FINITE GROUPS OCCURRING AS GROUPS OF INTEGER MATRICES

Transcription

1 FINITE ROUPS OCCURRIN S ROUPS OF INTEER MTRICES - Paa Bra IISER Pu I h roc w udy rou of arc I arcular w ry ad dr h obl ordr of arc of f ordr h rou L h rou of arc wh dra ro cra rul du o Mow rard h oro of h rou L ad L Q h rocd o dr h obl ordr of l h rou I would l o r y raud o y roc ud Prof B Sury for h ooruy o wor wh h udr h IS Sur Rarch fllowh H ac ad hl wa raly arcad ad addd o y rc I rcoz ha h roc would o ha b obl whou h Ida cady of Scc ad ha h for ra h fllowh Laly h roc would o ha bu whou h oao rodd by y coll h Ida Iu of Scc Educao ad Rarch Pu ad faculy cally Prof Rahura who wholhardly uor ay uch rarch daor a by h ud - -

2 INTRODUCTION Th rol of rou hory ly cral o oly o ahac bu alo ohr cc z hyc chry c hr hr yry of hr a rou h bacroud Horcally udy of rou ard a ruao of fly ay obc howr a a lar a h corbud o h dlo of a abrac hory byod h rou of f ubr uch aalyz a abrac rou dr way a a rou of arc rrao hory hrw lh arou fld of cc h rrao hory lay crucal rol uau chac ubr hory c Lalad cocur ad h laua of rrao hory I our r ror w ha dal wh f rou rrd a arc who r ar r Udr h roc ld ``F rou occurr a rou of r arc w udy f ubrou of L h rou of all rbl arc whch ha r r ad who r alo ha r r O of h faou clacal hor du o Mow ll u ha f rou l h rou L ad oly fly ay obl of f ordr for l or ubrou of h rou Th lad u o a h aur of h obl ordr ad how hy h ary wh Du o h fac ha hr a f ubr of h hrfor hr u o aal obl f ordr of ubrou ar by rcollc h bac of rou hory Lara hor [4]: If a f rou ha a ubrou H h h ordr of h ubrou H a dor of h ordr of a : H ad o b a hooorh fro h rou o h rou H f ad oly f y y If a hooorh a bco o-o ad oo ad o b a oorh Th a of a hooorh o H a ubrou of Had h rl of a hooorh h l a o h dy l fro a rou a oral ubrou of Udr rou oorh rou-horc ror ar rrd rou hooorh fro a rou o a rou H whch o-o a obl o dfy oorhcally wh a ubrou of H I arcular h follow rul how ha ry f rou ca b rardd a a rou of ruao Cayly Thor [4]: Ery rou of ordr oorhc o a ubrou of h rou of ruao or h rou S - -

3 L u ow loo a rou of arc udr ar ullcao Th rou L R h of all arc wh ral ubr r ad o-zro dra codrd udr h orao of ar ullcao ll l of h rou ha r udr ullcao c hy ar all rbl ad h dy udr ullcao h dy ar I Slarly h rou L Q h rou of all arc wh raoal r ad o-zro dra O df h rou L o b h rou of all arc wh r r ad dra or - No ha ach ar h ha a r whch alo a r ar I fac a ral ar ha a r ar whch alo ha r r f ad oly f h dra of h ar ± If a r ar ha a r ar B h h dra of ad of B ar r who roduc ; h l ha h dra u b or - Corly f h dra of a r ar or - h r of a ar by dd all h r of ado ar by h dra of ad No ha h ado ar of f ha r r wll alo ha r r Now w ca ro ha ay f rou ca b rardd a a ubrou of L for o Thor: y rou of ordr oorhc o a ubrou of L for h a PROOF : Cayly Thor a ha ay f rou of ordr ca b bddd h rou S Thu o how ha h rou bd h rou L uffc o how ha S bd h rou L Codr ay ruao S L T whr { } T : Q Q h caocal ba forq b h lar raforao dfd by df a a : S L by d S o h ar corrod o h raforao T wh rc o h ba - 3 -

4 obr ha h raforao ar of ach of h T a ar whr ach ry hr or ad wh acly o o-zro ry ach row ad colu Such arc ar calld ruao arc ad h of all uch arc dod by P alo obr ha P S Th ar T ha dra ual o h aur of h corrod ruao ha a hooorh a rc h corrod orao h cooo of ruao ad h ullcao of arc obr ow ha h o-o Codr for a S T T ff Thu ad : S L a o-o hooorh Thrfor bddd S S bddd h rou L By Cayly hor w ow ha h rou Thu h rou of ordr bddd h rou L or ohr word oorhc o a ubrou of L Th col h roof Ordr of f ubrou of L : ow obr ha h rou L for ay hr wll alway a ubrou of ordr ad alo a ubrou of ordr! uch a S I fac w wll obr h urr fac ha hr ay alo ubrou of ordr rar ha!! For al L C b a ubrou of L co of all daoal arc wh daoal r Th L B C P C Hr ad lwhr L B X Y whr X C ad Y P ad X XY X Y XX YY Y P Y Y L ] X C ad YY P XX YY B [ B - 4 -

5 L B L I B whr X Y X C X Y X X ad Y Y P C Y ad Y P P B Y Y L I C ad I P I I I B Thu B clod or addo coa h r of ach of l ad alo coa h dy l Thrfor B a ubrou of L Fro h rou roof w ca ha P S P S! lo C C Thu h cardaly of P ad C B C P! Thu hr a ubrou B of L wh ordr! B drd by h roduc of Srucur of fly rad abla rou rou ad o b abla f h rou orao alo coua o all l of h rou abla rou ad o b a fly rad abla rou f hr a f ub of uch ha h all ubrou of coa h whol of ; w wr Th a ha ry l of a f r lar cobaro of l of hr w wr h orao o addly Fudaal Thor of fly rad bla rou []: L b a fly rad bla rou Th r codo a r ad for all b dd for o r r afy h follow Th ro uu h codo a ad b - 5 -

6 - 6 - abla rou ad o b fr abla of ra f h rou of all -ul of r udr h orao of add ry-w For coc wor wh arc w wll rard a colu cor wh r r Th Fudaal Thor of Fly rad bla rou l ha a abla rou fr abla f a fly rad abla rou wh o oral l of f ordr I alo l ha ay ubrou of a fr abla rou fr wh ra l ha or ual o Th hor ro h urr fac ha f ubrou of Q L ar ally alrady ubrou of L Mor rcly w ro: Thor [] : If a f ubrou of Q L h coua o a ubrou of L PROOF : a f ubrou of Q L ad L r r F r r r F r F F F F obr ha of a ubrou Q L I Th I I H F H lo w fd ha H F F Thu h of raor for F foud o b

7 f adard ba for hch a f a of f ordr Thu h Fudaal Thor of bla rou ur ha F fr L d b a coo doaor for all h raor of F LCM of all doaor df df hr a o-o a bw f ad hooorh bw F ad hu hr a rory of a fr rou alo obr ha h hooorh u b o-o Thrfor F oorhc o a ubrou of Fro F w ca coclud ha F L : F b uch a oorh Now w df a lar raforao : Q Q by for wh a raforao ar C wh rc o h adard ba C Q h rrcd o : wh rrcd o F F For o C C C F C C C C Fro F F C C F C C F If Q h C C Q C ff b b fro h abo w ca ay C C C C lo C C a ubrou of L Q C C a ubrou of L h raforao ar for y f ubrou of L Q coua o a ubrou of L Hc h hor rod or Rduco od hooorh L b a r ubr - 7 -

8 Codr h wh addo odulo co of h o-a r o cd I o oly a rou a h rou orao bu alo ha aohr orao whch ullcao odulo ll h o-zro l ha ullca r If do h of all arc of z ad r fro addo ad ullcao orao of h ar ullcao orao ol boh h codr h rou L h rou of all rbl arc fro ; h arc ha dra whch o zro Each ry of a ar wh r r ca b rducd oulo h h orao o ay df a a whch wll b calld h rduco od for a r : For al f = h rrcd o L L L a rou hooorh : Th follow rul o oward dr a ubrou of f d L wh o oral arc of f ordr Prooo [] : L b a r ad uo ha I for o L If a r ad f I h I PROOF : Suo I wh I Th w ca wr I H for o o-zro ar H I dh whr d h cd of h r of H ad h cd of all h r of H I dh H dh ly Boal hor I dh d I I dh d H H dh dh dh H fr cacll coo r ad dd by d I h abo uao w fd ha dd ach r hrfor u dd H dd OR dd H - 8 -

9 ll r of H ha o coo facor Boh ar r dd Now dd ach r by hr or w H dh d H Slarly a abo w ca ay ha u dd H Th a coradco cd of H zro ar I H h Hc rod Now w ca ro a bauful clacal rul du o Hra Mow Mow Thor []: If a f ubrou of L h oorhc o a ubrou of L for all r I arcular h rou L coa uo oorh fly ay f ubrou PROOF : L I b a f ubrou of L For o r l b a a dfd a abo Suo ha Kr I I ad y I whr y N h ordr of ad I y ay hr b coo or r If coo h hr u o r dd y I y I y a all owr of u blo o b a ubrou y I I whr r ad y y I I I I ad y o h ordr of Th a coradco fro h rooo If y wa r h fro h rooo w coclud ha I aa a coradco Thrfor Kr I Thu h a c wh rrcd o h ubrou I ohr word oorhc o a ubrou of L - 9 -

10 B a f rou L ha oly a f ubr of ubrou Thrfor h rou L coa uo oorh fly ay f ubrou daly dduc: Corollary : Uo oorh L Q coa oly fly ay f ubrou POSSIBLE ORDERS OF ELEMENTS OF L : Th abo roof how u ha hr ar oly fly ay obl for ordr of l or ubrou of L ad L Q Th w ry o fd wha h obl ordr ar ad how hy ary wh Du o h f of h obl w ca alo ay ha hr u a aal obl ordr for ubrou of L For al : 6 3 Codr h ar Th ar a 4 4 ar wh ordr Thu w ca ay ha L 4 ha a l ad a ubrou of ordr wll lar rfy ha h aal ordr of ubrou L 4 Cyclooc Polyoal Codr a ar of ordr ; h l ha af h uao Th roo of h olyoal ar h h roo of uy Sc af h olyoal h alu of u alo afy h olyoal Thu h alu of ar h roo of uy Mal olyoal of a l h oc olyoal of h all dr whch wh aluad a h l zro Th al olyoal of h roo of uy ar calld h cyclooc olyoal - -

11 Th roo of h uao h h roo of uy ar by- co : Th abo of roo for a cyclc rou udr ullcao of col ubr Th raor of h rou ar calld h r h roo of uy Th r h roo of uy ar by rduc h of roo of uy by allow oly ho alu of whch ar co-r o d d co Th ubr of r h roo of uy by : whr h Eulr -fuco; h ubr of o r lr ha or ual o whch ar rlaly r o For r orc Th h cyclooc olyoal dfd by whr ra or d h of all r h roo of uy So ror of cyclooc olyoal ar a follow : Fro h dfo abo w ca ha h dr of by whr dfd a abo Fro h facorzao of w rou h facor ohr uch ha all of ordr d ar roud ohr By Lara Thor d u alway dd Sc ha ordr d hrfor a r d h roo of uy d d Thu w ca alo wr d d 3 Th cyclooc olyoal a oc olyoal or r [3] Proof : f a alu for fro h dfo of cyclooc olyoal - -

12 au ha a oc olyoal or r for all d Codr h olyoal d F d d d F ad lad r ha co-ffc c d ar oc olyoal or r By do alorh h r [ ] uch ha h oc ad F h r whr r or d r d F By rou hor w ha F Thrfor by uu of uo ad radr or C h Thu ad a oc olyoal d By rcl duco w ca ay ha ay cyclooc olyoal a oc olyoal or r 4 Cyclooc olyoal ar rrducbl orq [3] Thrfor h rrducbl facorzao of d d Coao arc of cyclooc olyoal h h hl of cyclooc olyoal w ca cra arc of cfd ordr L Codr a o r ad h cyclooc olyoal ca coruc a ar of ordr h rou of arc L b h coao ar for h cyclooc olyoal Rcall ha f a a a a olyoal h coao ar by a a C a a - -

13 I ca b chcd ha h characrc olyoal of coao ar C; o C = Fro h ror of cyclooc olyoal w ca ay ha h coffc of Thu a r ar a of h for abo lo c ad I ar r a rrducbl facor of L u au ha ha a al olyoal [ ] By h dfo of al olyoal a oc olyoal of h all dr for whch Codr o olyoal h r uch ha By do alorh of olyoal uch ha h r ad r or d r d Ealua a w r If r wll coradc h fac ha h al olyoal of Thrfor r a facor of ay uch olyoal whch wh alua a h zro ar Thu a facor of Sc a rrducbl olyoal Thrfor h la dr for whch af h olyoal h ordr of Now w ha a ar of ordr wh r r Now codr h ar h rou al ry ad coruc ha ar by u a cyclooc olyoal Eal : 4 If w facor 4 w fd ha Th coao ar of a follow- C By rfor a fw row orao o h 4 4 dy ar w oba h follow ar - 3 -

14 wh r ad 3 4 B B B ad B ar l of L Codr h ar BCB Coua of C h rou L ha I C I BIB B BC B BC BCB B CB B B C B BC BCB Thu h ordr of h a a C O alua BCB fd ha h a ar a h rou al Rar o coffc of cyclooc olyoal: Thouh for h fr 4 r for h coffc of blo oly o for 5 5 ha wo coffc a - I fac JSuzu rod h aaz fac ha ry r occur a a coffc o cyclooc olyoal Fally w fd a way o dr f for a r hr a ar of ordr ad z for a fd ad hu calcula h au obl ordr of a ar L for a fd Th follow hor h fal rul h drco [] MIN THEOREM : L wh dc r Th Q L - ad hc L - ha a l of ordr f ad oly f

15 for ohrw For h roof of h abo hor w d h follow la df a a : uch ha for wh for ohrw [] La: If N ad f d d h Proof : d a of dc dor of uch ha lcd d d d r Sc lcd d d r c c Sc c dd Codr h uo ara ou h coo r w ca oba a of c uch ha for ach c a dor of d c c d h c d S for ach c c d dfd a S : c ad cd c c Sc all h c ar arw rlaly r o ach ohr hrfor S S h y Sc c c c hrfor S Thu S S for a aro o h of - 5 -

16 - 6 - aal r owr facor of By aro h w ur ha ay w ca alway fd a uu uch ha dd c hou lo of raly w ca au ha dd c a w ca alway rarra h r o a o Sc b a ab for all b a S c for all Suo h lar o abo S c Thu c Suo h w ca ay S S c c Thu c Fro h abo rul ad w d Hc h la rod Now u h abo la w ca ro h a hor - PROOF OF THEOREM: Rcall ha ohrw for To ro h abo a w u how ha Q L ha a l of ordr f ad oly f Ca I:

17 - 7 - Suo ha a o r uch ha ad ad For ach w ca coruc a ar of ordr ad z do h h a ar a do h al abo Th w df a ar B uch ha B Th z of B ad h ordr of B lc Suo h B h drd ar If h I B h drd ar whr Ca-II: Suo ha ad Subu / / Thu fro h rou ca w ca ha / hr u o ar of ordr / blo o Q L Sc / odd h ha ordr Thrfor for ay o r h h hr u a ar of ordr h rou Q L ad hc L Corly uo ha Q L ha ordr for o L b h al olyoal of h ar L h rrducbl facorzao of b f f f Fro h rou rul w ow ha h facor of ay olyoal ha wh aluad a h zro ar Sc ordr of h I ad af h olyoal Thu a facor of Sc h rrducbl

18 facorzao of dc facor hu f f f By d d coar h wo facorzao for ach w for o d whch d u b a dor of Sc ha ordr lc d d d By rary dcooo lar or Q o a ar of h for hr h al olyoal of ach z of ach for ach d For ach l l b h Th l d d for ach Thrfor d l Fro h abo la d Hc w ha rod h a hor Hr a au corollary Corollary : L Q ha a l of ordr f ad oly f L Q do PROOF: Codr o o r uch ha L Q ha a l of ordr Bu h alo h rul ha Thrfor L Q ha a l of ordr h L Q alo ha a l of ordr Corly uo L Q ha a l of ordr whr ad a odd Sc ach odd h for ach Thu ach r hc wll r b ual o - 8 -

19 Thrfor ad L Q ha a l of ordr f ad oly f L Q do Hc h corollary rod Tabl wh ordr of l U h hor dcrbd abo ad follow corollar w ca ow fd h obl ordr of ubrou for a L Q or L ad hu w ca alo fd h aal obl ordr of ubrou h rou I a lar ar w ca alo ry ad loo a f ubrou of h rou L R Bu uch a ca h obl for a f ordr d o b f coclud h ror by obr h aal f ordr of l L for h fr fw r of [] aal ordr aal ordr

20 BIBLIORPHY: Ja Kuzaoch ad dry Palcho F rou of Marc ho Er ar Ir Th rca Mahacal Mohly -9 DS Du & RM Foo brac lbra 3 rd d Joh ly ad o 4 3 R Thaadura O coffc of cyclooc olyoal : Cyclooc fld ad rlad oc Pu 999 Bhaaracharya Pu 3 4 M r lbra Prc Hall