An introduction to groups


 Agnes Hawkins
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1 n introdution to groups syllusref efereneene ore topi: Introdution to groups In this h hpter Groups The terminology of groups Properties of groups Further exmples of groups trnsformtions
2 66 Mths Quest Mths Yer for Queenslnd Groups Up until now, when you used the term lger you proly thought of vriles nd opertions with those vriles. However, lger exists in wide vriety of forms, from oolen lger to group lger. In its most simple form it n e thought of s the siene of equtions. t this stge in your mthemtis studies you would e fmilir with the use of lssil lger to solve equtions suh s x 9 = 0, yielding the solutions of x = ±. However, in the 9th entury, mthemtiins grdully relised tht mthemtil symols did not neessrily hve to stnd for numers, if nything t ll! From this ide, modern or strt lger rose. Modern lger hs two min uses:. to desrie ptterns or symmetries tht our in nture nd mthemtis, suh s different rystl formtions of ertin hemil sustnes. to extend the ommon numer systems to other systems. In lger, symols tht n e mnipulted re elements of some set nd the mnipultion is done y performing ertin opertions on elements of tht set. The set involved is referred to s n lgeri struture. The symols my represent the symmetries of n ojet, the position of swith, n instrution to mhine or design of sttistil experiment. These symols my then e mnipulted using the fmilir rules used with numers. group is system of elements with omposition stisfying ertin lws. It is hoped tht this rief introdution to groups expnds your understnding of the verstile nd llenompssing onept of lger. The following setion of work fits into this field of study y virtue of the ft tht it dels with symols nd opertions. lgeri strutures Reserh the topi of lgeri strutures exmining erly lgeri systems tht developed in nient ivilistions suh s the Indin, ri, ylonin, Egyptin nd Greek. Highlight differenes nd similrities mong the vrious forms. ut first new tool to help you del with some notions used in groups. Modulo rithmeti Not to e onfused with the modulus of numer (see hpter on rel numers, R, where the modulus of, written = ), modulo rithmeti uses finite numer system with finite numer of elements. This is sometimes referred to s lok rithmeti euse of the similrities with reding the time on n nlog lok.
3 hpter n introdution to groups 67 onsider reding the time shown on the lok fe to the right. Whether it is m or pm we would sy it is o lok, ut in hour time the pm would e 00 hours. In effet we hve sutrted hours from the 00 ( hours) to give n nswer of. In this se we sy tht is the residue, or wht is left over when hours is sutrted from the. In modulo rithmeti the sme priniple is used exept tht the is repled y = = = nd so on. In our norml deiml system =, ut in modulo rithmeti the residue of differs from y (or multiple of ) nd nd re sid to e ongruent. Tht is, in modulo 5 rithmeti, the numers, 8 nd re ongruent nd in modulo rithmeti,,, nd 6 re ongruent numers. Using more preise terminology, ddition modulo 0 is written + 9 mod 0, mod 0, nd so on. (Note the revition of modulo to mod.) In mod, the numers 0 to re referred to s residues, s with 0 to 5 in mod 6. This informtion n e stored in tle, known s yley Tle. WORKE Exmple rw up yley Tle tht shows the residues using ddition modulo WRITE/RW rw n empty tle with 0,,, in + 0 the first row nd olumn nd put + sign in the top orner. 0 Strt working ross the first row = 0 et. nd do likewise with the first olumn. 0 0 The residues re the numers left over + 0 when is tken from the nswer (if the nswer is or greter). s you 0 0 omplete the tle note tht the nswers 0 re less thn. So, for + the residue 0 is 0. 0
4 68 Mths Quest Mths Yer for Queenslnd rememer rememer. Modulo rithmeti is like lok rithmeti where in mod 0.. The residues of modulo x re ll the whole numers less thn x.. ongruent numers in mod x ll differ y multiples of x. Groups SkillSHEET. WORKE Exmple List numers ongruent to: in mod 8 in mod 6. List the residues in: mod mod 9 mod. rw up yley Tle tht shows the residues for eh of the following: ddition mod 6 multiplition mod multiplition mod 5. The terminology of groups Mthemtis is often referred to s siene sometimes s the siene of ptterns. You will enjoy your studies of mthemtis more if you look for ptterns in ll the ides you explore. One suh ide is tht of groups. Group theory is pplied to mny res of siene suh s genetis, quntum theory, moleulr orits, rystllogrphy nd the theory of reltivity. In mthemtis, group theory is pplied to mny mthemtil models involving lger, numer theory nd geometry. In hpter you delt with different sets of numers within the Rel Numer System. Throughout your student life you hve used the opertions of ddition, multiplition, sutrtion nd division, finding squre root, reiprols, nd so on. These re exmples of opertions performed on numers tht re prt of ertin set. Opertions (suh s ddition) tht involve input vlues, for exmple +, re lled inry opertions. Those tht involve only one input vlue, suh s finding the squre root of numer (for exmple 8 ) re lled unry opertions. Others tht involve input vlues re lled ternry; for exmple, the prinipl, interest nd term of lon re the input vlues involved in lulting the mount of interest due on lon. (Stritly speking the multiplition involved is still rried out on pirs of vlues.) efinition of terms Groups tht we will del with onsist of system tht involves set of elements (often numers) nd inry opertion. Lower se letters:,, re used to refer to elements of the set nd the symol denotes whtever opertion is involved. For nonempty set of elements S = {,,, } involved in the inry opertion to e group, G = [S, ], the following properties must hold.
5 hpter n introdution to groups 69 losure n opertion is losed if the result of tht opertion is n element of the sme set s the inputs. For exmple + = 5 ould e written where S = {Rel numers} (or R) nd is the opertion of ddition. This opertion is losed euse 5 R ut onsider = where S = {Nturl numers} (or N) nd is the opertion of sutrtion. euse the result ( ) is not memer of the set of nturl numers this opertion is not losed. Tht is, the nswer is not prt of the initil set of nturl numers. ssoitivity If n opertion is ssoitive, the order in whih opertions re performed does not ffet the nswer. Often rkets re employed to determine the order of opertions. For exmple, onsider ( ) nd ( ) ( ) = 6 ( ) = = = In this se, oth nswers re the sme. Note tht only the position of the rkets hnges nd the order of the numers remins the sme. ut onsider the opertion of division: (0 ) nd 0 ( ) = 0 = =.5 = 0 Here the nswers re not the sme. ivision, like sutrtion, is not ssoitive. You would hve relised this in your erlier junior mthemtis studies. Identity For ll elements of set, if unique element exists in the set suh tht u = then u is the identity element (IE) for tht opertion. Tht mens tht there is only one element tht leves every element unhnged when the opertion hs een pplied. For exmple, + 0 = then 0 is the identity element for ddition (IE+) for rel numers. However, 0 = 0 so 0 is not the identity element for rel numers under the opertion of multiplition. Note: The one identity element must work for ll elements of the set so = 5 nd = 8. Inverse For eh element of set there is unique element suh tht = u where u is the identity element for tht opertion. Unique mens tht every element hs only one inverse.  = where is the identity element for multiplition (IE ) Therefore  is the multiplitive inverse of. Now onsider + = 0 where 0 is IE+; in this se is the dditive inverse of. However, note tht the set involved here would hve to e integers (tht is, oth positive nd negtive) not just whole numers euse {Whole numers}.
6 70 Mths Quest Mths Yer for Queenslnd We n now restte the definition of group. If the following properties hold for set of elements under ertin opertion :. losure. ssoitivity. existene of n identity element. existene of n inverse then the system under investigtion [S, ] is group. If fifth property, ommuttivity, lso holds, then the group is n elin group. ommuttivity If the order of the elements involved hs no effet on the outome, then the opertion is ommuttive. For exmple 5 = 0 nd 5 = 0 Hene multiplition with rel numers is ommuttive. Note the stted ondition, with rel numers euse you hve lredy worked with mtries where multiplition in not ommuttive. However, onsider 0 = 5 nd 0 = 0. So division is not ommuttive. You would e fmilir with other opertions s well tht re not ommuttive. WORKE Exmple Find the identity element nd inverse for the opertion defined s = + +. n identity element is n element tht, when involved in n opertion with nother element does not hnge the vlue of tht element. n inverse is n element tht, when involved in n opertion with nother element results in the IE for tht opertion. WRITE Let + + = (where = IE) therefore = IE = Therefore + + = (where is the inverse of nd = IE from prt ) + = = WORKE Exmple Find the identity element for the opertion defined s + =. n IE is n element tht, when involved in n opertion with nother element does not hnge the vlue of tht element. WRITE + = let = IE Squre oth sides: + = therefore = 0 nd = 0 therefore IE = 0
7 hpter n introdution to groups 7 History of mthemtis NIELS HENRIK EL (80 89) uring his life... Lord yron, the English poet, writes on Jun. Npoleon onprte eomes emperor of Frne. Jenptiste Lmrk, the Frenh iologist, proposes tht quired trits re inherited y individuls in popultion. Niels el ws one of the most produtive mthemtiins of the 9th entury. orn in Norwy on 5 ugust 80, y the ge of 6 he hd strted his privte study of the mthemtis rememer rememer of Newton, Euler, Guss nd Lgrnge. s the sole supporting mle of his fmily t 8 he tutored privte pupils while ontinuing his own mthemtil reserh. y the ge of 9 he hd proved tht there ws no finite formul for the solution of the generl fifth degree polynomil. He died of tuerulosis on 6 pril 89, two dys efore the nnounement of his posting s professor to the erlin university. His life in poverty stnds in ontrst to the regrd with whih he is held in his field; the term elin group eing nmed in honour of el. His studies on group theory were entrl to the development of strt lger. Questions:. How did el finnilly support his fmily?. Whih property do groups ering his nme exhiit? set S forms group under the opertion if nd only if (iff) ll of the following re true:. it is losed under ; tht is, the result is n element of S. the order in whih opertions re performed hs no effet on the results; tht is, it is ssoitive. there is only identity element (IE), u, suh tht u =. there is unique inverse for every element suh tht = u, where u = IE. 5. If the property of ommuttivity lso holds, then it is n elin group. The terminology of groups + Show tht = is not losed with respet to whole numers. Stte its identity element. If n opertion is defined s + determine whether this is losed if nd re whole numers. (Rememer you only hve to find one exmple where the opertion is not losed to disprove sttement.)
8 7 Mths Quest Mths Yer for Queenslnd WORKE Exmple WORKE Exmple Find the identity element nd the inverse for the opertion on rel numers where = +. Wht is the identity element of the opertion = + if nd re rel numers? 5 The opertion = is defined for positive rel numers nd. Find the identity element for this opertion. + 6 evelop proof to show tht = hs no identity. 7 n opertion is defined with respet to n ordered pir of integers s (, ) (, d) = (d +, d). Show tht (0, ) is the identity element for the opertion. 8 Show tht = ( + ) hs no identity for rel numers. Properties of groups WORKE Exmple Verify tht the set of integers forms group under ddition. Is this group elin? Wht numers re involved? ll positive nd negtive integers nd 0 re involved so stte the set nd opertion. While you n think of tul vlues for the integers (, 0 ) your nswer should use only vriles, with onstnts used s exmples only. Test eh of the properties in the sme order eh time to help you rememer the tests. iii The sum of ny integers is n integer. iii The order in whih the opertion is performed hs no effet on the result. iii Sine 0 Z, IE+ exists. iv Sine Z ontins ll positive nd negtive whole numers, the inverse is Stte tht the system forms group under the onditions stted. WRITE Let Z = {,,, } e the set of integers; the opertion is ddition. iii The opertion is losed: + = where, nd Z iii The opertion is ssoitive: ( + ) + = + ( + ) iii The identity element exists: + 0 = iv The inverse exists: + = 0 Thus the set of integers forms group under ddition.
9 hpter n introdution to groups 7 If the group is elin we need to show tht this opertion is ommuttive. WRITE ommuttivity + = + Therefore the group is elin. Note tht the test for ommuttivity is performed lst euse the first properties re neessry to stte tht it is group in the first ple, efore it is shown to e elin. This group, G = [Z, +] is n infinite group, hving n unlimited set of elements. You will lso del with finite groups whih hve ountle numer of elements. WORKE Exmple 5 Verify tht the set of odd integers does not form group under ddition. Wht numers re involved? The set of odd integers inludes 5,,,,, 5 Stte the set nd opertion. WRITE S = {,,, } is the set of odd integers. The opertion is ddition. Test the properties s shown in worked exmple. There is no need to proeed ny further with tests to verify the system is group s it is not losed. losure: + S Let = nd = = 8 nd 8 S Therefore G [S, +] The set of odd integers does not form group under ddition. WORKE Exmple 6 onstrut yley Tle for [{, i,, i}, ] nd determine whether this onstitutes group. Set up the empty tle. WRITE i i i i ontinued over pge
10 7 Mths Quest Mths Yer for Queenslnd omplete the tle. Rememer from hpter on omplex numers tht i = nd i i =. WRITE i i i i i i i i i i i i Test the group properties losed set, ssoitive, identity element nd multiplitive inverse. The nswers n e otined from the tle. (Multiplition y leves ll elements unhnged.) Stte your onlusion.. ll the results re memers of the originl set {, i,, i}. This is losed set.. The set is ssoitive e.g. ( i) i = i i = nd (i i) = =. The identity element, IE =. Multiplitive inverse: there is (IE ) in every row of the tle so eh element hs unique inverse. Therefore, the system is group. Note tht the yley Tle is symmetril out the leding digonl. The tle ould e flipped over on the leding digonl nd remin unhnged. This mens tht the order of opertions will not ffet the results, tht is, tht the opertion is ommuttive. Therefore this group is lso elin. i i i i i i i i i i i i WORKE Exmple 7 Leding digonl onstrut yley Tle for [{mod 5}, +] nd determine whether it is n elin group. eide wht numers re present in mod 5 nd omplete yley Tle of residues. WRITE
11 hpter n introdution to groups 75 Test for the group properties. Test for ommuttivity WRITE. ll results re memers of the originl set. So, the set is losed.. ddition with whole numers is ssoitive.. The identity element, IE+ = 0 exists.. There is 0 entry in eh row euse eh element hs orresponding element tht, when dded, results in 0 (IE+). So, there is n dditive inverse. Therefore the system forms group. ddition mod 5 is ommuttive s shown y the symmetry out the leding digonl. For exmple: + 0 = nd 0 + = nd + = nd + = Therefore the group is elin. Note: There re 9 xioms tht relte to opertions nd whole numers tht require no proof: they re ssumed to e true. The ssoitivity sttement in the exmple ove relied on one of these xioms nd you n stte tht these xioms hve een used. They re given here with no explntion.. losure Lw of ddition. ommuttive Lw of ddition. ssoitive Lw of ddition. Identity Lw of ddition 5. losure Lw of Multiplition 6. ommuttive Lw of Multiplition 7. ssoitive Lw of Multiplition 8. Identity Lw of Multiplition 9. istriutive Lw of Multiplition over ddition, where ( + ) = + rememer rememer. To determine whether set forms group under n opertion ( ) test eh of the four properties; tht is, test whether it is losed nd ssoitive, whether there is n identity element nd unique inverse.. To determine whether the group is elin, show tht the opertion is ommuttive (e.g. = ).
12 76 Mths Quest Mths Yer for Queenslnd Properties of groups SkillSHEET. WORKE Exmple WORKE Exmple 5 WORKE Exmple 6 WORKE Exmple 7 Verify tht the set of rel numers, [R, +], forms group under ddition. Is this group elin? onsider the set of even numers (n) where n ±Z. oes this form group under ddition? (Note: 0 {even numers}) oes it form group under multiplition? oes the set of powers of form group under: ddition? multiplition? Verify tht the set of even integers does not form group under division. 5 onstrut yley Tle for [{mod 5 exluding 0}, ] nd determine whether this onstitutes group. 6 rw up yley Tle for the set of even powers of under ddition. oes this form group under ddition? oes this form group under multiplition? 7 onstrut yley Tle for [{mod }, ] nd determine whether it is n elin group. 8 etermine whether eh of the tles elow forms group. d d WorkSHEET. 9 onstrut yley Tle for the set = [{5, 0, 0}, lowest ommon multiple of, ] oes this set form group? 0 The movements of root re restrited to no hnge (N), turn left (L), turn right (R), turn out (): { N, L, R, }. onstrut yley Tle nd show tht this set of movements forms group.
13 hpter n introdution to groups 77 pplition of groups permuttions symmetry of squre (or ny other shpe) my e written s permuttion y hnging the positions of the verties. For exmple, referring to the figure t right, we ould write: 5 P = 5 5 to the position of vertex, nd so on., whih mens tht vertex goes The only other two permuttions llowed here re: P = 5 nd P = etermine whether these permuttions form group under the opertion mening followed y. Further exmples of groups trnsformtions onsider ll the trnsformtions tht shpe ould undergo. Rottions out its entre nd refletions out its xes of symmetry involve hnges in the verties only. refully exmine the digrm elow. Mke sure you understnd the symols nd the new positions of the verties. Rottions ntilokwise: R R R Refletions: R V in the vertil xis of symmetry R H in the horizontl xis of symmetry R R in the top right digonl R L in the top left digonl R 0 no hnge.
14 78 Mths Quest Mths Yer for Queenslnd R 0 R L R L R 90 R V R L R R R R R H R 80 R H R 70 R V Therefore the set of ll trnsformtions or symmetries is given y the set {R 90, R 80, R 70, R V, R H, R R, R L, R 0 } nd the inry opertion tht omines ny two of these trnsformtions is referred to s omposition, where one opertion follows nother. ll the omputerised movements involved in sreen nimtions re sed on similr ompositions of trnsformtions. WORKE Exmple Find the result of R 80 R v. 8 rw the initil squre with lelled verties. WRITE/RW Trnsform the squre using R rottion ntilokwise. Lote vertex nd move it 80 ntilokwise. ll other verties follow in order round the squre.
15 hpter n introdution to groups 79 WRITE/RW For R V mrk vertil xis of symmetry in this figure (from step ) nd reflet or flip the squre out this xis. Reposition the verties one side t time, nd. 5 This mthes with single trnsformtion representing R H. The result is R H. Funtions onsider funtions f(x) = x, g(x) = x, h(x) =  nd k(x) = (where x 0). x  x When these funtions re involved in omposition of funtions suh s g[h(x)], the funtion h(x) is sustituted s the inner funtion into the outer funtion whih is g(x). Tht is g[h(x)] =  where  (the inner funtion) is sustituted into g(x) whih is (x). x x Similrly, k[g(x)] = where g(x) = x (the inner funtion) is sustituted into ( x) k(x) =  (the outer funtion). Tht is, k[g(x)] = =  = h(x). x ( x) x WORKE Exmple 9 Show tht funtions f(x) = x, g(x) = x, h(x) =  nd k(x) = form group under x  x omposition. omplete yley Tle for these ompositions. WRITE f g h k f g h k f g h k g f k h h k f g k h g f ontinued over pge
16 80 Mths Quest Mths Yer for Queenslnd Test the group properties. Stte your onlusion. WRITE losure: yes ll results re elements of the originl set. ssoitive: yes for exmple (f g) h = g h = k f (g h) = f k = k Identity element is f(x) Inverse: yes f(x) ours in every row nd olumn. omposition of these funtions forms group. History of mthemtis RTHUR YLEY (8 895) uring his life... Thoms Edison invents the phonogrph. Slvery is offiilly olished throughout the western world. lfred Noel invents dynmite. rthur yley, fmous English mthemtiin, ws orn on 6 ugust 8. His pulished mthemtil ppers re lssis nd inlude disussions on the onept of ndimensionl geometry. t the ge of 5 he egn prtising lw whih he ontinued to do until 86. In his spre time he wrote more thn 00 mthemtil ppers. In 86 he epted professorship in mthemtis t mridge University. One of his most fmous nonmthemtil omplishments ws his role in hving women epted t mridge. Like Niels el (see pge 7), mny of his reserh topis re now used in strt lger nd group lger, s well s in work with mtries nd the theory of determinnts. The yley Tle is nmed fter him. He died on 6 Jnury 895 hving reeived mny demi distintions. His totl works fill volumes of out 600 pges eh testimony to his prodigious life nd study in mthemtis. Questions. Wht is one of yley s most signifint nonmthemtil omplishments?. List four fields of mthemtis whih feture in yley s work. rememer rememer The inry opertion tht omines ny two trnsformtions (for exmple, rottion nd refletion) is lled omposition, when one opertion follows nother.
17 hpter n introdution to groups 8 Further exmples of groups trnsformtions rw yley Tle for the rottion of n equilterl tringle. Lel eh vertex. oes it form group? Is it elin? rw yley Tle for the refletions of n equilterl tringle through eh of the verties R 0, R V, R L, R R. oes it form group? R R R L R V WORKE Exmple 8 Explin wht the following digrms represent out the group shown elow. F R 0 R 0 F esrie the symmetries of the following figures, using fully nnotted digrms. nonsqure retngle nonsqure rhomus n ellipse 5 onsider n infinitely long strip of Hs, printed on trnsprent pper, s shown elow..h H H H H H. esrie the xes of symmetry of this group. 6 Lote the xes of symmetry for the following figures. WORKE Exmple 7 omplete yley Tle for the omposition of the following funtions. 9 f (x) = x f (x) =  f (x) = x f (x) = x x where f f = f [f (x)] oes this omposition form group?
18 8 Mths Quest Mths Yer for Queenslnd 8 Show tht the set of ll mtries forms group under mtrix ddition. d i Give one exmple of mtrix tht does form group under mtrix multiplition. ii Give one exmple of mtrix tht does not form group under mtrix ddition. i Give the ondition for mtries to form group under mtrix multiplition. ii Show tht these mtries form group under mtrix multiplition. 9 Show tht the set of mtries ,,, forms group under mtrix multiplition z 0 Show tht the set of mtries of the form z, where z is omplex numer, forms group z z under mtrix ddition under mtrix multiplition. ssume z + z 0. S is the set of ll mtries suh tht 0 0, where z is nonzero omplex numer. z z Show tht 0 0 is the identity element under mtrix multiplition. oes the set form group under mtrix multiplition? = i 0, where i =. The set T onsists of positive powers of suh tht 0 i T = n where n is positive integer. Find ll the elements of set T. oes the set T form group under mtrix multiplition? Some pplitions of group theory o the residues of {0, } mod form group under ddition? teher of strt lger intended to give typist list of 9 integers tht form group under multiplition modulo 9. Insted, one of the 9 integers ws omitted so tht the list red:, 9, 6,, 5, 7, 79, 8. Whih integer ws left out? Show tht {,, } multiplition mod is not group ut {,,, } multiplition mod 5 is group. Give n exmple of group elements nd with the property tht 5 The integers 5 nd 5 re two of integers tht form group under multiplition mod 56. List ll integers.
19 hpter n introdution to groups 8 6 If the following tle is tht of group, fill in the lnk entries. e d e e d d e e 7 Prove tht if G is group suh tht the squre of every element is the identity, then G is elin. 8 Exmine whether rottions nd refletions s stted erlier in this setion, form elin groups. 9 Quternions The onept of set of elements lled quternions ws first developed y the Irish mthemtiin, Willim Hmilton (see pge 8). Quternions re ordered sets of four ordinry numers, stisfying speil lws of equlity, ddition nd multiplition. Quternions re useful for studying quntities hving mgnitude nd diretion in threedimensionl spe nd this hs enled gret dvnes in quntum theory, reltivity, numer theory nd group theory. The numers re, i, j nd k nd hve the following properties: = i = j = k = ijk = i = i j = j k = k ij = ji = k i(jk) = (ij)k = ijk ll rel nd omplex numers do ommute with i, j, nd k ut they re not ommuttive with eh other. Follow this exmple tht shows tht jk = i ijk = from the definitions i ijk = i multiply oth sides y i on the left (or premultiply y i) i jk = i ssoitivity jk = i from the definitions jk = i premultiply oth sides y jk = i euse multiplition etween these elements is not ommuttive it is essentil tht ll multiplition is done from prtiulr side of n expression nd to perform this multiplition on oth sides of the equl sign. You must respet the order of plement of terms in this system. Show tht i jk = kj ii ki = j Show tht i = i If q = s + wi + vj + yk nd p = m + ni + oj + jk, find the produt of the two quternions.
20 8 Mths Quest Mths Yer for Queenslnd 0 Puli Mtries The ides introdued in the setion on quternions ove n e extended to represent mtries. One set is: = 0 i = 0 j = 0 k = While the mtries for i nd k might look little dunting, they n e simplified y repling the elements with omplex i. The lst three of these mtries re used in the study of quntum theories to explin nd predit the ehviour of eletrons. They re lled the Puli Spin Mtries nd students of hemistry will ppreite the importne of the spin of eletrons in tomi onding nd the strength of different mterils. vrition of these mtries used in the study of nuler physis is shown elow: P = 0 Q = 0 R = 0 S = T = 0 i V = i 0 U = i 0 W = 0 i i 0 0 i 0 i i 0 On exmintion of the first nd seond rows of the mtries ove you will notie tht the seond row is refletion of eh mtrix in the first row, multiplied y i. onstrut yley Tle to disply the results of mtrix multiplition using these 8 mtries. rrnge them in the order given; tht is, from P to W. etermine whether the totl set forms group. Mrk off the top lefthnd orner. Exmine this setion of the tle nd show tht this suset forms group. This is n exmple of sugroup, where suset of group forms omplete group of its own. Internet serh The rel life pplitions of groups re quite omplex. Use the internet to reserh this field of study. Inlude list of distint topis nd more detiled report tht highlights the use of group theory.
21 hpter n introdution to groups 85 summry Groups Modulo rithmeti is like lok rithmeti where = in mod 0. The residues of modulo x re ll the numers less thn x. ongruent numers in mod x ll differ y multiples of x. The terminology of groups set S forms group under the opertion if nd only if (iff) ll of the following re true: it is losed under, tht is, the result is n element of S the order in whih opertions re performed hs no effet on the results, tht is, it is ssoitive there is only identity element (IE), u, suh tht u = there is unique inverse for every element suh tht = u, where u = IE. Properties of groups set forms group under n opertion if elements of the set re losed nd ssoitive, nd there is n identity element nd unique inverse. The group is n elin group if the opertion is ommuttive (e.g. = ). Trnsformtions The set of ll trnsformtions (for exmple, rottions nd refletions) nd the inry opertion tht omines ny two of these trnsformtions is referred to s omposition.
22 86 Mths Quest Mths Yer for Queenslnd HPTER review test yourself HPTER etermine whether the following re groups: {, } under multiplition {0, } under ddition. etermine whether the following re groups: the set {,,, 5, 7, 8} under multiplition modulo 9 the set {0,,,, } under multiplition modulo 5 the set {,, 6, 8} under multiplition modulo 0 d the set {0,, } under ddition modulo. etermine whether eh of the following form groups: the set of integers where p q = p + q the set of positive rtionl numers where p q = p  q. Show tht the set of ll integers forms n elin group under the opertion = +. There re two lights in room, one on the eiling nd one on the wll with possile sttes for the two lights oth on, oth off, wll light on only, or eiling light on only. There re possile hnges of stte: no hnge, oth hnge, eiling light hnge nd the wll light hnge. These hnges re denoted y N, W, nd W respetively. Show tht the set {N,, W, W} forms group with respet to the opertion followed y. 5 Wht property of group is displyed in yley Tle if: the elements re symmetril out the leding digonl the sme element does not pper more thn one in ny row or olumn the identity element ours only one in eh row or olumn. 6 etermine whether the following re groups: the set of integers, modulo n under ddition the set of integers, modulo n under multiplition the set of integers, modulo n, exluding 0, under multiplition d the set of integers, modulo n, exluding 0, under multiplition, if n is prime. 7 etermine whether the set of ll moves tht n e mde y knight on hessord forms group or not. 8 Verify tht the set m 0, where m 0 forms group under mtrix multiplition. 0 m Verify tht ll p q mtries form group under mtrix ddition. 9 Show tht the following set of mtries forms group under multiplition i 0 0 i i 0 0 i i i 0 0 i i 0 0 etermine whether or not the following funtions form group under omposition of funtions. ssume tht they re ssoitive. x x + f (x) = x f (x) =  f (x) = + x f (x) = f 5 (x) = f 6 (x) = x + x x + x
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