Chapter 4: Algebra and group presentations

Transcription

1 Chapt 4: Algba and goup psntations Matthw Macauly Dpatmnt of Mathmatical Scincs Clmson Univsity Math 4120, Sping 2014 M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

2 Ovviw Rcall that ou infomal dfinition of a goup was a collction of actions that obyd Ruls 1 4. This is not th odinay dfinition of a goup. In this chapt, w will intoduc th standad (and mo fomal) dfinition of a goup. W will also spnd tim convincing ouslvs that both dfinitions ag. Along th way, w will us multiplication tabls to btt undstand goups. Finally, w will lan about goup psntations, an algbaic dvic to concisly dscib goups by thi gnatos and lations. Fo xampl, th following is a psntation fo a goup that w a familia with: Do you cogniz this goup? G = a, b a 2 = 1, b 2 = 1, ab = ba. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

3 Mo on Cayly diagams Rcall that aows in a Cayly diagam psnt on choic of gnatos of th goup. In paticula, all aows of a fixd colo cospond to th sam gnato. Ou choic of gnatos influncd th sulting Cayly diagam! Whn w hav bn dawing Cayly diagams, w hav bn doing on of two things with th nods: 1. Labling th nods with configuations of a thing w a acting on. 2. Laving th nods unlabld (this is th abstact Cayly diagam ). Th is a 3d thing w can do with th nods, motivatd by th fact that vy path in th Cayly diagam psnts an action of th goup: 3. Labl th nods with actions (this is calld a diagam of actions ). Motivating ida If w distinguish on nod as th unscambld configuation and labl that with th idntity action, thn w can labl ach maining nod with th action that it taks to ach it fom th unscambld stat. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

4 An xampl: Th Klin 4-goup Rcall th ctangl puzzl. If w us hoizontal flip (h) and vtical flip (v) as gnatos, thn h is th Cayly diagam labld by configuations (lft), and unlabld Cayly diagam (ight): Lt s apply th stps to th abstact Cayly diagam fo V 4, using th upp-lft nod as th unscambld configuation : v h vh Not that w could also hav labld th nod in th low ight con as hv, as wll. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

5 How to labl nods with actions Lt s summaiz th pocss that w just did. Nod labling algoithm Th following stps tansfom a Cayly diagam into on that focuss on th goup s actions. (i) Choos a nod as ou initial fnc point; labl it. (This will cospond to ou idntity action. ) (ii) Rlabl ach maining nod in th diagam with a path that lads th fom nod. (If th is mo than on path, pick any on; shot is btt.) (iii) Distinguish aows of th sam typ in som way (colo thm, labl thm, dashd vs. solid, tc.) Ou convntion will b to labl th nods with squncs of gnatos, so that ading th squnc fom lft to ight indicats th appopiat path. Waning! Som authos us th opposit convntion, motivatd by function composition. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

6 A goup calculato On nat thing about Cayly diagams with nods labld by actions is that thy act as a goup calculato. Fo xampl, if w want to know what a paticula squnc is qual to, w can just chas th squnc though th Cayly gaph, stating at. Lt s ty on. In V 4, what is th action hhhvhvvhv qual to? h v vh W s that hhhvhvvhv = h. A mo condnsd way to wit this is hhhvhvvhv = h 3 vhv 2 hv = h. A concis way to dscib V 4 is by th following goup psntation (mo on this lat): V 4 = v, h v 2 =, h 2 =, vh = hv. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

7 Anoth familia xampl: D 3 Rcall th tiangl puzzl goup G =, f, gnatd by a clockwis 120 otation, and a hoizontal flip f Lt s tak th shadd tiangl to b th unscambld configuation H a two diffnt ways (of many!) that w can labl th nods with actions: f f 2 f f 2 2 f 2 f Th following is on (of many!) psntations fo this goup: D 3 =, f 3 =, f 2 =, 2 f = f. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

8 Goup psntations Initially, w wot G = h, v to say that G is gnatd by th lmnts h and v. All this tlls us is that h and v gnat G, but not how thy gnat G. If w want to b mo pcis, w us a goup psntation of th following fom: G = gnatos lations Th vtical ba can b thought of as maning subjct to. Fo xampl, th following is a psntation fo V 4: V 4 = a, b a 2 =, b 2 =, ab = ba. Cavat! Just bcaus th a lmnts in a goup that satisfy th lations abov dos not man that it is V 4. Fo xampl, th tivial goup G = {} satisfis th abov psntation; just tak a = and b =. Loosly spaking, th abov psntation tlls us that V 4 is th lagst goup that satisfis ths lations. (Mo on this whn w study quotints.) M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

9 Goup psntations Rcall th fiz goup fom Chapt 3 that had th following Cayly diagam: On psntation of this goup is G = t, f f 2 =, tft = f. H is th Cayly diagam of anoth fiz goup: It has psntation G = a. That is, on gnato subjct to no lations. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

10 Goup psntations Du to th afomntiond cavat, and a fw oth tchnicalitis, th study of goup psntations is a topic usually lgatd to gaduat-lvl algba classs. Howv, thy a oftn intoducd in an undgaduat algba class bcaus thy a vy usful, vn if th inticat dtails a hamlssly swpt und th ug. Th poblm (calld th wod poblm) of dtmining what a mysty goup is fom a psntation is actually computationally unsolvabl! In fact, it is quivalnt to th famous halting poblm in comput scinc! Fo (mostly) amusmnt, what goup do you think th following psntation dscibs? G = a, b ab = b 2 a, ba = a 2 b. Supisingly, this is th tivial goup G = {}! M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

11 Invss If g is a gnato in a goup G, thn following th g-aow backwads is an action that w call its invs, and dnotd by g 1. Mo gnally, if g is psntd by a path in a Cayly diagam, thn g 1 is th action achivd by tacing out this path in vs. Not that by constuction, gg 1 = g 1 g =, wh is th idntity (o do nothing ) action. Somtims this is dnotd by, 1, 0, o N. Fo xampl, lt s us th following Cayly diagam to comput th invss of a fw actions: f 2 f 2 f 1 = bcaus = = f 1 = bcaus f = = f (f ) 1 = bcaus (f ) = = (f ) ( 2 f ) 1 = bcaus ( 2 f ) = = ( 2 f ). M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

12 Multiplication tabls Sinc w can us a Cayly diagam with nods labld by actions as a goup calculato, w can cat a (goup) multiplication tabl, that shows how vy pai of goup actions combin. This is bst illustatd by diving in and doing an xampl. Lt s fill out th following multiplication tabl fo V 4. h v h v h v v h h h v v h v Sinc od of multiplication can matt, lt s stick with th convntion that th nty in ow g and column h is th lmnt gh (ath than hg). M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

13 Som maks on th stuctu of multiplication tabls Commnts Th 1st column and 1st ow pat thmslvs. Why? Somtims ths will b omittd (Goup Explo dos this). Multiplication tabls can visually val pattns that may b difficult to s othwis. To hlp mak ths pattns mo obvious, w can colo th clls of th multiplication tabl, assigning a uniqu colo to ach action of th goup. Figu 4.7 (pag 47) has xampls of a fw mo tabls. A goup is ablian iff its multiplication tabl is symmtic about th main diagonal. In ach ow and ach column, ach goup action occus xactly onc. (This will always happn... Why?) Lt s stat and pov that last commnt as as thom. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

14 A thom and poof Thom An lmnt cannot appa twic in th sam ow o column of a multiplictaion tabl. Poof Suppos that in ow a, th lmnt g appas in columns b and c. Algbaically, this mans ab = g = ac. Multiplying vything on th lft by a 1 yilds a 1 ab = a 1 g = a 1 ac = b = c. Thus, g (o any lmnt) lmnt cannot appa twic in th sam ow. Th poof that two lmnts cannot appa twic in th sam column is simila, and will b lft as a homwok xcis. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

15 Anoth xampl: D 3 Lt s fill out th multiplication tabl fo th goup D 3; h a sval diffnt psntations: D 3 =, f 3 =, f 2 =, f = f 2 =, f 3 =, f 2 =, f = f. 2 f f 2 f 2 f 2 f f 2 f f 2 f 2 f f 2 f 2 2 f f f 2 f 2 f f f f 2 f 2 f 2 f f 2 2 f f f 2 Obsvations? What pattns do you s? Just fo fun, what goup do you gt if you mov th 3 = lation fom th psntations abov? (Hint: W v sn it cntly!) M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

16 Anoth xampl: th quatnion goup Th following Cayly diagam, laid out two diffnt ways, dscibs a goup of siz 8 calld th Quatnion goup, oftn dnotd Q 4 = {±1, ±i, ±j, ±k}. i k j k 1 i j 1 1 j k i j i 1 k Th numbs j and k individually act lik i = 1, bcaus i 2 = j 2 = k 2 = 1. Multiplication of {±i, ±j, ±k} woks lik th coss poduct of unit vctos in R 3 : ij = k, jk = i, ki = j, ji = k, kj = i, ik = j. H a two possibl psntations fo this goup: Q 4 = i, j, k i 2 = j 2 = k 2 = ijk = 1 = i, j i 4 = j 4 = 1, iji = j. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

17 Moving towads th standad dfinition of a goup W hav bn calling th mmbs that mak up a goup actions bcaus ou dfinition quis a goup to b a collction of actions that satisfy ou 4 uls. Sinc th standad dfinition of a goup is not phasd in tms of actions, w will nd mo gnal tminology. W will call th mmbs of a goup lmnts. In gnal, a goup is a st of lmnts satisfying som st of poptis. W will also us standad st thoy notation. Fo xampl, w will wit things lik h V 4 to man h is an lmnt of th goup V 4. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

18 Binay opations Intuitivly, an opation is a mthod fo combining objcts. Fo xampl, +,,, and a all xampls of opations. In fact, ths a binay opations bcaus thy combin two objcts into a singl objct. Dfinition If is a binay opation on a st S, thn s t S fo all s, t S. In this cas, w say that S is closd und th opation. Combining, o multiplying two goup lmnts (i.., doing on action followd by th oth) is a binay opation. W say that it is a binay opation on th goup. Rcall that Rul 4 says that any squnc of actions is an action. This nsus that th goup is closd und th binay opation of multiplication. Multiplication tabls a nic bcaus thy dpict th goup s binay opation in full. Howv, not vy tabl with symbols in it is going to b th multiplication tabl fo a goup. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

19 Associativity Rcall that an opation is associativ if panthss a pmittd anywh, but quid nowh. Fo xampl, odinay addition and multiplication a associativ. Howv, subtaction of intgs is not associativ: 4 (1 2) (4 1) 2. Is th opation of combining actions in a goup associativ? YES! W will not pov this fact, but ath illustat it with an xampl. Rcall D 3, th goup of symmtis fo th quilatal tiangl, gnatd by (=otat) and f (=hoizontal flip). How do th following compa? f, (f ), (f) Evn though w a associating diffntly, th nd sult is that th actions a applid lft to ight. Th moal is that w nv nd panthss whn woking with goups, though w may us thm to daw ou attntion to a paticula chunk in a squnc. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

20 Classical dfinition of a goup W a now ady to stat th standad dfinition of a goup. Dfinition (official) A st G is a goup if th following citia a satisfid: 1. Th is a binay opation on G. 2. is associativ. 3. Th is an idntity lmnt G. That is, g = g = g fo all g G. 4. Evy lmnt g G has an invs, g 1, satisfying g g 1 = = g 1 g. Rmaks Dpnding on contxt, th binay opation may b dnotd by,, +, o. As with odinay multiplication, w fquntly omit th symbol altogth and wit,.g., xy fo x y. W gnally only us th + symbol if th goup is ablian. Thus, g + h = h + g (always), but in gnal, gh hg. Uniqunss of th idntity and invss is not built into th dfinition of a goup. Howv, w can without much toubl, pov ths poptis. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

21 Dfinitions of a goup: Old vs. Nw Do ou two compting dfinitions ag? That is, if ou infomal dfinition says somthing is a goup, will ou official dfinition ag? O vic vsa? Sinc ou fist dfinition of a goup was infomal, it is impossibl to answ this qustion officially and absolutly. An infomal dfinition potntially allows som tchnicalitis and ambiguitis. This asid, ou discussion lading up to ou official Dfinition povids an infomal agumnt fo why th answ to th fist qustion should b ys. W will answ th scond qustion in th nxt chapt. Rgadlss of whth th dfinitions ag, w always hav 1 =. That is, th invs of doing nothing is doing nothing. Evn though w havn t officially shown that th two dfinitions ag (and in som sns, w can t), w shall bgin viwing goups fom ths two diffnt paadigms: a goup as a collction of actions; a goup as a st with a binay opation. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22

22 A fw simpl poptis On of th fist things w can pov about goups is uniqunss of th idntity and invss. Thom Evy lmnt of a goup has a uniqu invs. Poof Lt g b an lmnt of a goup G. By dfinition, it has at last on invs. Suppos that h and k a both invss of g. This mans that gh = hg = and gk = kg =. It suffics to show that h = k. Indd, h = h = h(gk) = (hg)k = k = k, and th poof is complt. Th following poof is lgatd to th homwok; th tchniqu is simila. Thom Evy goup has a uniqu idntity lmnt. M. Macauly (Clmson) Chapt 4: Algba and goup psntations Math 4120, Sping / 22