Ta hay chang minh rang can lily tiara f it khi A chay khap Z
chi cho ta ba phin tit phan bit la e, fl, f2. Thht tray ta hay
ley ma thy thita thy y . Chia A cho 3 ta throe A= 3q + r, 0 4 r< 3
r la du cia phap chia A cho 3. Ta 'cat
[At = = •fr, = (fb q irc
Nhung f = e , nen
Vi r lay met trong ba gia tri 0, 1, 2, nen fl la mot trong ba phan to f = e, fl = , ff = 12 . Nhu fly nhdm con A3 sinh ra bat /3 gam ba 'align tu, d6 la e, f va f2 . Nghien mitt
fp ta cd
fi = = e.
Bang If Wan twang tu, ta cuing c6 nhdm con sinh ra bdi 12 gam ha phan tit
e, fl, fr Vay nhdm con sinh ra bdi 12 truing vai nhdm con sinh ra bat /3.
Con fp 14, 15 la nhang chuygn tri nen ta c6 ngay T23 . T 24 . e.
Cling If Wen nhu tren ta talc e, 13 1 la nhdm con sinh ra bai /3 , {e, /4 } la nhdm con sink ra hat 14, {e, 15 } la nhdm con sinh ra bai /5 .
Cu6i sung, vi = e vdi MO A e Z nen nhdm con sinh ra beg e la nhdm con tam uniting {e}.
Nhdm S3 khang doe sinh ra bat met phan to nao cast n6 ca, trey 83 khang plied la xyclic. Con cat nhdm con kg tren cita 53 dell la xyclic.
2) Nhdm Ong cat s6 nguyen Z la xyclic vi cat phan tit obi no la ate bet dm 1, fly 1 la met phgn tit sinh aim Z. Ta cuing cc' thg cot ate phgn tit aim Z la the bet aim -1 cho nen -1 Ming la mat phgn t& sinh cda Z
Ngoai 1 vb. -1 ra, Z khang can phgn tit sinh nao khac. Z la met nhdm xyclic ve ham, can cat nhdm xyclic bong vi du
1) la hau han.
Gia sit X let met nhdm, e la phgn tit truing lap aim X va a la met Win tit aim X. Neu kitting c6 met s6 nguyen during n nao sao cho a" = e thi nhdm con sinh ra bat a la v6 han, vi a2 x d' vai A m p. 'Prong trtiang hap trail lai, goi m la se nguyen during be nhat sao cho am = e, thg thi nhdm con sinh ra bat a :co m phgn tit : = e, a l = a, a2 , , (clang minh trong tu nhu trong vi cht 1)).
Dinh nghia 5. Gia sit a IA met phan tit bat ki cim met
them X va A la nhem con sinh ra bai a. Phan ta a g9i la co cdp ut5 hqn ndu A va han • trong trueng . hop nay khang 06 met
s6 nguyen duang n nao sao cho an = e. Phan tit a goi IA cd cdp in nett A cci cap m ; trong truang hop thy in /a se nguyen duang be nhat sao cho = e.
Mat phan tiictEX co cap 1 khi va chi khi a = e.
3. Nh6m con chugn tic va nhOm thuang
Gia sit A la met nhcim con aim met Mt X, ta hay dinh
nghia quan he trong tap hop X nInt sau : mei x, y E
A, x y ngu va chi ngu fl y E A.
B6 de 1. Quan trong X /2/ mat quan he Wang duang. Chang minh. la phan xa vi f i x = e e A. - la dal sting vi the x I y E A thi y l x = (f l y) 1 E A. Cudi env - la bac cau vi neuz i y va y l z E A thi x I z = (f l y)(y -l z) E A. •
Vol mai phan to x E X, ta Id hieu lip thong duang chla x IA Fe va Id hieu be then cim X gam cac 'Man tit co clang xa vii a chay khdp A la xA, the la x4 = xa I a E A I.
B6 d6 2. 1 = xA.
Chang mink. Truac hat ta hay chdng minh x C xA. GM, sit
y e x. fly x y, tdc la z 1y la met phan tit a nao de dm
A, do d6 y = xa E xA. Bay gib ta chtIng rainh bao ham ngttac lai : xA C x. Mugn vay ta lay met phan to y thy g tha xA.
y cti dang xa vie a la met phfin td nao d6 ciut A. Tit' y = xa,
to suy ra E A, fly x y, tdc la y n
Dinh nghia 6. CO.c be phan xA gpi la Mc lop trai cua ahem con A trong X Thong to, Mc 16p phdi Ax dm A trong X la cac be phan ma the phan tit co dang la ax vie a E A.
Cling nhtt dai vii cac lop trai, ta co thg cluing minh cac lap
phal. cim A la can liqttudng duang theo quan he thong duang : x y ngu NA chi age E A.
Tit cac b6 de 1 Ira 2 ta suy ra
Hg qua. (lid sit x ua y la hai pan ta tiny y caa nhom X thl thi :
(i) xA = yA nen utt chi Mu y e A.
(ii) xA fl yA = 0 Mu va chi Mu x y 0 A.
Tap hop thud:1g dm X tren quan ha tilting dtiong goi la
tap hop finking dm nhdm X tren nhom con A, ki higu la XIA. Caa pilau to caa XIA la cac lap trai xA.
Bay gib ta hay lip dung cac kat qua trail de" nghien ca) quan hg gills) cap cim mgt nhdm hUu hat va cap oda mat nhdm con ena nd. Ts. cd
Dinh Ii 9 (Dinh II Lagranggio). Cap caa mot nhdm X has hgn la bpi caa cap caa moi nhdm con caa no.
Chang mirth. Gift sit X co cap la n va A, mat nhdm con bat
ki aim nd, cd cap la m. Trade hat ta hay cluing mink MO lop
trai xA, x E X, dau cd s6 plias' tit la m. Milan vay ta hay icdt A = {x i , x2 ,..., xm }
va cac phan t6
Cac phiin to nay la phan bigt, vi nth) cd qu i = xx2 °hang han, thi x i = x2 . Dd la tat ca cite plain tit dm lop trai xA.
Nhu vay xA cd m pilau tit Vi X la hitu him nen s6 cam lop trai xA la him han, goi 1 la s6 cac lap trai xA. Do cac lap trai la rbi nhau nen ta cd n = ml.
'Nogg to ta cling cd I la s6 cac !Op pilaf. Ax. n
S6 I cac lop trai xA (hay Cop phiti Ax) goi la chi .96 caa nhdm con A trong X.
VI MO Oat) tit x cua mgt nhdm X sinh ra mgt nhdm con cd cap bang cap cim x, nen
He qua 1. Cap caa mot phan to thy S caa mot nhom Mu hon. X la Mc caa cap clia X.
Vi min plign tit x # e Goa mat nhdm X dela sinh ra mat
nhdm con cd cap Ithimg nh6 hon 2, nen :
Ha qua 2. Moi nhOm hau han co cap nguyen t6 dau ld xyclic ad duce sinh ra ben mat phan td bat ki, khac glean hi
trung lap, cda ;thorn.
Vi du. Ta hay lay nhdm cac phgp thg S3 cd cap 6. Vi du rave 2 oho ta bit the plain tit coa nd ed cap hoac la 1, hoac la 2, hoae la 3, do la nhUng uac etla 6. Bay gib ta hay lay mat nhdm con dm S3 chang han nhdm con A = e, f3 1 Dinh li Lagranggia cho ta bigt s6 the lap trai tha A la. 3. Ta hay thanh lap the lop trai dd. Ta co
eA = fee e, ef3 = f3 1 f1A = {fi e = 11 13 = f2A = { {/fie = 12, 1213 = f5} f3 A = f3 E eA f4A vi [4 e ft A. f5 A = f2 A, vi 15 E 12 A.
Bay gib to hay xet cac Cop phai eim A :
Ae = ee d e, f3 e = 13 1 = Af3 Af t = {eft = ft , 13 11 = f5 ) = Afs Aft = (eft = f2, f3f2 = fs} = Afx Chti y ta cd fi A m Aft , f2 A m Aft .
Tth ye tthang hop tang quat, gia set X la mat nhdm va A la mat nhdm con cita nd. Ta dude tap hop thuong X/A ma cac phan to la the lap trai xA, x E X. 'Pa mut% trang N cho XIA
mat phdp toan dg nd tra thanh mat nhdm. Nhung ta lai mugn phdp Oen de khang phai la thyg ma suy ra tic phdp Wan cim
X, au the ta mu6n cho trung ging via cap (xA, yA) lap trai xyA, thg thi thong ling dd co phai la mat anh xa tit X/A x X/A clan X/A khang ? De tra lei eau hai do, ta 'cat vi du tram Chang
han ta cho tuong dug voi cap (ft A , f1 A) lop 4A. Vi = Jr, ,
nen ffel. = f2 A. Mat khdc vi f1 A = f4 A, nen cling vitt quy tax Wong ling ta c6 lop tral eA = 124 A Wong Ong vet cap
(f4 A, f4 A) = (ft A , f1 A). Nhung f2 A m eA, vay quy tem Wen
khang cho ta mot anh xa. Mu6n quy Sc 416 cho ta met anh
xa, ta can an Binh cho nhdm con A eta X milt then kiOn, ma
ta got la diet' hen chutin tic.
Djnh nghia 7. Mat 'them con A rem mot nhdm X gyi la chudn Mc thu oh. chinaux l axEAvoimoictEAvaxEX.
Djnh 0 10. NeU A la mat nhdm con chudn Mc cact mat nhdm X, thi :
(i) Quy Mc cho tuong ring v6i cap (xA, yA) lop trai xyA la mat anh m tit X/A x X/A din XIA
XIA cling vol phep town hai mg& (xA, yA) •—n xyA
la mot nhdm, goi Ia nhdm thuong elm X trOn A.
Cluing mink. (i) D4 cluing minh quy tac dd la met anh xa thi ta phai chting minh rang nail x i A = xA va yi A = yA thi x i y i A = xyA, hay theo he qua oda bit de 2, thu x 1x1 e A Ira
y -1 y i e A thi (xy) 1 (x i y i ) = y 1 x 1 x i y i e A. Bat x = a, ta co a E A theo gia thiat. Xet tich y 1 x 1 x1 y1 = y-1 ayi . Ta
c6 the vier : y 1 ayi = (fl ay)(y -1 Nhung y -1 ay E A vi A la chuan the va y lyi e A theo gia thiat, vay (y -1 ay)(y -l y i ) E A, the la (xy) (x i yi ) E A. Ta ki hiOu cal hap thanh eta xA
va yA la xA.yA.
(ii) D4 thit nghtem tinh chat kat hop eta phep than hat ngOi
trong XIA, ta hay lay ha phan tit toy y x, y, z cua X The thi (xA.yA)zA = xyzA = xA.(yA.zA).
Do c16 phep toan hai ngai da cho la kat hop.
, Do thd nghiem su ton tai cua met don vi trai, ta hay set lap eA = A, trong d6 e la phan trung lap. Ta cd
eA.xA = exA = xA
ved .M9i lap trai xA e X/A. Vay eA = A la met dam vi trai.
Cugi ding vie m9i xA E X/A, ta cd = x I xA = eA.
Theo dinh li 5, X/A Ming veil phop teen hai ngOi cla cho trong X/A la met nhdm, vii eA lit Olen tit trung lap oh'. x I A la
nghich den cua xA. n
Dinh li sau day cho ta bit met dinh nghia thong during clia nh6m con chugn the
Dinh li 11. GM' sit A Itt mat nhom con Giza mat nhom X. Cdc dieu kien sau dtty /a Wong duang
a) A la Malin Mc. b) xA = Ax vai mai x
Cluing minh. Ta hay chdng minh a) keo theo b). Gia su xa, a E A, la met phan til by y cua xA. Vi A la chugn Mc nen y -I ay E A WO mei y E X. Lay y = x -I , ta cd xax-1 E A. Dat xax I = a' E A, to cd xa = a'x E Ax. Vay xA C Ax. DA° lai gia su ax, a e A, la met 'Man ta thy y mia Ax. Vi A la deign the nen x I ax E A. Dat x 1 ax = a' E A, ta cd ax = xa' E xA. Vay Ax C xA. Kgt ban xA = Ax.
Bay gib ta clung minh b) keo theo a). GM sit a va x la hai
phan tit thy y theo thil tu dm A va X. Ta cd ax E Ax. Theo
gia thigt Ax = xA, nen ax e xA, tile la ax = xa' voi a' la met phan to nao dd dia A. Ta suy ra x -I ax = a' e A. Theo dinh
nghia 7, A la chugn Sc. n
Do dinh li tram tit gib ngu A la chugn the thi ta khang phan Net lap trai, lap phai oils. A va goi met lap trai (hay met lap
Vi dt.t. 1) Trong met nhdm X, the nhdm con tam thtiOng {e} va X la chugn the