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Algebraic Structure Part 2 DSTL
A subgroup is a subset of a group that is also a group. A normal subgroup is a subgroup where left and right cosets are equal. The intersection of two normal subgroups is also a normal subgroup. A permutation is a one-to-one mapping from a set to itself. Permutations form a group. A cyclic permutation has a single generator element. The length of a cycle of an element in a permutation is the order of that element. A ring is a set with two binary operations, addition and multiplication, satisfying certain properties. Integral domains have no zero divisors. A field has nonzero multiplication and every nonzero element has a multiplicative inverse.
Algebraic Structure Part 2 DSTL
- 1. Page No. Topic.ALgehraicSnuckut.LUnikIL.Date..1.0l2.22a 5ubauaupl ASubgraupi a Sabsek tof graup TOuP_ that_atsie the atuu group eguitmenk. HmusE elomenh hueLre ConkaLnhe lclentiky elemet H a subgroup e a OY A Subquoup H a quoup 6that des nat include the einu gutp selyi ia knousn asa mope Subaraup. TE clendked HC H<G- H&G- AamplL= he gMitap_C iDkqus eupped wlh Suhgraups aneatal Numbet eLuepid uilh addiho
- 2. Page No, Topic. 2 Date.. sse uduo_aD tlumunt 1upl= Le G bea mulHpliahve RLaan 4,Lhan TOup and Eq non=ne4ave Cmallest inlkqeH n u Said to be odey o_eumens e idnttky elamnt aiae goup_G) Mulhpli catd EemensE addihuTlunHly elument G= Lw,w uemulkhplicaH grap w3. (w Ou) 3 04) = 2
- 3. Page No. Date. 3 Topic. AneluMLnE_whch an qenemalr a0 9uoup Coulud aLeneuator eneuuoY- entiue elhment_atl called allelemenl a _Can be uepresenkd Lora quoup4,*),an Ling PoLu Lxcmpl= a,lL2,3} t4 m odulo ccddi ho Hou to jnd_madwlo adeliltn 4MLLLLplit hon Xm a Xm b a+th i 0tb m i atb m_ i Xpa b ab ab<m abm abZ m CXuo0 2 2 m 2 3 X 2 m
- 4. Page No. Topic. ****************** Date.. Cyclic lmauplA_qmup [lq, i Said tohe Cyclicqroup it Conkains& CLt-liasE one qenuaker emink. Ok A ycl eNLaleol b a Single elhmeut uoLp aqTOupthaEcan be C L23,02tu d a yclu Grou (modulo addihon) S- 1t1H|=3_ S +1+}+1+| 25-4)1 O S7 u=p1_ UaCenuator 2 moduldo 2212 f4 O 23Gy o7 02- Cddihon 21B-4 840 O2, qrt not 3 26Xy 2 enuatoY 3 124 0 L3 u CyenvuatY SkyRider
- 5. Page No. 5 Topic... Date... Er2|hese o- Complux numbeHA 1,-1, -1 ondu mulliplitatad iu c Cylia group Ci wsed opowoL ZeuO ,3 hintu Uagenuater a y- y-i24 Dodntty ELument alse a genu aber Mdil- a d_a aenuator 4aCyclic group G i alle aaeneuate la Hene he au twoIencuCher 4-1,which CoLnscll thu CyLlic Noh2- A Cucu'c_qrOup alwayA an abelian group CyCiG but not eveuy a belian qroup u a rOUP SikRide Ruttonal Numbeu Undeu adcli ton ànot aycic but aabeltan
- 6. Page No. 6 Topic. Date.... *********************** Prowethat q(0L23},t4}- agroLp i closLunepropeuyE Modulo Addikicn 0+|=1 2+3= S4= 1+2 =3 5- 2 3 2 3 O 2 2 3 O atb éq 3 2 CloSuu þropuy Sakictitd4 ASSa.ciakie PropeyE ab l2(2 ty 3) = ( y 2) ty3 145,4 3tu3 2 2 AssoticheDTOP Aky Rahsjird
- 7. Page No. Topic. Date.. Ldentik ElLmut 2te Tdinkk_elimut u zuo (zeuo alho ens gien Set) 2 ty 0o 2 3 tu 0- E3 =0_40C4 Sastieed Iner Elimut a+ a e l=0 t43 0 3 = O Satinl puDpLiu ou Jatdied gien_ AIL 4auu Se a qrmp t (moduloiddih
- 8. Page No. Topic.CoSes.... Dat8 CosETS= G 3 MuLplicakie qroup Subgoup o G At pe a ony element o then Ha fha:hCH Right CaSet of_Hin - aH=fah h Ch?-Let coset o Hin 4 (a deu i oup H+a= htq:heHG atH= ath:héH4 Ly-L G= Set 0 inleg ens H= Set_o InkgeanA Cboh ar adoihe Hta= hta heh Shaly, heH3 2 L-1, L-} H L-14 (mulhplitakie) Hi =fhiheh -i3coset ider
- 9. Page No Topic.oxMal. ukgrau..9 Date bepiaihon-Let NA Hhe h EN Subgroup o g7oupC and aEq N nn N hen Nàcalled noYmal Subgroup of q- Aloxmal Subgroue canalko be dinoltd by HX, AEG. Ln=l n N N 1fermulkplu (ati ve L N IN Henuilu a NOYma Subaroup apply wth al elumeh ofCa Mn N E)= N mulHplcahu aol-e iourhy elumot eou N NOma Suh-g3oup o ond onl it ond onyik 1N7 cN
- 10. Page No Topic. Date. -A Sub-qpoupA o a a1oup Nmal nd ony AA N _1E4 Lat Nd=N xeG q (equal4 lonkunyn) then Nua Namal subgraupo4 Conv enael-Lot N be a nma Subarp o (o A (x1) cN N CN iu) N) _C 2N N omQ 4(_ we ondud hat- 2N = N
- 11. Page No. Topic.. Date....n.. ss) heatm-- The inkouecHon otwo ormal Sub- graups A a group i anarMal Sb-goup aoLe ond k_be two namal Subgroup o H,K 2 Subqou|2%L HOEu aso a uba10uD 2be any eloment o(q and n be any elumant HAE Hnonmal Subgroup nEH= 2nM eH Simularly t nék 2n C n HAKL HOEisd nrmal Subgroup o lq
- 12. Page No. Topic..ermulaktan..qr.aup. 12 Date. ontomappinq Petmuta Hon 1-Ua one-to-one s2s_ S 02, nLontain n elumen S (a (o) t C03) C 2 a 3 f(on) imap an 02 a3 o) f0) ffos) lOn DeHMtahon 2 3 S=,233 then iB PhmuaHoD pO F7 23 3 2 1L 2 3 2 23 23 23 31 2 /12 3 13 2 2 3 23
- 13. Pago No 1S Topic. Date...s40ssssse Tolalpossible permulaHoo o S u 3L Tola passilb peumutakon onu SetE wih nm ben o elumLk Tolel no aa elumanloang &t eDeg o PeamulaHon menhoned exompl 3(Deqrer o PeTmuka hon)_ Hun D Set Set oC Peu mulaksns LermA ar0up fenmutaan Toup o SqmmeMC qoup CYCLIC feumutah'on L2 S 3 eumulakon u uclic 24 S G 3 C2 43)=f-Cuu Permul lengho yc =4
- 14. Pogo No. 14 Topic. lengln o_Cycle aloelemunl tn uy_ud Ex2Cycl C 3 4 26) upeienk q peumulaHon o deqrie 9 on aSeE S ConAIS Hng o h elumnli L2 3,u, S,61L,9Then Pumutahon wtll be /L34 2 G s 7 6= 3 42 6I S 7 (34 2 lengn o yu C3Peu Peumutahon 4= 23 46S7 T23 4 S6 Cyc C23 6) =9 1engh Pemutahan h L 2 3 /293 3S6 not not cuc 2no ud eam Deliwho -H_pumukahon o h ype 0 92030 7 Coulltd a yic pmuleuhon ora_ucl su wAually denokd by h Symbal (a,Q2 On) skyRide
- 15. Anand Date: Page: Lagnange s hearem_E he nd.eu eacb gubgmup o a Hnil gn0upU_a diHser_o uoaatup [uao= Let H be the Subgnocp o Aailr 9up ancl OCH)En (mn) Suppoil hi,h2 hm be m dishnct mm of H hen Ha aigh Colet of H Cn G. e haH Ha- ha,h2a hanay Ha m dushnd mimbs h 0=hjaL Lwich A aCohaclichC o6 hhj by1gn CancoLI Cuhba Any two dishinc Tght Coselz_ (qare Any olistntt Gu inui Gadp
- 16. Anand Date Pug he Dumbeuo dishnttght CoSel6 o Hto C wil "be_pnia Sen_eQual tot theonion okdisHnct mght cotelh a equal h C=Ho U Ho U Un olG Ha: -lHa,1t Ho21 Ho n he orde aLLanelemat gauhat Th Ord o gToupqraLp i tne numbee a ili QLLmena S ag2o1p i'ninih i Corclincliy
- 17. Anand Date: Page: Contpe Ring A setR Soud lo t Rng R CR _da abelion g1up LR,) 0a emiqaup Dishibuhon la SouhsH K denokd by a.bt)_ =ab*a Catb)C 0-c+} 22 Ring wih Uwk el1 + 0-0.l=q Commutaie Ring fa,beRt) 0.bsba Sek q Tokegeu R=So 1!,t2,T3 R,t)_ao abelion gagp R, ) a CotnRelel emiq1oup i'spibuloo 4w 2(3tu) 2:3t 2.4 V_tommculaliie the Ping Cdcup ng Litn ON X CF does N haI (k) abebon (R, Semugeup pshibui law Ring win Comaulalie
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