x'(xx") = x'e, vay (x'x)x" = x'e
hay
tile IA
'Prong trdong hop phep toan hai ngOi. cua nhdm Id hieu bang datt (dau. Ong +), thi phan to c161 xdng duy nhat eda x ki datt (dau. Ong +), thi phan to c161 xdng duy nhat eda x ki higu IA x-I (-x) va con ged la nghich dcio cua x (d61 eim TU dinh nghia cim phan tit nghich dao (phan to dal) ta tuiuglUch
(s 1 ) 1 = x, ( -( -x) = x). Ngu nhdm la aben va phop town cua
nhdm ki hieu bang dgu . (Mu +) thi phtin to xy-1 =
(x + (-y) = (-y) + x) ki hieu la xly (x - y) ya goi la thitong caa x teen y (hieu the x va y).
Dinh li 2 Gnat gian uerc). Trong mat nhom, thing thzic xy -Y xz (yx = zx) keo they clang that y = z.
Nhu vey, ta bau trong met nhdm luet gian be thue hien dirge yid moi Agri tit
Ch'ing mirth. Nhan ben trai hai ve cda tang xy = xz
yeti x-1 , ta cd
t (xy) = x 1 (xz)
hay (x 111 x)y = (x l x)z
hay ey = ez
tilt la y = z. n
Dinh li 3. Trong mat nhom, plutong trinh ax = b (xa = b) to nghiem duy aka x = a lb Ix = ba l 1.
Chang mirth. Ta thtly ngay gia tri x t= a -lb la nghiem ena
phvong trinh vi a(a -15) = (aa 1 )1, = eb = b. Dd la nghiem duy
nhat, vi ngu c la met nghiem cita phttong trinh nghia la
ac = ax = b, ta guy ra c = x theo het glen ado. Chung
minh toting tv cho phuong trinh xa = b. n
Dinh li 4. Thong mat nhom ta to (xy) 1 = y lx 1 udi x, y C¢ hai pit hi bat ki caa nhom.
Cluing mink. Ta co (xy)(y -lx -i ) = x(yy lx 1 = xx t = e lx 1 )(xy) = y 1 (x lx)y = y ly = e c I (xy) 1 = y lx -1 . n 45
Dinh li nay m6 rang the khac cho tich m re thy y nhan
to (x ix2 xn ) -1 = x - 1 x -1 x -1 Mc Met
(ar ) l
vai ince se tii nhien khac 0 ; nguai ta quy the vigt Oen to do
dual dang a', va mat khdc dat a" - = e, nhu thy ta da loan thanh vie xac dinh a cho m9i se nguyen A. 1n van cd (dec gia
hay chting minh diet' do)
al ai = a?+”, (a))/' =
vek moi A, p E Z. Clni g : ki hien lit dung cho phep nhan, trong
triamg hop phep Ong ta vigt Aa.
DU6i day ta hay dua ra met s6 dinh nghia thong diking cha nhdm.
Dinh li 5. hide nem nhom X ta met nhom ndu va chi nits hai &Su kien sau duce than man :
1 ° . X co mat don vi &di e.
2° . Ved moi x E X, co mOt x' E X scro.cho .x'x = e.
Cluing mink. Hign nhien hai dieu kith tract la can, ta hay Chung minh cluing la du. Gia sit cd cac Mau kith 1 ° va 2° .
Lay met phein tit thy y x E X. Theo 2 ° cd met ,x' E X sao cho x'x = e. Van theo 2 0 cd met x" E X sao cho x"x' = e.
Ta co
xx' = = x"x'xx' = x"ex' = x",(' = a Mat khac, ta lai ea
xe = x(x'x) = (xx')x = ex = x.
Ket luan e 14 don vi caa X va x' la nghich dais tha x. fly X la met nhdm. n
'Pa thing ditgc met dinh li Wang tit nth ta thay the' don vi trail bang don vi phai va phan to x' trong dieu kith 2 ° dang
Dinh li 6. MN nOa nhom khcic rdng X la mot nhom niu ud chi niu cac phuong trinh ax = b vb. ya = b co nghCkm trong X Si moi a, b E X.
Cluing mink. Dinh Ii 3 oho ta biet dieu Mgt la can ta hay chiing minh no la du. Vi X la khac rang nen cd mat plign
to a E X. Gia sV e la mot nghiem eta pinning trinh ya = a. 'hi hay cluing minh e la don vi trai eta X. Gia sic b IA mat phgn to toy y eta X. Goi c la mat nghtem eta phuang trinh ax = b. 'Pa cd
eb = e(ac) = (ea)c = ac = b.
Vey e la mat don vi trai caa X. Bay gib ta hay cluing minh vol moi b E X, cd b' E X sao cho b'b = e. Mutin fly ta xet phuong trinh yb = e. Sat ten tai nghigm eta phuong trinh nay cho to b'. Nhu \ray the dieu Alan 1 0 va eta dinh II 5 dttoc them man, ta cd X IA mot nhOm. n
2. Nh6m con
Gia su A la mat 60 Olen eta mat nhdm X, ta cd thg ea xy E A voi moi x, y E A, tile la A la mat bb phan an dinh
eta X (§1, 1, dinh nghia 2). Nhu fly phep toan hai ng6i trong X earn sinh mat Olen toan hai ngid trong A. Nhu A cling yen phdp toan cam sinh la mat nhdm, thi ta It A la mat nhom con eta nhdm X. Chang han ta cd 140 Olen the s6 nguyen Z lit
mat nhdm con eta. nhdm Ong the s6 him ti Q. Ta chn y bio phan {-1, 1} tuy lap thanh mat nit deg vOi phep nhan, nhung khang phai la mat nhdm con eta nhcim cOng cat s6 nguyen Z.
Dinh nghia 2. Mat he Olen an dinh A eta mat nhdm X la mat nhom con eta X neu A ding vai phdp toan earn sinh IA
mat nhdm.
Dinh ii 7. Mat 60 phOn A dm mot nixes X la mOt nhom con czict X niu tta chi niu cac diku /den sau day than man :
1) Vat moi x, y E A, xy E A.
2) e E A, Si e lit phan to trong ldp cita X. 3) Vai moi x E A, x 1 E A.
Cluing mink. At cd. Gia sit A lit met nhdm con dm X. Theo
dinh nghia nhdm con, ta cd ngay 1 0 . Goi e' m phan to trung lap oda nhdm A. Ts co
e'a = a yea mai a E A
Mat khac ta Ming cd
ea = a vi e la pilau to trung lap cu. X.
Do dd
e'a = ea
hay e' = e
vi luat gian dee thoc hien &toe vdi moi phtin td cim nhdm X.
Vity ta cd 2 °. Cue' cang goi x' lit nghich dao trong nhdm A cua
mat pilau td x E A. Vay ta cd
xx = e = 1 1x.
Thoc hien Mat gian doe, ta dime x' = x i . Dieu nay thong
minh 3° .
DA. Gia sit A la mat be phan cim X thda man the diet; kien
da cho. VUi 1 °, A lit mot nita nhdm, vi phep toan hai ngti da
ciao trong X la ket hop. 2 ° va 3 ° not len nda nhdm A co phtin
to trung lap va moi pilau tit dm A co nghich dito trong A. Vay A la met nhdm, do (Id A la met nhdm con cda X. n
H9 qua. Gid set A la nay 60 plain khac r6ng cart met nhom X Cac &On kin sau clay la Wang daring :
a) A la mot &Wm con &La X.
b) Vdi moi x, y E A, xy e A Mt E A.
c) Vdi moi x, y E A, xy t E A.
Cluing mink. Theo dinh li 7 ta cd ngay a) keo theo b). Hien
nhien b) keo theo c). Ta chdng minh c) keo theo a). VI A x 0 nen cci mat x E A. Theo c) ta cde=xx l eA. Vay diet' Men 2 ° cda dinh li 7 them. man. GM sd x E A. Van theo c) ta co
x -I = ex I E A. Ta duge dieu kien 3 ° can dinh li 7. Duel ding, gia sa x, y E A. Vi y -I E A, nen c) cho x(y -I ) -1 = xy E A. Nhu fly dieu kien 1 ° Ma dinh Li 7 them man. Do do . A la met
nhdm con eim X. n
VI du. 1) Be phan mZ gem the sic nguyen la bei dm met
se nguyen m cho trubc la mat nhdm con etla nhdm Ong the se nguyen Z. That fly, tip dung diet' kien c) Ma he qua trail,
ta ce mZ let nhdm con vi mx - my = m(x - y) E mZ, tUc la
hieu ctla hai phan to thuec mZ lai thfle mZ.
2) 'Prong met nhdm X, be phan A gem the luy thita n it eim met Oen tit a E X la met nhdm con ctla X That fly, as (eta) = al a = a1-8 E A. Dec biet the bOi Ma met se
nguyen lit met nhdm con Ma nhdm Ong the so` nguyen Z. Nhd fly, vi du 1) chi la met trubng hop ciac biet cua vi du 2).
3) BO phan { e 1 chi gem c6 phan tit trung lap va 130 Olen X la hai nhdm con dm met nhdm X. Nglibi ta goi chting la the ?them con tam thabng cart X.
Dinh Ii 8. Ciao can mat ho btft ki nhung nhdm con can mat thorn X la mat nham con caa X.
Chang mink. X& mot hg Mt ki (A,), e I nhfing nhdm con caa X va goi A la giao cua cluing. Trude Mt ta ce A s 0, vi phiin tit trung lap e caa X thubc A veil 1110i a e I, do de e E A. Bay gib ta lay hai phan to bet Id x, y E A. Vi x, y E A,
nen x, y E Act voi MOI a E I. Vi the 21a la nhung nhdm con
nen xy 1 E A veri 1110i a E 1, do de xy 1 e A. He qua caa
dinh Ii 7 cho ta Net A la nhdm con caa X. n
Gift sit U la met be phan Ma met nhdm X. The thi U chtla trong it nhat met nhdm con cat*. X, cu the X. Theo dinh Ii 8, giao A cos. tat ea the nhdm con Mut X chtla U la met nhdm con cart X dada U. Do la nhdm con be nhat cita X chtla U
(quan he thd to la quan he bao ham).
Dinh nghia 3. Gia. sit U la met be phan caa mot nhdm X. Nhom con A be nhat elm X chile. U goi la nhOnt con sinh ra
bei U. 'Prong traimg hop A = IC, ta ndi rang U la met he sinh tha X vA X aye sinh ra bed U.
Neu U = { a ), a a X, thi de ding they rang ahem con A sinh ra bid U cd the phin tit la the luy thaa all, A e Z. That
ny, then v1 do 2) be phan gem eac•lay thita ce l la ma nhdm
con coil X. Han naa, d6 lit nhdm can be nhtit tha X chtta {a} vi m9i nhdm con can X ehda { a thi deo chda the lily thiia
tha a Ngubi ta goi A lit nhdm con sinh ra bit a.
Dinh nghia 4. MOt tilt X goi la xyclie neu v.& chi Su X
dam Binh ra b&i mOt phi]) tit a E X. Phan tit a goi la mOt phan to sinh tha X.
Nhu tray ma nhdm IL la xyclie nett vb. chi Su the phin tit cAa n6 la cac lay thiia a1 , A a Z, caa met phtin tit a a X.
Vi dy„ 1) X& nhdm the phep the 83 ma the phin tit la
e = fy = 1 2 (1 2 1 2 (3 2 3 3) 1) ' s. 11 13 3 = (2 1 2 3 (1 3 3 1.)' 2 4. ) = (3 1 2 1 3 2) e " 3 = (1. 2 2 1 3 3)
lb hay nghien cdu the phan tit can nhdm con shah ra bbi ft , c6