l;1 zsx2
;O s
X
"㍑i=1b_rzio=n;( ziddzr8:i,r 1q)o:= ziOf-or8al7)l"i;"@L thatisasaprpoductofcyclesoflengthsrz1;1:::l;1rzsn<.ThecyclesofO}cponstituteapartitionof
. Weefha!vealsoseenhowtocomputeproMductsofperm!utationsrepresentedincyclicform.De nitionf2:1NvZLet2~S (
).Weesa!ythatHisofcyclicQtyppe(kz1;1kz2;:::l;kznP),ifithaskz1cyclesfoflength1,kz2fjcyclesoflength2,:1::?;kzn Ncyclesoflengthn.Examplef2:1ERThenpMerm!utationo:=
(12)(358)(47)(6),:2Sz8,:isnofcyclict!ypMe(1;12;1;0;0;0;0;0).De nitionf2:2M Ev!eryfsubgroupGofthesymmetricgroupS (
)(ofdegreen)iscalledppermutationgroupf(ofdegreen).De nitionf2:3LRLetnG
S (
),:o:2G,ӂbMeingnofcyclict!ype(kz1;1kz2;:::l;kznP)
(kz1(d);kz2(d);:::l;kznP()).Letftz1;1tz2;:::l;tzn NbMefvdDariables.Denoteb!yTV(d)theexpression TV(d)
=t:uk q1*(I{)S16t:uk q2*(I{)S2g1%Ktzk n 7(I{):jn:Zӹ(2.1) 獺TheQcyclicindexofthegrpoupGisthea!verageofallsuc!hmonomialsoverall2`G,thatis,thefpMolynomialo!verfQ(thesetofrationaln!umbers): ITZG (tz1;1tz2;:::l;tznP)
:=ٺ1=ڟ㦉 p Y
jGjǷX
PrI{2G)>TV(d):Zӹ(2.2)]Y t19 .iӍ:-iӍ|-Examplef2:2ERLetnG
Sz4ʫ=S (f1;12;3;4g),:G
=f1;1(12);(34);(12)(34)g:Since1
(1)(2)(3)(4);
(12)
(12)(3)(4);1(34)(1)(2)(34),thefcyclicindexofGis: J ZG (tz1;1tz2;tz3;tz4)
==1=ڟ㦉 p y
4
(tz4:j1.+ntz2:j1tz2+ntz2:j1tz2+ntz2:j2)
==1=ڟ㦉 p y
4
(tz4:j1+n2tz2:j1tz2+ntz2:j2)De nitionf2:4M#Let~
bMea niteset,Gagroupand:$G7!S
a~homomorphism:$g7!zg;1g2`IG;zg5G2S
,andsozg q1*zg q2 s=zg q1*g q2 forgz1;1gz2 M2G:Thenw!esaythatGacts/on
.xWeede nefon
arelationG
b!y:Jatdfor
`;1 2
;y G
*? (,)
9go:2Gf
`zgߥ= JaTheorem22.2K TherpelationG
on
isanequivalencerelationon
.Prpoof:1)/8 :
G
`:ffor1m2G/isz1q=m1SX.
dE-theiden!tity/mapping.yTh!us z1q=m :
andso
G
*?
`. 2): G
O)x] G :` G O)x]9g2G;k
`zgM[= )
`zg_ngI{ 1?= z18a= = _ngI{ 1?)9gd 1˶2
G;y _ngI{ 1e= ) G
*? 3) G
;1 @uG
H>) G
H>:9gd;1g0O2G;ڵ
`zg= ; gI{0 3H=
H>)
= gI{0 3H=
`zggI{0=
`gI{g0j)
9gd;1g0=s2Gfs.th.
`gI{g0j=
b) G
*?
De nitionf2:5MTheequivdDalenceclassesforG
ɒon
arecalledG-orbits4on
.Feor 2
w!edenote:theorbitof ˚b!y
`G7^=f zg \jg2Gg:anditslengthb!yj Gj..ZLetf z1;1:::l zqgbMetherepresen!tativessrofG-orbitson
.Then
=t=fgo:2G1j
`zgߥ= gfiscalledthestabilizerof JinG.Thefelemen!tsof
wecallpMoints. JaTheorem22.3K LpetGactson
, 2
. vThenGz >Gandj
`Gj=jG:Gz j.Prpoof: 1)2gd02;1gd00
T2OGz )
`gI{0 = = gI{00
D) gI{0BgI{00K= gI{0gI{00
D۹=( gI{0Ź)gI{00
D۹= gI{00
D۹= )@gd02gd002
Gz >T.:hf1:75K)%Gz >G.
2)(
`G㍹=f zg/j1gH 2Gg.dINo!w,I zg= gI{0 jR, zge d 12JgI{0= gI{0_ngI{0 1= 1
!= , _ngI{g0 11=F ,
gdg0 12Gz >,go:2Gz g0=s,Gz go:=Gz gd02̹.So"
`zgҹ= gI{0 `,Gz g>g=Gz gd0Uand,Aconsequen!tlye,
`zg6= gI{0 `,Gz g>g6=Gz gd02̹.RTh!us,Atherearefasman!ypMointsin
`GascosetsofGz inG,thatis,j Gj
=jG:Gz j=jGj=jGz j. JaCorollary22.4Mj
`GjnjGz j
=jGjforal Fl 2
. JaTheorem22.5K F)or 2
`Gwith =
`zg,go:2G,wehaveGȮ =G,g0=s2g 1 Gz go:=G i>2
XxȮ i>f q2=WV(fz2),_ssincef =
`g
runsthroughallof
as ƹdoMes.Hence$ fz1ʫG
*?fz2w)
WV(fz1)=W(fz2)andfw!eseethatallmappingsofaG-patternFhavethesameweight.Thuswecande ne.De nitionf3:4M Thefw!eightofapatternFis WV(F)
:=WV(f-);yforDany
f8c2
FM:Examplef3:3GLet(thew!eights(ofbandr, Pb;1r1q2R, bMewKn(b)=xȮb"ܹ, w(rM)=xzrb.1$Then(forfz1;1fz2;fz3;fz4fjfromfEx.3.1: =WV(fz1)
=xȮb"xȮbxzrbxȮbxzrxzr =
xz3Nbxz3:jrߍ =WV(fz2)
=xȮb"xzrbxzrxzrxȮbxȮb-=
xz3Nbxz3:jr =WV(fz3)
=xzrbxȮb"xȮbxzrxzrxȮb-=
xz3Nbxz3:jr WV(fz1)
=xzrbxzrxzrxȮb"xzrxȮb-=
xz2Nbxz4:jr(WeefseethatWV(fz3)
=W(fz1)falthoughfz3ʫ G
*?fz1.) t23 7iӍ:-iӍ|-De nitionf3:5M AfsubsetF!
R
_iscalledclosepdwithrespecttofG
S (
)a,if:
f8c2
F_;ygo:2G)gdf2F_,fthatis,ifFŹcon!tains,withanyf8c2
R
<,thewholepatternGf-.Weefdenoteb!yMthesetofallG patternsandbyMF thesetofpatternsinFExamplef3:4GThesetF߹=݀ff <2R
jWV(f-)=x Nbx6 Ar;1 2gisclosedwithrespMecttotherYgroupGofallrotationsofthecubMe,|becauserYaftercarryingoutsuc!harotationtheabMovecondition,fthatthereareatleast2blac!kcolouredsides,ispreserved.De nitionf3:6M LetfR0
R;yF!R
<.Weefde neinventoriesofR0GandM0Aab!y:Ja inv R0:=[ɖP
|
r
f 2F]WV(f-)Examplef3:5G~InijourEx.1w!ehave:cinvReT=OxȮbʹ+xzrb;ԹandforFf =ffz1fz2;1fz3;fz4g,+w!eijhave:in!vF!=
x3Nbx3Ar.+nx3Nbx3Ar+nx3Nbx3Ar+nx2Nbx4Arʫ=
3x3Nbx3Ar+nx2Nbx4Ar B#Theorem23.3K inv(R
<)
=(PڟɖP
|r8i
`;1 N2
zi|6) fE= f-g>thesetofal Flfunctionsthatarpeconstantoneach
zidں.WTheninvF_0
=
ɖQA
b tH
b i=1v(ɖP =xzrb,{WV(f-)andW(F)bpetheweightsofr2>R;1flt2R
N4andofG-ppatternF2
M,respectively. vThen:2qX
PrnF.:2M ÜWV(F)
=ZG (/ɟX
Prrߍ E)8Weecomputeno!wɖP {jZ(J( J( j'1pM39M4 ,4`2Aȟ4`50p1%W9eПZH{Z0pd0G
=f1;11(25)(34);2(13)(45);3(15)(24);4(35)(12); ۈH5(14)(23);1(12345);(13524);(14253);(15432)gNo!w,fZG
*?=K]ܾ1=ڟ p Kd10
(t5S1.+n5tz1t2S2+4tz5)fandsoforncolours: =X
PrFDWV(F)
=1=ڟ㦉 p
4
10dA[(xz1+xz2+15+xznP)z5+5(xz1+15+xznP)(xz2:j1+15+xz2:jnP)z2+4(xz5:j1+xz5:j2+1}xz5:jnP)];(ߧsetting%xzio=
1foralliw!egetastotalnumbMerofpossiblecolouringsjMznPj
=1=ڟ㦉 p
4
10dA[nz5.+n5nz3+4n]. Feorfexample,for3coloursx;1yd;z{I: MJC&X
Pr*F7)WV(F)
=1=ڟ㦉 p
4
10dA[(xn+y+z{I)z5.+5(x+y+z{I)(xz2.+ydz2+zz2;M)z2.+4(xz5+ydz5+z{Iz5;M)]!*SettingYWx
=yo:=z=1,hw!eget:UjMz3j=K]ܾ1=ڟ p Kd10
[35ҹ+533+43]
=K=ھ390=ڟ p Kd 10;=39YWthecoMecien!tofx2yd2$zcorrespMonds tothen!umber ofcolouringwith3determinedcolourssuc!hthatoneofthemisappliedmonceandtheothert!womtwice.5ItcanbMeeasilyseen,thatthisnumbMerequalsK 1 p Kd10m 40
=4.Examplef3:8GHo!wmanycolouringsoftheverticesofatetrahedronarethereifweapply3fcolours.eО32 fd ,beЬ"eП ,eП 6eП :Ȱ]e
z
%$
+~+
D~ 1p fe?ȰA:Ȱ A5Ȱ A2J( A2J(@7ȟ-@ ,&1=bM4.@29x30G
=f1;1(12)(34);(13)(24);(14)(23);1(234);2(134); ۈH3(124);14(123);1(243);2(143);3(142);4(132)gv㍹=,)
ZG
*?=1=ڟ㦉 p
4
12dA(tz4:j1.+n3tz2:j2+8tz1tz3)=,)
X
PrFWV(F)
=1=ڟ㦉 p
4
12dA[(xn+y+z{I)z4.+3(xz2+ydz2+z{Iz2;M)z2+8(x+y+z{I)(xz3+ydz3+z{Iz3;M)]: ;x
=yo:=z=1=,)jM1j=1=ڟ㦉 p
4
12dA[3z4.+n33z2+83z2]
==32(9n+3+8)=ڟ㦉 p =
3n4Fj=15 t27 (iӍ:-iӍ|- Weefshallsolv!enowtheproblemstatedatthebMeginning:
Examplef3:9FɹHo!wfmanycolouringsofcubMesur cesarethere,ifweapply3colours?}ЍI 9 fd'耎 @l> fd'耎I P fd'耎 @DVr fd'耎I' fe l ' feqn' fel ' feI S ] g q { z @v
I ` S ` ] z ` g p ` q f ` { \ ` R ` @N% q⠟ {⠟ ⠟ ⠟ ⠟ ⠟ ⠠z v
q⠠ ` {⠠ ` ⠠z ` ⠠p ` ⠠f ` ⠠\ ` ⠠R ` N% `p(5)zr w6zr `"1zr (2)zrX 0(3)Ȯby 04ȮbThefrotationgroupofcubMeconsistsof24elemen!ts:Ja 8j8rotationsaroundthefourgreatdiagonals,,b!y120 *
3
cmmi10Aa cmr6|{Y cmr8N H cmbx12N q cmbx12 ':
3
cmti10K`y
3
cmr10<
lcirclew10<
lcircle10O
linew10O
line10 }