Incidence Matrices of Directed Graphs of Groups and their up-down Pregroups

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1 SQU Jounal fo Sn ( Sultan Qaboos Unvst DOI: Indn ats of Dtd aphs of oups th up-down Poups Wadhah S Jassm Dpatmnt of athmats Fault of Sn Soan Unvst Soan Ebl Iaq Emal: wadhahasm@mathssoandud ABSTRACT: Th am of ths wo s to v a dfnton of th ndn mats of th dtd aph of oups onstut an up-down poup of th ndn mats of th dtd aph of oups thn v an alothm fo th up-down poup of th dtd aph of oups Kwods: Indn mat of -labld aph; up-down poup; dtd aph of oups ndn mat of a dtd aph of oups مصفوفات الوقوع لبيانات الزمر الموجه و أعلى - أسفل ما قبل زمرها وضاح س جاسم الملخص: هدف بحثنا هذا هو اعطاء تعريف لمصفوفات الوقوع لبيانات الزمرالموجه وأعلى أسفل ما قبل زمرها بناء أعلى أسفل ما قبل زمره لمصفوفات الوقوع لبيانات الزمر الموجه ومن ثم اعطاء خوارزمية لبناء أعلى أسفل ما قبل زمره لبيانات الزمر الموجه الكلمات المفتاحية: الزمر الموجه مصفوفة الوقوع للبيان المحمول بعناصر ا لمجموعه أعلى - أسفل ما قبل زمره بيانات الزمر الموجه ومصفوفة الوقوع لبيان Intoduton I n [] w av th dfnton of th ndn ats of - Labld aphs In [2] [3] w av th dfnton of th dtd aph of oups onstutd aph of oups fo poups dtl fom th odd t of poups fom that dtd aph of oups w onstutd th up-down poups thn w showd thos two poups a somoph In [4] Rmln av an ampl of a poup P of fnt hht; h sad but Jm Sha I spnt a v lon vnn wth th omput vfd th poup aoms I ba ths pont n mnd In [2] [3] w hav a dt mthod to obtan ampls of poups n th fom of up-down poups fom an dtd aph of oups but somtms thos aphs of oups a la thn wll ta a lon tm to fnd thos up-down poups In [] w dfnd th ndn mats of -labld aphs Th man am of ths wo s to psnt th dtd aph of fnt oups n tms of th ndn mats of -labld aphs so that b addn tan ondtons to allow th ndn mats of th -labld aph to b mo onfdnt wth th dfnton of th dtd aph of oups; w an thn wt a omput poam to od all lmnts of th up-down poup of that dtd aph of oups as an applaton of th ndn mats of -labld aph Thfo ths pap s dvdd nto s stons In ston 2 w v th bas onpts of aphs poups ndn mats of -labld aphs In ston 3 w v th dfnton of ndn mats of dtd aphs of oups In ston 4 w onstut th up-down poup of th ndn mats of th dtd aph of oups In ston 5 w dfn an alothm on th ndn mats of th dtd aph of oups so w an thn wt a omput poam fo ths alothm 2 Bas onpts 2 Poups Th da of poups os ba to Ba [5] th dfnton of poup was vn ndpndntl b Stallns [6] n 97 Th tho of poups has bn dvlopd b [4] Stallns [6] Hoa [7] Hoa Jassm [3] oths W now tun to th onal dfnton of poups [6] 40

2 INCIDENCE ATRICES OF DIRECTED RAPHS Lt P b a st wth an lmnt P a mappn of a subst D of P P nto P dnotd b ( W shall sa that s dfnd nstad of ( D Suppos that th s an nvoluton on P dnotd b suh that th follown aoms hold: P: = fo all P P2: fo all P P3: If s dfnd thn s dfnd ( P4: f z a dfnd thn (z s dfnd f onl f (z s dfnd n whh as th two a qual w wll sa z s dfnd P5: Fo an w z n P f w z a dfnd thn th w o z s dfnd Hoa [7] showd that w ould pov aom P3 abov b usn th follown poposton aoms P P2 P4 Dfnton 22 [7]: Fo an P put L( {a P: a s dfnd} W wt f L( L( f L( L( L( L( ~ f L( L( It s la that ~ s an quvaln laton ompatbl wth Th follown sults a tan fom Stallns [6] Rmln [4] (S [7] fo shot poofs Poposton 23 ( If o thn a dfnd ( If a a a dfnd thn (a(a s dfnd f onl f s dfnd n whh as th a qual B usn aom P5 abov (whh wll b dnotd b P5( Rmln [4] povd ondtons P5( P5( of Lmma 24 blow Lmma 24 [7] Th follown ondtons on lmnts of P a quvalnt: P5( If w z a dfnd thn th w o z s dfnd P5( If a a a dfnd but s not thn a < a < P5( If s dfnd thn o Thfo w wll sa P s a poup f t satsfs aoms P P2 P4 th ondtons of Lmma 24 abov Th unvsal oup of a poup P [3] s dnotd b U (P has th follown psntaton P; whnv s dfnd fo P Now f P s a poup thn (P s t - l patal odn; that s P/~ has a mnmum lmnt fo an z n P z z w hav o oov Rmln n [4] dfnd that fo an lmnt n P w sa that has fnt hht n 0 f th sts a mamal totall odd subst { 0 n } of P suh that 0 n H also showd that th lmnts of P fom an od t (dnotd b O whos vts [] a th quvaln lasss of th lmnts of P und ~ whos ds a fomd b onn ah vt [] of hht n > 0 to th unqu vt [] of hht n satsfn [ ] [ ] all ds of O a dtd awa th bas vt [ 0 ] of hht 0 In [8] Stallns onstutd an { a b} up down poup fo a f oup F natd b of nfnt hht h showd that U(P th unvsal oup of a poup P s somoph to F In [23] w av th dfnton of a dtd aph of oups whh onssts of a dtd aph Y wth a bas vt v a spannn t T whos ds a dtd awa fom th bas vt v toth wth a oup v fo ah vt v fo ah dtd d Y a suboup of ( ( whh s mbddd n b whh s dfnd b ( a a a wh s th ( v Y T v labld of th d It s dnotd b W also onstutd a dtd aph of oups of P dtl fom th od t O of P thn showd that th fundamntal oup ( Y T v v of a aph of oups s somoph to U(P W onstutd an up down poup Q dtl fom th ( v Y T v dtd aph of oups of a poup P w showd that U(Q s somoph to ( v Y T v U ( Q U ( P thn that 4

3 WADHAH S JASSI 25 Indn ats of Labld aphs In [] w av th dfnton of th ndn mats of Labld aphs (wh an -labld aph s a dtd aph wth ah d labld b an lmnt of th subst of th oup F natn th oup F som dfntons sults latd to t Rall that fom aph tho th dtd aphs a wthout loops baus w annot dfn th ndn mats of dtd aphs Th ndn mats of dtd aphs a wth n vts m ds ( t s n m [ ] mats n m wh suh that: 0 f f f v s v ( notndn v ( wth Sn all ds n Labld aphs a labld th ndn mats of th dtd aphs do not dal wth th labln of ds w wll put mo ondtons on th ndn mats of dtd aphs as blow to obtan th dfnton of th ndn mats of th - Labld aphs { a b} Dfnton 26: Lt b an Labld aph wthout loops (wh thn th ndn mat of th Labld aph s an n m f v [ ] ndn mat n m ( labls wh wth nts suh that 0 f v f v s ( not ndnt wth labls ( th Cal ost aph NB Indn mats of Labld aphs wll b dnotd b ( F of th oup F natd b F (H of th Cal ost aph labld aphs (H of th suboup H of F th podut of o aphs Fo ampl: th Cal aph of th suboup H of F th o aph ~ ( H ( K a - { a b} Now f th Labld aph has loops wth labln a o b thn hoos a md pont on all ds labld a o b to ma all of thm two ds labld aa o bb sptvl Thfo n th st of ths wo w wll assum that all Labld aphs a wthout loops ( Dfnton 27: Lt b an ndn mat of Labld aph If wth non zo nts n suh that thn mat of Labld aph ( ( dosn't ontan an ow s alld a foldd ndn Now w v th bas dfntons som sults on th ndn mat of Labld aph vn n [] ( Lt b an n m [ ] ndn mat of Labld aphs lt b a ow a ( olumn n sptvl If s a non zo nt n th ow thn s alld an ndn ow wth th olumn at th non zo nt f th non zo nt thn th ow s s( alld th statn ow (dnotd b of th olumn th ow s alld th ndn ow ( dnotd b ( of th olumn f If th ows a ndnt wth olumn at th non zo nts sptvl thn w sa that th ows a adant If h a two dstnt olumns n ( suh that th ow s ndn wth th olumns h at th non zo nts h h sptvl (wh thn w sa that h a adant olumns Fo ah olumn th s an nvs olumn dnotd b s( ( ( s( suh that Th d of a ow of ( as 42

4 ( INCIDENCE ATRICES OF DIRECTED RAPHS s th numb of th olumns ndnt to d( s dnotd b If th ow s ndnt wth at last s alld a banh ow If th ow th dstnt olumns h at th non zo nts thn th ow s ndnt wth onl on olumn at th non- zo nt all oth nts of a zo thn th ow s alld an solatd ow A sal n s a fnt squn of fom 2 S 2 2 s ( wh ( s( 2 S Th statn ow of a sal 2 2 s th statn ow of th olumn th ndn ow of th sal S s th ndn ow of th olumn w sa that S s a sal fom to S s 2 s( S ( S a sal of lnth fo If thn th sal s alld a losd sal If th sal S s dud losd thn S s alld a ut o a l If has no l thn s alld a fost ndn mat of Labld aph Γ Two ows n ( a alld onntd f th s a sal S ( n ontann oov ( s alld onntd f an two ows n ( a onntd b a sal S If ( Labld aph Γ Lt b a subaph of Γ thn of ows olumns of hav th sam mann n pop subndn mat of ( If ( b at last on sal S n ( ( ( s a onntd fost thn a substs of as th do n ( ( ( ( ( ( f s a olumn n ( If ( ( A omponnt of s a subndn mat of ( ( thn s alld spannn t of f an ndn mat of - Labld aph Γ ( ( ( s alld a t ndn mat of ( s alld a subndn mat of f th st s( ( thn ( ( thn ( s alld a s a mamal onntd subndn mat of v two ows n ( a ond s alld spannn ndn mat of ( s a spannn t ndn mat Th nvs of ( ( s Now b dt alulatons th dfntons abov w an pov th follown sults Lmma 28: If olumns ( s a t ndn mat of Labld aph Γ wth n ows thn 3 Indn mats of dtd aphs of fnt oups ( has n Dfnton 3: An ndn mat of a dtd aph of fnt oups onssts of an ndn mat of - ( labld aph (T wth a spannn t mat of - labld aph a bas ow toth wth a fnt oup fo ah ow a fnt oup fo ah olumn suh that: ( Th olumns of a dtd awa fom ; 2 Eah olumn oup s a suboup of ( ; 3 Eah olumn oup s mbddd n t ( b a fd monomophsm dfnd b ( a a a s( = s th non- zo ntan of of ( / ( T It s dnotd b ( ( Y ( T NB: An ndn mat of a aph of oups ma b mad nto an ndn mat of a dtd aph of oups (T that b hoosn a bas ow an ontaton on olumns thn dntfn wth th ma of ( und th ( lvant monomophsm 43

5 Fo ah dtd olumn s( olumn wth statn ow a th nts of th olumn NB W wll dnot that ( ( t n b (wh (T s qual to o - f WADHAH S JASSI ( ( s( suh that f lt ( b lt t s( wh (T s n ( t a nonzo nts of ( t t ( b th nvs t suh Eampl: In ths ampl w wll v a dtd aph of oups thn onstut th ndn mat of ths dtd aph of oups ( ( Y ( T Lt th dtd aph of oups ( v Y T v b as follows: { a a} V 7 V4{ a b a b} { a} 8 { a} V 6 { a} 7 { } 4 V { b } 3 b V 5 3 { b} { } 6 5 {} 2 { b} V V {} V { } {} 2 b {} { b} { b} {} {} {} { a} { a} Fu Th dtd aph of oups onstutd n [3] p72 of th poup vn n [4] p ( ( Y ( T {} Th ndn mat of th abov dtd aph of oups { b} { b b} {} { a b a b} { a} { a a} s as blow: Fu 2 Th ndn mat of th dtd aph of oups vn n Fu abov 44

6 INCIDENCE ATRICES OF DIRECTED RAPHS 4 Th up-down poup of an ndn mat of a dtd aph of fnt oups In ths ston w onstut th up- down poup of th ndn mat of a dtd aph of oups as blow; Lt ( Th fundamntal oup of ; a b th ndn mat of a dtd aph of oups ( ( ( a a (Y has th follown psntaton: ( T ( ( Y ( T lt ( Y/ ( T Now fo ah dtd olumn lt b also dnotd b dnot th nvs olumn wth s( ( ( s( ; also lt b of fom w n n n wh 2 n 2 n s a ut at wth ows 2 n sa wh ah s n A wod of ths fom an subwod of t s dud f t ontans no subwod a ( o a wh a (a If t dos ontan suh a subwod w an usn th latons substtut o a sptvl to obtan a shot wod of th vn fom psntn th sam lmnt Thus ah lmnt of th fundamntal oup s psntd b a dud wod w of ths fom Its nvs s psntabl b th wod w dfnd n th usual wa oov b [3] th dud wod psntn an lmnt s unqu modulo a susson of ntlavn a a h substtutn ( h fo a o v vsa fo an Lt ( b th ndn ( ( ( mat of th dtd aph of oups Y T q lt 2 n b an upwad ( 2 sal n whh s a fnt squn of olumns dtd awa of th bas ow Lt th ows of th sal q b w n A wod of tp q s a wod 2 2 n n n wh n v wod w must b dud s( s th non- zo nt of th statn ow of th q olumn Now Lt 2 q 2 h b upwad sals n ( both statn at w Lt w h h q b wods of tp q sptvl wh h Th wod w s alld an ntal subwod of th wod w wttn w w f hn fo f s an lmnt of fo ah Lmma 4: Th laton " s an ntal subwod of " s both tanstv t ndn mat l that s w w thn w w f w w w w thn th w w o w w Poof: Th sult follows dtl fom th dfnton Now lt q 2 ndn at th sam ow lt tp q q sptvl suh that th lmnts q 2 w h w w h b upwad sals n ( both of thm statn at w h b wods of a n thn suh a wod ww s alld an up down wod b Fo ampl th wod s an up-down wod fom Fu 2 dvd fom th upwad sals q wth ows 5 q wth ows 2 4 & 5 Q( ( Lt b th st of all up-down wods of th ndn mat of a dtd aph of oups ( Rdun an up-down wod n ( vs anoth up-down Thfo w assum that suh a wod s dud W us w w to dnot an up-down wod Lmma 42: Lt w w zz ( ww ( zz Q( ( b dud up-down wods thn s n f onl f w s an ntal smnt of z o z s an ntal smnt of w Poof: Sn th wods w w zz ( ww ( zz btwn th last of w th fst on of z a both dud so duton an onl ta pla n th wod 45

7 oov ( ww ( zz ( ww WADHAH S JASSI dus to an up-down wod f onl f th all ( zz n w o all n z a lmnatd whn puttn n dud fom Ths happns f onl f w s an ntal smnt of z o z s an ntal smnt of w sptvl Now w show that Q( ( L( ww Dfn Q( ( satsfs ondtons { uu ; uu P 2 Q( ( s a subst of P It mans to b shown that Q( ( Q( ( } ww Q( ( s a poup Sn P 4 ww fo Lmma 43: Lt w w zz b dud up-down wods thn w z Poof: Suppos that w z To show w w zz w must show that ( uu ( zz Q( ( s n uu Q( ( fo som of u o u s an ntal smnt of z ( z u u z o Lmma 4 w hav w u u w o thn b Lmma 42 aan Q( ( ( zz L L( ww Thfo s a subst of Hn w zz Q( ( Thom 44: s a poup Q( ( Q( ( ( ( so P satsfs ondton 5 as bfo mpls that w w zz L( zz s a subst of L ww ( thn b Lmma 42 z s an ntal smnt sptvl Sn w w z ( uu ( ww If n th as b s dfnd n P 5 ( of Lmma 24 ( ww ( zz Poof: To show s a poup w wll show that satsfs ondton Thfo lt w w zz Q( ( b dud up-down wods n suppos that Q( ( dfnd n Hn b Lmma 42 w hav z w o w z Thus b Lmma 43 w w o zz ww ( Thfo ondton P 5 Q( ( of Lmma 24 holds n Dfnton 45: Th st of all up down wods of th ndn mat of a dtd aph of oups alld th up- down poup of ( ( Y ( T s zz Q( ( th Indn mat of a dtd aph of oups wh Q s th up- down poup of th dtd aph of oups as shown n [2] [] Thom 46: Th Unvsal oup of ( fundamntal oup ( Poof: Sn v lmnt n Q( ( Q( ( (T spans (Y s dtd awa fom Q( ( ( ( podut of lmnts of s an lmnt n (t s dnotd b ( v lmnt of oov th patal multplaton n U( Q( ( ( ( ( Q( ( s s somoph to th sn th t ndn mat an b wttn as a mpls th latons of 5 An alothm fo th up-down Poup of ndn mats of dtd aphs of oups Lt ( ( ( Y ( T b th ndn mat of a dtd aph of oups thn w us th psntaton of th dtd aph of oups of an up-down poup to wt down all th lmnts of th up-down poup of that aph of oups b appln th follown alothm Th stps a vn blow: w I Fnd all up wods 2 n n n of tp upwad sals q ( n n n wh n s th non- zo ntan of th ow whh s th statn of th olumn as dfnd abov thn pod stp II; w w II If two up wods 2 n n n 2 m m m ndn at th sam ow ( ow ontans non-zo ntans of foms thn mas on of thm an up wod w sa 2 m m w m 2 n n n mas th oth up wod down wod b hann th dton of all olumns ts ntan w to b n n n b dntfn thm w t an up-down wod w w m m m n n n n n n (wh Thn pod to stp III; 46

8 INCIDENCE ATRICES OF DIRECTED RAPHS w 2 If th up wods 2 n n n nd at an solatd ow thn han th dton of all olumns w ts labl to b n n n b dntfn thm wth th ow that both of thm nd wth thn w t an up-down wod ww 2 n n n n n n = 2 n n n n n n n n n ( wh Thn pod to stp III; III If th s no oth up-down wod thn stop w Poposton 5: All up wods 2 n n n of tp upwad sals n ( ( ( Y ( T w a sam as all up wods 2 n n n of tp ( upwad paths n Y T v ( Poof: Sn all vts v ds n Y T v a psntd b ows olumns n ( assoatd vt oups v d oups a psntd b ow oups olumns oups sptvl wth ntans of th labld ( of th ds of Y T v suh that f T f Y T Thfo th dton th labln of olumns of ( a sam as n ( w Hn all up wods 2 n n n of tp upwad sals n ( a w sam as all up wods 2 n n ( n of tp upwad paths n Y T v Poposton 52: Th alothm must stop Poof: Sn th sz of ( s n m all vt oups d oups a fnt so ( s fnt ndn mat B stp I w t all dud up- wods b stp II w t all up- down dud wods thn b stp III w wll t all up- down dud wods Sn th on - labld aph dos not ontan loops so th st of all dud up-down wods s fnt thn th alothm must b stop aft a fnt tm 6 Conluson W hav vn a nw applaton fo th ndn mats of -labld aphs Ths applaton s th ndn mats of dtd aph of fnt oups Thfo w hav addd tan ondtons to allow th ndn of - labld aphs to b mo onfdnt wth th dfnton of th dtd aph of fnt oups B ths wa w an wt a omput poam to od all lmnts of th up- down poups of that th dtd aphs of fnt oups Rfns Jassm WS Indn ats of -Labld aphs an applaton Sultan Qaboos Unvst Jounal fo Sn Hoa AH Jassm WS Dtd aphs of oups th up-down poups Fault of Sn Bulltn Sana'a Unvst Jassm WS Poups aphs of oups PhD Thss Bmnham Unvst Rmln F Poups Bass S tho Aman athmatal Sot mos Ba R F Sums of oups th nalzatons III Aman Jounal of athmats Stallns JP oup tho Th dm anfolds Yal onoaphs 97 7 Hoa AH Poups Lnth funtons Podns of th Cambd Phlosophal Sot Stallns JP Adan oups Poups Essas n oup tho SRI Publatons 8d B S stn Abdu KA Rpsntn Co aphs Nolas's Alothm S Thss Bahdad Unvst Chswll I Abstat Lnth funtons n oups Podns of th Cambd Phlosophal Sot Jassm WS Dtd Co aphs th up-down poups Al-ustansa Jounal of Sn (4: Lndon RC Lnth funtons n oups athmata Snava S JP 968 oups dsts Coll d Fan Tanslaton Ts Spn Vla 980 Rvd 0 Sptmb 205 Aptd 7 Dmb