trong dd log x la logarit eo so 10 So. x VI log(xy) = log x + log y,
nen log la mat clang Su. Bang eau nay eon la mat song anh nen la mat (tang Su.
4) Gilt S A la mat nhdm con chuan tac cila mat nhdm X.
Anh xa
h : X XIA x h(x) = xA
la mat dang cau to nhdm X den nhdm thuong Th4t fly, h(xy) = xyA = xA. yA = h(x) h(y). Eking Su nay can la mat toan eau, goi la toes °du Minh tat.
5) Gal th X vit Y la hai nhdm thy g, anh xa
Y
x e,
Vol e lit phat tit trung lap caa Y, la mat clang eau goi la Bong eau tam thetemg.
6) Nall f : X —> Y la mot clang cau to nhdm Alf clan nhdm Y
thi anh xa nguoc : 12. cfmg la mat Bang eau That fly, ta cd mat song anh, ta chi con cluing mink f I la mat
d6ng cau. Gia th y, y' la hai phan tif thy g eim Y. Dat x = r(y), x' = f- 1 ta ed f(x) = y va f(x') = y'. Vi f la mat clang cau nen f(xx') = 1(x) f(x) = yy'. Do do (1 (yy) xi = r' (y)r'(y?). fly r" la mat dOng cau. Ta Joao
hai nhdm X va Y la clang eau vat nhau, va ta vial X = Y, nau
co mat clang eau tit nhdm nay ten nhdm kia.
Dinh nghia 9. Gia th f : Y la mat (tong Su to nhdm
X clan nhdm Y, the phtin trung lap ciaa X va Y duoc ki hieu thee the to la ex va ey . Ta ki hiau
Imf = f(X)
Keel x E f(x) = ey = j i (e y)
va goi Imf la dnh cua clang cau f, Kerf la hat nhdn aim dOng
eau f
Sau day, ta se dtta ra mat s6 tinh chat eda tong eau.
Dinh li 12. Gid szi X, Y, Z la nhung nhOm vez f : X —> Y .vd g : Y Z Ca nhung clang c6u. The thi anh xe tick
gf : X Z
tong la met dong cau. Dec Wet tech cda hai clang eau Id met demg colt.
Cluing minh. Gia oft a, b la hai phan td thy y caa nhan X. 'Pa cd, do f va g la nhung &Ong cal; gf(ab) = g(f(ab)) = g(f(a) fib)) = g(f(a)) ef(b)) = gf(a)g f(b). n
Dinh Ii 13. GM sit f : X Y Id met ang cdu tit met niter) X den met niter?) Y. The thi :
f(ex) = ey
(ii) f(x I ) = ff(x),F I me mai x E X.
Cluing minh. (i) Gia sit x la ma phan to thy 3 am X.
Ta cd f(ex)f(x) = ftexX) = f(x) = ey/c(x)
Vay Be x) f(x) = ey f(x) hay flex) = ey sau khi that hiOn luat
OM) vac.
i) Ta cd
f(x -1 )/(x) = f(x-lx) = f(ex) = ey = B(x)1 (x).
soy ra
fix = Kx/1 -1 n
Dinh li 14. Gici silt f : X Y Id met ang cclu tit met nhent X din met nhOm Y, A la met nhom con am X vet B la met nhOm con chudn Mc cad Y. The thi :
(I) f(A) Id met nhom con elm Y
(ii) r 1 (B) let met nhom con chuein Mc ata X.
Chang mink. (i) Tratdc ha f(A) x 4) vi A la ma nhOm con nOn ex E A, do dd e y = f(ex) e f(A). Ta hay lay hai phan hi
thy 3 : y, yi E BA). Vi y, y i E f(A), nen cd x, xt E A sao cho
= f(x)f(x1 1 ) = f(xx i-1 ). Vi A la nhdm con nen xxT I E A, do dd yy1 1 = f(xx1 1 ) e f(A). Ta suy ra f(A) la nhdm con cim Y.
f1 # 0, vi B la nhdm con nen ey E B, do do
1(ex) = e y E B, ta suy ra ex e f 1 (B). Bay gip; ta Idy hai phAn
tit thy y. x, x 1 e fl (B), ta cluing minh xx.T 1 E f l (B). MuO'n
vay ta 'Mt f(xx i i ). ah co f(xx i l ) = f(x)f(x 11) = f(x)f(x i ) -1 . Nhung f(x), f(x 1 ) E B va f(x)f(x 1) -1 E B vi B IA nhdm con. Vey f(xxT 1 ) e B, tdc IA xxi 1 E f I (B), do dd rl (B) IA nhdm con
MIA X. Cu6i cimg ta chiing minh nd IA chudn tac Muen fly gia sit a e [ 1 (B) va x E X. Xet
f(x -l ax) = f(x I) f(a)f(x) = f(x) -1 f(a)f(x) E B vi f(a) e B va B
la chudn Sc. Do dci x 1 ax E ft (B) v6i moi a e r1 (B) va moi x E X. Vey f1 (B ) chudn Sc. n
Tit dinh h. 14 ta cd he qua tat khac
H5 qua. Gid szi f : X —. Y la mot Ong ceiu tit met nhom X den mot nhom Y. The thi Imf mot nhom con cart ,Y pa Kerf la mat nhom con chudn tar ceta X.
Dinh li 15. Gid sti f : X —.le la mot dung edu tit met Ahem X deit mot nhom Y The thi :
(i) f lit mot town anh neu yet chi niu Imf = Y. (ii) f la mot don anti neu utt chi /Mu Kerf = ex}. Ch'ing mink. (i) Suy ra tit dinh nghla caa town anh.
(ii) Gia su f la mat don anh. Voi mei phan to y E Y co
nhieu nhdt mot phAn tit x sao cho f(x) = y. Vey Kerf = { c}. Dad lai gia su Kerf = Xet hai phdn tit.x, x 1 e X sao cho f(x) = f(x 1 ). 'llt suy ra f(x) f(x 1 ) 1 = ey. Nhung
f(x) f(x) 1 = f(x) fix 1 ) I = f(xx 1-1 ). fly f( ) = ey,