Discrete Structure: Algebraic Structure PART 2 complete notes as per syllabus of AKTU, Lucknow

1. Page No.
Topic.ALgehraicSnuckut.LUnikIL.Date..1.0l2.22a
5ubauaupl ASubgraup i 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.. ss e
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.
An eluMLnE_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 lmaupl A_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|he se 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....
***********************
Prowe that 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.... Dat 8
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-qpoup A 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.
onto mappinq
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 alo elemunl 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 hea rem_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 dishntt ght 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 set R 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

18. (18 Anand
Date:
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19. February 2021
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21. February
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