Lecture 2: Cayley Graphs

Transcription

1 Mth 137B Professor: Pri Brtlett Leture 2: Cyley Grphs Week 3 UCSB 2014 (Relevnt soure mteril: Setion VIII.1 of Bollos s Moern Grph Theory; 3.7 of Gosil n Royle s Algeri Grph Theory; vrious ppers I ve re n nnot rememer.) This week s leture, s well s next week s leture, re ll out the interply of grph theory n lger. Speifilly, we re going to evelop Cyley grphs n Shreier igrms to stuy vrious kins of groups, n to prove some very eep n surprising theorems from strt lger! Beuse this ourse oes not ount group theory s prerequisite, we shoul first, um, efine wht groups re: 1 Group Theory: Exmples If you ve me it to this lss, you ve worke with groups efore! However, you my hve never seen them tully efine. Let s fix tht: Definition. A group is set G long with some opertion tht tkes in two elements n outputs nother element of our group, suh tht we stisfy the following properties: Ientity: there is some ientity element e G suh tht for ny other g G, we hve e g = g. In other wors, omining ny group element g with the ientity vi our group opertion oes not hnge g! You know mny ojets like this: if we work with the rel numers R n think of ition s our group opertion, then 0 is our ientity, s 0 + x = x for ny x. Similrly, if we onsier the rel numers gin ut tke our opertion to e multiplition, then 1 is our ientity, s 1 x = x for ny x. Inverses: for ny g G, there is some g 1 suh tht g g 1 = e. In other wors, if we strt t ny group element g, we n lwys fin something to omine with g using our group opertion to get k to the ientity! Agin, you know severl ojets like this: with R n ition, the inverse of ny numer x is just its negtive x, while if we onsier the set of nonzero rel numers n multiplition, the inverse for ny x is just 1/x. Assoitivity: for ny three,, G, ( ) = ( ). In other wors, the orer in whih we group omintions together oesn t mtter, s long s the sequene tht we hve those ojets groupe together in oes not hnge! I.e. we n omine with, or first fin n then omine tht with. Agin, most of the nturl opertions you re fmilir with (ition, multiplition) re ssoitive: it is perhps more interesting to point out some things tht re 1

2 nonssoitive. For exmple, exponentition is nonssoitive opertion: 2 (34 ) = , while (2 3 ) 4 = 8 4 = It ers noting tht this oes not sy tht = : tht is ifferent property, lle ommuttivity, n is not property tht groups nee to hve (s we will show in the exmples!) Groups tht re ommuttive re lle elin groups, fter the mthemtiin Niels Henrik Ael. We list numer of exmples of groups, s well s some nonexmples: Exmple. As note ove, the rel numers with respet to ition, whih we enote s R, +, is group: it hs the ientity 0, ny element x hs n inverse x, n it stisfies ssoitivity. Nonexmple. The rel numers with respet to multiplition, whih we enote s R,, is not group: the element 0 R hs no inverse, s there is nothing we n multiply 0 y to get to 1! Exmple. The nonzero rel numers with respet to multiplition, whih we enote s R,, is group! The ientity in this group is 1, every element x hs n inverse 1/x suh tht x (1/x) = 1, n this group stisfies ssoitivity. Exmple. The integers with respet to ition, Z, + form group! Nonexmple. The integers with respet to multiplition, Z, o not form group: for exmple, there is no integer we n multiply 2 y to get to 1. Nonexmple. The nturl numers N re not group with respet to either ition or multiplition. For exmple: in ition, there is no element 1 N tht we n to 1 to get to 0, n in multiplition there is no nturl numer we n multiply 2 y to get to 1. Exmple. GL n (R), the olletion of ll n n invertile rel-vlue mtries, is group uner the opertion of mtrix multiplition. Notie tht this group is n exmple [ of] 0 1 nonelin group, s there re mny mtries for whih AB BA: onsier 0 0 [ ] [ ] [ ] [ ] [ ] = versus = Exmple. SL n (R), the olletion of ll n n invertile rel-vlue mtries with eterminnt 1, is lso group uner the opertion of mtrix multiplition; this is euse the property of eing eterminnt 1 is preserve uner tking inverses n multiplition for mtries. Exmple. Z/nZ = {0, 1,... n 1} is group with respet to the opertion of ition mo n: the ientity in this group is 0, every element x hs n inverse n x (euse x + (n x) = n 0 mo n,) n ition is ssoitive. 2

3 Exmple. Consier regulr n-gon. There re numer of geometri trnsformtions, or similrities, tht we n pply tht sen this n-gon to itself: i.e. there re severl rottions n refletions tht when pplie to n-gon o not hnge the n-gon. For exmple, given squre, we n rotte the plne y 0, 90, 180, or 270, or flip over one of the horizontl, vertil, top-left/ottom-right, or the top-right/ottom-left xes: (rotteryr0 ) (rotteryr90 ) (rotteryr180 ) (rotteryr270 ) (fliproverrhorizontl) (fliproverrvertil) (fliproverrul/drrigonl) (fliproverrur/dlrigonl) Given two suh trnsformtions f, g, we n ompose them to get new trnsformtion f g. Notie tht euse these two trnsformtions eh iniviully sen the n-gon to itself, their omposition lso sens the n-gon to itself! Therefore omposition is wellefine opertion tht we n use to omine two trnsformtions. (rotte y 270 ) (flip over vertil) = (flip over UR/DL igonl) 3

4 Notie tht the trivil rottion y 0, when ompose with ny other mp, oes not hnge tht mp: so, uner the opertion of omposition, rottion y 0 is n ientity! Similrly, notie tht performing the sme flip twie in row returns us k to the ientity, so every flip hs n inverse given y itself! (I.e. if f is flip, f f = i: i.e. f = f 1.) As well, if we rotte y k egrees, rotting y 360 k egrees results in totl rottion y 360: i.e. rottion y 0. So ll rottions hve inverses s well! Finlly, notie tht euse funtion omposition is ssoitive, this opertion is ssoitive s well: onsequently, the olletion of ll symmetries of regulr n-gon forms group uner the opertion of funtion omposition! Exmple. (Z/pZ) = {1,... p 1} is group with respet to the opertion of multiplition mo p whenever p is prime; hek this if you on t see why! Exmple. The symmetri group S n is the olletion of ll of the permuttions on the set {1,... n}, where our group opertion is omposition. In se you hven t seen this efore: A permuttion of set is just ijetive funtion on tht set. For exmple, one ijetion on the set {1, 2, 3} oul e the mp f tht sens 1 to 2, 2 to 1, n 3 to 3. One wy tht people often enote funtions n ijetions is vi rrow nottion: i.e. to esrie the mp f tht we gve ove, we oul write f : This, however, is not the most spe-frienly wy to write out permuttion. A muh more onense wy to write own permuttion is using something lle yle nottion. In prtiulr: suppose tht we wnt to enote the permuttion tht sens 1 2, 2 3,... n 1 n, n 1, n oes not hnge ny of the other elements (i.e. keeps them ll the sme.) In this se, we woul enote this permuttion using yle nottion s the permuttion ( n ). To illustrte this nottion, we esrie ll of the six possile permuttions on {1, 2, 3} using oth the rrow n the yle nottions: 1 i : (12) : (13) : 2 3 (23) : (123) : (132) : 4

5 Beuse the omposition of ny two ijetions is still ijetion, we hve in prtiulr tht the omposition of ny two permuttions is nother permuttion: so our group opertion oes inee omine group elements into new group elements. Composing ny mp f with the ientity mp i(x) = x oes not hnge the mp f, so i(x) is n ientity element; moreover, ny ijetion f hs n inverse funtion f 1 suh tht f f 1 is the ientity mp. Therefore, this forms group! Exmple. The free group on n genertors 1,... n, enote is the following group: 1,... n, The elements of the group re ll of the strings of the form k 1 i 1 k 2 i 2... k l i l, where the inies i 1,... i l re ll vli inies for the 1,... n n the k 1,... k l re ll integers. We lso throw in n ientity element e, whih orrespons to the empty string tht ontins no elements. Given two strings s 1, s 2, we n ontente these two strings into the wor s 1 s 2 y simply writing the string tht onsists of the string s 1 followe y the string s 2. k opies Whenever we hve k {}}{ in string, we think of this s eing..., i.e. k opies of. If we hve multiple onseutive strings of s, we n omine them together into one suh k : for exmple, the wor 3 2 is the sme thing s the wor 6. Finlly, if we ever hve n 1 or n 1 ourring next to eh other in string, we n simply reple this piring with the empty string e. For exmple, the free group on two genertors, ontins strings like , 12, 1 2 4,... As esrie erlier, we ontente strings y simply pling one fter the other: i.e = As esrie ove, we typilly simplify this right-hn string y neling out terms n their inverses, n grouping together ommon powers of our genertors: = = This is group! In prtiulr, ontention is ssoitive, the empty string e is lerly n ientity, n we n invert ny wor k 1 i 1 k 2 i 2... k l i l y simply reversing it n swithing the signs on the k i s: i.e. k 1 i 1 k 2 i 2... k l i l k l i l 5... k 2 i 2 k i 1 = e 1

6 2 Group Theory: Aitionl Definitions When we work with groups, few onepts re prtiulrly useful to think out: Definition. Given group G with n opertion sugroup of G is suset S of G suh tht S is group in its own right using the opertion from G. Note tht this mens tht omining ny two elements in S must remin in S, the inverse to ny element in S must lie in S, n the ientity element of G must e in S. Definition. Given group G, we sy tht it is generte y some olletion of elements 1,... n G if we n rete ny element in G vi some omintion of the elements 1,... n n their inverses. Note tht some groups hve multiple ifferent sets of genertors: i.e. Z, + is generte oth y the single element 1 n lso y the pir of elements {2, 3} Definition. In our ove isussion, we hve primrily efine groups y giving set n n opertion on tht set. There re other wys of efining group, though! A group presenttion is olletion of n genertors 1,... n n m wors R 1,... R m from the free group 1,... n, whih we write s 1,... n R 1,... R m. We ssoite this presenttion with the group efine s follows: Strt off with the free group 1,... n. Now, elre tht within this free group, the wors R 1,... R m re ll equl to the empty string: i.e. if we hve ny wors tht ontin some R i s sustring, we n simply elete this R i from the wor. You hve tully seen some groups efine vi presenttion efore: Exmple. Consier the group with presenttion n. This is the olletion of ll wors written with one symol, where we regr n = e: i.e. it s just e,, 2, 3,... n 1. This is euse given ny string k, we hve k = l for ny k l mo n. This is euse we n simply ontente opies of the strings n, n s mny times s we wnt without hnging string, s n = e! You hve seen this group efore: this is just Z/nZ with respet to ition, if you k times {}}{ reple with 1 n think of s k. 6

7 Often, we will give group with presenttion in the form 1,... n R 1 = R 2, R 3 = R 4,...,... R m 1 = R m, euse it is esier sometimes to think of sying tht ertin kins of wors re equl rther thn other kins of wors re the ientity; this is equivlent to the group presenttion 1,... n R 1 (R 2 ) 1, R 3 (R 4 ) 1,...,... R m 1 (R m ) 1. With these efinitions set own, we n tully strt to o some grph theory: 3 Cyley Grphs n Groups Definition. Tke ny group A long with generting set S. We efine the Cyley grph G A,S ssoite to A s the following irete grph: Verties: the verties of G A re preisely the elements of A. Eges: for two verties x, y, rete the oriente ege (x, y) if n only if there is some genertor s S suh tht x s = y. If this hppens, we eorte the ege (x, y) with this genertor s, so tht we n keep trk of how we hve forme our onnetions. We onsier few exmples here: Exmple. The integers Z with the genertor 1 hve the following very simple Cyley grph: =1 This is not hr to see: we hve one vertex for every element in our group (i.e. every integer,) n n ege (x, y) for eh pir x, y suh tht x = y + 1, y efinition. Beuse this is Cyley grph, we lel eh of these eges with the genertor tht rete tht ege: for this grph, euse there s only one genertor this is pretty simple (we just lel every ege with 1.) Exmple. The integers Z with the generting set {2, 3} hve the following Cyley grph: =2 =

8 Agin, our verties re just the integers. However, this time we hve two genertors: the genertor 2 onnets ny two integers tht iffer y 2, while the genertor 3 onnets ny two integers tht iffer y 3. Notie tht this grph is not the sme s the grph ove: in generl, group n hve mny mrkely ifferent Cyley grphs epening on the genertors tht you pik for it. Exmple. Consier the symmetri group S 3 with genertors (12), (123). First, we lulte how these genertors intert with our group elements when ompose together: group elt. genertor i (12) (13) (23) (123) (132) (12) (12) i (123) (132) (13) (23) (123) (123) (23) (12) (13) (132) (123) We n use this tle to rete the Cyley grph for this group n generting set: (132) =(123) (23) =(12) (12) (13) i (123) Exmple. Consier the group given y the presenttion, 3 = 2 = () 2 = i. Beuse we o not know ll of the elements in this group he of time, it is not neessrily ovious how to rete this group s Cyley grph; unlike in our erlier exmples, we nnot simply write own ll of the verties n then rw eges orresponing to our genertors. Inste, to fin the Cyley grph orresponing to this group, we n use the following proeure: 0. Strt y pling one vertex tht orrespons to the ientity. 1. Tke ny vertex orresponing to group element g tht we urrently hve in our grph. Beuse our grph is Cyley grph, it must hve one ege leving tht vertex for eh genertor in our generting set. A eges n verties to our grph so tht this property hols. 8

9 2. If some wor R i is wor tht is equl to the ientity in our group, then in our grph the pth orresponing to tht wor must e yle: this is euse if this wor is the ientity, then multiplying ny element in our group y tht wor (i.e. tking the wlk on our grph orresponing to tht wor) shoul not hnge tht element (i.e. our wlk shoul not tke us somewhere new, n therefore shoul return to where it strte!) Ientify verties only where solutely neessry to insure tht this property hols t every vertex. (This is the omputtionlly iffiult prt of this lgorithm. In generl, fining the Cyley grph, or even more simply etermining whether two ritrry wors in presente grph re equl, is n uneile prolem: it is provle tht no lgorithm exists tht will lwys solve this prolem. Look up things like the hlting prolem if you wnt more exmples of suh things.) So: if we o this here, we woul strt y rwing the following grph. = = i We ege/vertex pirs to oth of these e verties,, tht orrespon to our genertors. Notie tht the reltion 2 = i tells us tht our -ege leving must return to i, n tht none of our other reltions pply t this urrent stge (s they orrespon to wlks of length t lest 3.) 2 = = i Now, we rw new ege from the verties,, 2. Notie tht the reltion 3 = i tells us tht the -ege leving 2 returns to the ientity, n tht the reltion 2 = i tells us 9

10 tht the ege leving returns to. Furtheromre, the reltion = i, long with the oservtions tht 2 = i = 1, 3 = i 2 = 1 gives us numer of new reltions: = i = 1 = 2, n therefore the -ege leving goes to 2. Furthermore, this lso tells us tht the -ege leving 2 goes to, euse the wlk orresponing to 2 strting from must return to. = i = 1 =, n therefore tht the -ege leving goes to. Furthermore, this lso tells us tht the -ege leving goes to, euse the wlk orresponing to 3 strting t must return to. This gives us the following grph: 2 = = i At this stge, we hve stisfie our seon property (tht there is n ege leving eh vertex for eh genertor,) n we hve only ientifie verties when solutely fore to o so y our reltions. From visul inspetion, it is ler tht we stisfy the three reltions 3 = 2 = = i t every vertex; so this is the Cyley grph orresponing to our group! 10