groups1 - A combinahrin‘ We“ 9‘ Jfll’i Orijwuw...

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Unformatted text preview: A combinahrin‘ We“ 9‘ Jfll’i Orijwuw jmrs were. defined as "all svmndries 04 an object“ To be, «nu-dz, we. take an “95:,ch +0 be. a Fir (395) where x h a s¢£ L-Hw. groundsd) And S is a shwiurc. defined on X. E : S null 51 A raph on X A .. a gym-Fife» 9‘ X a permub'fiono‘ X I Evan, Avhmorfh'WM any? smsfies +hm worn-hes: ® it con‘hias fie IdefiHy pamuhtiom, trivkuy, ® it conhins +h¢ 'mvcrse .4 20.9“ o? it‘ damn-ks- i') «(9:5 +hen Sums), @ it 'Is cloud vndu composition - 34' «(9: S And 6"“):5’ the,“ c(1‘l'($3)55, Er our Sauna! definifion 6" a "yr-cup" we are More Johan-1. A fermuéo'éim 5523' 'IS t set G 0‘ Ptrmd'hd’ions of some Stt X which satisfies 0) —@. Givgn "W Pumvhfion (dc-Fined hen .s bijufion) of x, there is A (Mm-oi wen, +0 Apply H: to an object. Th0. Permuh'tion 11' of X is an gvhmorfm of (x5) if «M = S. The Mpmsrggl’ 0F 095) is +he set A“ ($.57 of 0-“ avfomorph'lsms 09 (NS). we will +4". 4-H; as our Hr“ definition of- a "Jroup‘ Fir-5+ though, 53 +his rally More gaunt? Or, is ever, wmu{d’o‘on group an AU'EOMOPPHSM gravy? (0" Some ob) act) hposaim: Eva-7 yumvt-dion are»? i; an .v'hmorpkism group. Proof: L“ (a be a PeerhfI‘on. group on the. 5d: X=fx.,...,xn§. Dding S: §(1erJ,...,1l'(xn)) : “16‘. China: Avf (X,S) : G, For we 6, «(a = i («no»), ...,«Lch..»)-.we6} 37 © “4 G). G = Item” =tec," S ° 11’“) = {(1r(11"(1:Lx.7))‘...,v(w"(t¢l,»); 1:663 : 2 (um. um: tad : 5' 5° ‘1 SAu{(X,S). Now consider some 1V¢AV*L"")' By (D, 6 «Mums 'H'W- “‘WM'I: 50 S urd'm'ns (Kum‘n). The «Hon o-F 11' on 5 sends 'I'MS +uyle +0 («L1.).....1r(%n7), 5° «(-4. yrev'wxj Avt(X,S)SG. I Is +his de-FMJHM am, More flank-«L? No‘o £373.13 Theorem: Every «hi-rad grog? is 'ISomorphic +0 5 pzrwh't'lon 3"“? Proo'h La», 6 be. an abshad- _ Jrovr- we mn+ +o Jim! m RSomoc-phic ytruukstion an»? "I Some $0.1: X. Lu: X=G, and 6’s? (as-.34»; when. 9300 = .9" For a.“ KGX (=5). In the (aft "00‘, DYck defined 4" absfm't amp as a. set 6; with em operation - 'Hwt Saé'lsfies cs ociafivH’ '- ® ’ ‘70-‘07:qu G) Herd-“7‘ 32:6 soc-k flack er-j'e‘j © 'Inversesz . Vaec, [33"efi Sula 'Hltf 3.3" = ‘7'“, 9 9.. This was no‘l’ thL-received. Kit-3m "+he diséd vanfaje o-F Hue «NH-ad- Me+hod is “we if" I’ll-ails +0 O—ncovrage +kou’hé. Bums: G has an 'Idenh'w‘y etemnt e, i‘ 31th the." 93m =3 at h = flan, So 93 at P“. we «(59 have. (9,~ moo = 949.00) =93“ - x) : TLH-x) =(3.h)')t = 93mm. Verifyinj +he. pgmohfion group o‘x‘mMs ®'® $or G. is EASY, and ‘H'Ns G 26" a flrMu'i'ul'ion ( R260.“ Hut '1‘ N '\S 0. normal Subaru», a; 6, 'then G can b0. broken info flu groups: N and 6/“, fin tad-{uri- amp. If we. find u'rth a finite grow, we. can cord-5mm. +his bregkin 7'0".“ wth we and up wi I. A collection 0‘ Sign; (groups ui+hou+ normal why-ours), The. Sudan-Han Thtortm s¢ys Hut «he set '09 s'nmple your; we and up wH'H is unique" I.“ 'er I'M-Os, work on +51: Chautauqua“ took 9-“, AM ‘Ihdktted 'H'mf d'ker (PorAdk grout” M33h+ en‘s-6. In sacral, the nvmbe: o-F Llama-ts and 'Her orders woo“ to. known, but one neeJeJ +9 Ad'vdty confirmi- ‘HR at“? +9 prove 'Hmé if €Xi$+edu THQSQ Hue, mos-NJ unfit-0:414 us au+oMorphisM group; 61‘ graphs. Therefire, we. retuy only <Qf€ .boui- +he meat; 53M?"- 3'°"f‘- These have Bun “uni-Pied info I8 iNFinit'e 'FQM'IUCQ and 26 QRtyHons, cod.le §yor¢di¢ 3223?}.- The $irs+ 0; these is be 4:0an Unfit [160, no O'H‘DQFS Mr: 9nd... liijmn - Sims 3527i # 4-| 4"} 5' Z, 000 etemcnts Conshvcfid 'm Hé? as, essenfiaflyl the. awl'omorph'nsm are»? a; o. 22— rejulu- (“My uric; has “are; 7.7-) Jrapk. with \00 “races, and Hans... \loo 2430.5. The vah. 'w a¢4ually +he IMF?“ graph um +hzso. round-tr: Sud-t ‘Hfiafi o it does n.+ conhin K3, and o wen, pair o-F non-neiahhoeinj vufices shut womanly 6 “MM” Afighbors. AFQ 0“ “Mic groups iSomorPkic to aomorPMSm jroufs of graph-5.? Frock’c's Theorem: Ves. We ska-d. a ‘Proo-F. First we. Svfposc C1 is a are»)? And SSGVQI The Codie», dijrofk. DUMS) has Varéices 6, and 'Fer and. Jet. and se S, Here .55 ‘0 e430. ‘t‘om 3 to 3.5 (Abdul by 5. mm 6-; thé's Theorem mg: Take 5 +9 be a gravy wH’h M Mats, and sci: S:G\‘¢3=$suw’s”'l~s' We, Know «Hut 6 is isomorphic 4.. fig au+oM°P7KISMS es- VG.” uhid-s pnsewe adj: Labels. How can we; buiLd an \mdired'el graph. with Precisdy +hesa au-foMor'PUSMS? 5' u " Repace, b7 +he Sadat-k J o—r;\—0——IO h. db Li'l'h : Lo! ‘ quces FO-C'tfi: 0 is Canflcctd 'I“ MA on“! W S Jame.”le <1 (Lu, every 21th "9 C1 is a yroduefi o4 ck.th of S) PJ= x H 3-»: is an M4omorfhisM 04 'b (a, 5). © The ao‘l‘omorPMSMS o; 6 which Present fidje Lobds are yredsdj +he au+omorPHSMs 99®- Therein: +h€$€ “‘h' MMPMSMS Forth 9* 3""? \Souorphk +0 G. 'Wooh Exerci 3:. ~— ...
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groups1 - A combinahrin‘ We“ 9‘ Jfll’i Orijwuw...

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