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ĐạI Số ĐạI CƯƠNG








R
Im
R/Ker

huế 2007

B
`
AI T
ˆ
A
.
P CHU
.
O
.
NG I – NH
´
OM
1.
Trˆen tˆa
.
pho
.
.
p Q c´ac sˆo

´
h˜u
.
utı
’
, x´et ph´ep to´an ∗ x´ac d¯i
.
nh nhu
.
sau:
∀a, b ∈ Q,a∗ b = a + b + ab.
a) Q c`ung ph´ep to´an ∗ c´o pha
’
il`amˆo
.
t nh´om khˆong? Ta
.
i sao?
b) Ch´u
.
ng minh Q \{−1} c`ung ph´ep to´an ∗ ta
.
o th`anh mˆo
.
t nh´om.
2.
Ch´u
.
ng minh tˆa
.

pho
.
.
p G = {(a, b) | a, b ∈ R,b=0} c`ung ph´ep to´an k´y hiˆe
.
u
nhˆan
∀(a, b), (a

,b

) ∈ G, (a, b)(a

,b

)=(ab

+ a

,bb

)
l`a mˆo
.
t nh´om v`a H = {(a, 1) | a ∈ R} l`a mˆo
.
t nh´om con cu
’
a G.
3.

Cho G = R
∗
× R (v´o
.
i R l`a tˆa
.
pho
.
.
p c´ac sˆo
´
thu
.
.
cv`aR
∗
= R\{0})v`a∗ l`a ph´ep
to´an trˆen G x´ac d¯i
.
nh bo
.
’
i:
(x, y) ∗ (x

,y

)=(xx

,xy


+
y
x

).
a) Ch´u
.
ng minh r˘a
`
ng (G,∗) l`a mˆo
.
t nh´om.
b) Ch´u
.
ng to
’
r˘a
`
ng v´o
.
ibˆa
´
tk`yk ∈ R,tˆa
.
pho
.
.
p H
k

= {(x, k(x−
1
x
)) | x ∈ R
∗
}
l`a mˆo
.
t nh´om con giao ho´an cu
’
a G.
c) H˜ay x´ac d¯i
.
nh tˆam Z(G)cu
’
a G.
4.
Trˆen tˆa
.
pho
.
.
p G =[0, 1) = {x ∈ R | 0 ≤ x<1}, x´et ph´ep to´an ⊕ nhu
.
sau:
∀x, y ∈ G, x⊕ y = x + y − [x + y](o
.
’
d¯ˆay [x + y] l`a phˆa
`

n nguyˆen cu
’
a x + y).
a) Ch´u
.
ng minh (G,⊕) l`a mˆo
.
t nh´om abel.
b) Ch´u
.
ng minh r˘a
`
ng ´anh xa
.
f : G −→ C
∗
x´ac d¯i
.
nh bo
.
’
i f (x) = cos 2πx +
i sin 2πx, l`a mˆo
.
td¯ˆo
`
ng cˆa
´
u nh´om, trong d¯´o C
∗

l`a nh´om nhˆan c´ac sˆo
´
ph´u
.
c kh´ac 0.
5.
Ch´u
.
ng minh r˘a
`
ng mˆo
.
t nh´om m`a khˆong c´o nh´om con thu
.
.
csu
.
.
l`a nh´om d¯o
.
n
vi
.
ho˘a
.
c l`a nh´om cyclic c´o cˆa
´
p nguyˆen tˆo
´
.

6.
Cho G l`a mˆo
.
t nh´om v`a H l`a mˆo
.
t nh´om con chuˆa
’
nt˘a
´
ccu
’
a G sao cho H ⊂
Z(G). Ch´u
.
ng minh r˘a
`
ng nˆe
´
u G/H l`a mˆo
.
t nh´om cyclic th`ı G l`a nh´om abel.
7.
Cho G l`a mˆo
.
t nh´om nhˆan v`a H l`a mˆo
.
t nh´om con cu
’
a G.Ch´u
.

ng minh:
a) Nˆe
´
u[G : H]=2th`ıHG.
b) Nˆe
´
u HGv`a [G : H]=m th`ı a
m
∈ H, ∀a ∈ G.
Typeset by A
M
S-T
E
X
8.
Cho G l`a mˆo
.
t nh´om nhˆan, A v`a B l`a hai nh´om con cu
’
a G.K´yhiˆe
.
u:
AB = {ab | a ∈ A v`a b ∈ B},BA= {ba | b ∈ B v`a a ∈ A}.
Ch´u
.
ng minh r˘a
`
ng AB l`a mˆo
.
t nh´om con cu

’
a G khi v`a chı
’
khi AB = BA.
9.
Cho G l`a mˆo
.
t nh´om, A,B,C l`a c´ac nh´om con cu
’
a G.Ch´u
.
ng minh:
a) A ∩ B l`a mˆo
.
t nh´om con cu
’
a G.
b) A ∪ B l`a nh´om con cu
’
a G khi v`a chı
’
khi A ⊂ B ho˘a
.
c B ⊂ A.
c) Nˆe
´
u C ⊂ A ∪ B th`ı C ⊂ A ho˘a
.
c C ⊂ B.
10.

Cho G l`a mˆo
.
t nh´om nhˆan c´o t´ınh chˆa
´
t: ∀x ∈ G, x
2
= 1, v´o
.
i 1 l`a phˆa
`
ntu
.
’
trung ho`a cu
’
a nh´om G.Ch´u
.
ng to
’
r˘a
`
ng:
a) G l`a mˆo
.
t nh´om aben.
b) Nˆe
´
u G l`a nh´om h˜u
.
uha

.
n th`ı tˆo
`
nta
.
isˆo
´
tu
.
.
nhiˆen n sao cho sˆo
´
phˆa
`
ntu
.
’
cu
’
a
nh´om G b˘a
`
ng 2
n
.
11.
Cho G l`a mˆo
.
t nh´om v`a A,B,C,Kl`a c´ac nh´om con cu
’

a G.Ch´u
.
ng minh
r˘a
`
ng:
a) Nˆe
´
u A ⊂ C th`ı AB ∩ C = A(B ∩ C). (Lu
.
u´yr˘a
`
ng AB khˆong nhˆa
´
t thiˆe
´
t
l`a mˆo
.
t nh´om con cu
’
a G.)
b) Nˆe
´
u A ⊂ B, A∩ K = B ∩ K v`a AK = BK th`ı A = B.
12.
a) X´et tru
.
`o
.

ng Z
13
c´ac sˆo
´
nguyˆen mˆod¯ulˆo 13. H˜ay lˆa
.
pba
’
ng nhˆan cu
’
a Z
13
.
Ch´u
.
ng to
’
r˘a
`
ng Z
∗
13
= Z
13
\{0} l`a mˆo
.
t nh´om cyclic.
b) X´et tru
.
`o

.
ng R c´ac sˆo
´
thu
.
.
c. Khi d¯´o R
∗
= R \{0} c´o pha
’
i l`a mˆo
.
t nh´om
cyclic khˆong?
13.
Trong nh´om nhˆan C
∗
c´ac sˆo
´
ph´u
.
c kh´ac khˆong, h˜ay x´ac d¯i
.
nh nh´om con
cyclic sinh bo
.
’
i phˆa
`
ntu

.
’
x ∈ C
∗
, trong d¯´o
a) x = −
√
2
2
+
√
2
2
i,
b) x = cos
4π
7
+ i sin
4π
7
.
14.
Cho S
3
l`a tˆa
.
pho
.
.
ptˆa

´
tca
’
c´ac ho´an vi
.
cu
’
atˆa
.
pho
.
.
p {1, 2, 3}.
a) H˜ay lˆa
.
pba
’
ng nhˆan cu
’
a S
3
,ch´u
.
ng to
’
S
3
l`a mˆo
.
t nh´om.

b) T`ım tˆa
´
tca
’
c´ac nh´om con chuˆa
’
nt˘a
´
ccu
’
a S
3
.
c) Cho G
1
v`a G
2
l`a hai nh´om c´o cˆa
´
plˆa
`
nlu
.
o
.
.
t l`a 24 v`a 30. Cho G
3
l`a nh´om
khˆong giao ho´an v`a l`a a

’
nh d¯ˆo
`
ng cˆa
´
ucu
’
aca
’
G
1
v`a G
2
. Mˆo ta
’
nh´om G
3
(qua
ph´ep d¯˘a
’
ng cˆa
´
u).
15.
X´et nh´om Q c´ac sˆo
´
h˜u
.
utı
’

v´o
.
i ph´ep cˆo
.
ng thˆong thu
.
`o
.
ng. Ch´u
.
ng minh r˘a
`
ng:
3
a) Q khˆong l`a nh´om cyclic;
b) Q/Z c´o d¯˘a
’
ng cˆa
´
uv´o
.
i Q khˆong?
16.
K´y hiˆe
.
u H =


mb
0 1


∈ GL(2, Z
7
) | m, b ∈ Z
7
,m= ±1

, trong d¯´o
GL(2, Z
7
) l`a nh´om nhˆan c´ac ma trˆa
.
n vuˆong cˆa
´
p2kha
’
nghi
.
ch lˆa
´
yhˆe
.
sˆo
´
trˆen
tru
.
`o
.
ng Z

7
c´ac sˆo
´
nguyˆen mˆod¯ulˆo 7. Ch´u
.
ng minh r˘a
`
ng:
a) H l`a nh´om con cu
’
a nh´om GL(2, Z
7
) c´o 14 phˆa
`
ntu
.
’
.
b) Mo
.
i phˆa
`
ntu
.
’
cu
’
a H c´o thˆe
’
viˆe

´
td¯u
.
o
.
.
c duy nhˆa
´
tdu
.
´o
.
ida
.
ng A
i
B
j
, trong
d¯ ´o 0 ≤ i<7, 0 ≤ j<2v`aA =

1 1
0 1

,B=

−
1 0
0 1


.
17.
Cho G l`a nh´om nhˆan d¯u
.
o
.
.
c sinh bo
.
’
i hai phˆa
`
ntu
.
’
x v`a y v´o
.
i c´ac quan hˆe
.
:
x
3
= y
2
=(xy)
2
=1.
a) X´ac d¯i
.
nh c´ac phˆa

`
ntu
.
’
cu
’
a nh´om G v`a lˆa
.
pba
’
ng nhˆan cu
’
a G.
b) T`ım tˆa
´
tca
’
c´ac nh´om con cu
’
a nh´om G.
18.
Cho G l`a nh´om v´o
.
i ph´ep nhˆan ma trˆa
.
n, d¯u
.
o
.
.

c sinh bo
.
’
i hai ma trˆa
.
nhˆe
.
sˆo
´
thu
.
.
c A =

01
−10

v`a B =

01
10

.
a) X´ac d¯i
.
nh c´ac phˆa
`
ntu
.
’

cu
’
a nh´om G.
b) T`ım tˆa
´
tca
’
c´ac nh´om con cu
’
a G.
19.
Cho G l`a mˆo
.
t nh´om nhˆan v`a n l`a mˆo
.
tsˆo
´
nguyˆen du
.
o
.
ng sao cho
f
n
: G −→ G : x → x
n
l`a mˆo
.
t to`an cˆa
´

u nh´om. Ch´u
.
ng minh r˘a
`
ng:
a) x
n−1
y = yx
n−1
, ∀x, y ∈ G.
b) V´o
.
i n =3,Gl`a mˆo
.
t nh´om aben.
20.
Cho G l`a mˆo
.
t nh´om sao cho c´o mˆo
.
tsˆo
´
nguyˆen n>1 thoa
’
m˜an (xy)
n
=
x
n
y

n
, ∀x, y ∈ G.Go
.
i G
(n)
= {x
n
| x ∈ G} v`a G
(n)
= {x ∈ G | x
n
=1}.
Ch´u
.
ng minh r˘a
`
ng:
a) G
(n)
Gv`a G
(n)
G.
b) G/G
(n)
∼
=
G
(n)
.
21.

a) Cho H l`a nh´om con cu
’
a nh´om nhˆan C
∗
= C \{0} gˆo
`
m c´ac sˆo
´
ph´u
.
cc´o
mˆod¯un b˘a
`
ng 1, R
∗
+
l`a nh´om nhˆan gˆo
`
m c´ac sˆo
´
thu
.
.
cdu
.
o
.
ng. Ch´u
.
ng minh r˘a

`
ng
C
∗
/H d¯ ˘a
’
ng cˆa
´
uv´o
.
i R
∗
+
.
4
b) Cho f : G −→ H l`a mˆo
.
t to`an cˆa
´
u nh´om, M l`a mˆo
.
t nh´om con chuˆa
’
nt˘a
´
c
cu
’
a H , N = f
−1

(M). Ch´u
.
ng minh r˘a
`
ng N l`a nh´om con chuˆa
’
nt˘a
´
ccu
’
a G v`a
G/N d¯ ˘a
’
ng cˆa
´
uv´o
.
i H/M.
22.
Ch´u
.
ng minh r˘a
`
ng:
a) Nˆe
´
u G l`a nh´om cyclic th`ı Aut(G) l`a nh´om aben.
b) Nˆe
´
u G l`a nh´om cyclic cˆa

´
p p nguyˆen tˆo
´
th`ı Aut(G) l`a cyclic cˆa
´
p p − 1.
23.
Cho f : G −→ K l`a mˆo
.
td¯ˆo
`
ng cˆa
´
u nh´om. Ch´u
.
ng minh r˘a
`
ng:
a) Nˆe
´
ucˆa
´
pcu
’
a G l`a h˜u
.
uha
.
nth`ıcˆa
´

pcu
’
a f(G) chia hˆe
´
tcˆa
´
pcu
’
a G.
b) Nˆe
´
u H l`a nh´om con c´o chı
’
sˆo
´
n trong G, Kerf ⊂ H v`a f l`a to`an cˆa
´
uth`ı
f(H) c´o chı
’
sˆo
´
n trong K.
24.
Cho G l`a mˆo
.
t nh´om, C
g
: G −→ G l`a ´anh xa
.

v´o
.
i g ∈ G x´ac d¯i
.
nh bo
.
’
i C
g
(x)=
gxg
−1
.Go
.
i Aut(G)={f : G −→ G | f l`a d¯˘a
’
ng cˆa
´
u } , Inn(G)={C
g
| g ∈ G}.
Ch´u
.
ng to
’
r˘a
`
ng:
a) C
g

l`a mˆo
.
ttu
.
.
d¯ ˘a
’
ng cˆa
´
u, Aut(G)l`amˆo
.
t nh´om v´o
.
i ph´ep to´an ho
.
.
p th`anh
v`a Inn(G)l`amˆo
.
t nh´om con chuˆa
’
nt˘a
´
ccu
’
a Aut(G).
b) Z(G)={a ∈ G | ax = xa , ∀x ∈ G} l`a mˆo
.
t nh´om con chuˆa
’

nt˘a
´
ccu
’
a G
(go
.
i l`a tˆam cu
’
a nh´om G)v`aG/Z(G)
∼
=
Inn(G).
25.
Cho G l`a mˆo
.
t nh´om, v´o
.
i x, y ∈ G,k´yhiˆe
.
u[x, y]=x
−1
y
−1
xy (go
.
i l`a giao
ho´an tu
.
’

cu
’
a x v`a y). Go
.
i[G, G] l`a nh´om con cu
’
a G sinh ra bo
.
’
itˆa
.
p {[x, y] | x, y ∈
G}.Ch´u
.
ng minh r˘a
`
ng:
a) [G, G] l`a nh´om con chuˆa
’
nt˘a
´
c nho
’
nhˆa
´
tcu
’
a G sao cho G/[G, G] l`a aben.
b) [xy, z]=y
−1

[x, z]y[y, z], ∀x,y,z∈ G;
c) Nˆe
´
u[G, G] ⊂ Z(G) (tˆam cu
’
a G) th`ı v´o
.
i a ∈ G, ´anh xa
.
f : G −→ G x´ac
d¯ i
.
nh bo
.
’
i f(x)=[x, a] l`a mˆo
.
td¯ˆo
`
ng cˆa
´
u. T`ım Ker(f).
d) H˜ay x´ac d¯i
.
nh [S
3
,S
3
], trong d¯´o S
3

l`a nh´om c´ac ho´an vi
.
cu
’
a3sˆo
´
1, 2, 3
v`a ch´u
.
ng minh S
3
/S

3
∼
=
Z
2
.
26.
Cho G l`a mˆo
.
t nh´om, a ∈ G l`a phˆa
`
ntu
.
’
c´o cˆa
´
ph˜u

.
uha
.
n n.Ch´u
.
ng minh
r˘a
`
ng v´o
.
imo
.
isˆo
´
nguyˆen du
.
o
.
ng m,cˆa
´
pcu
’
a phˆa
`
ntu
.
’
a
m
l`a

ord (a
m
)=
n
(m, n)
,
trong d¯´o (m, n)l`au
.
´o
.
c chung l´o
.
n nhˆa
´
tcu
’
a m v`a n.
27.
a) Cho G =<g>l`a nh´om cyclic cˆa
´
p 168. T`ım cˆa
´
pcu
’
a phˆa
`
ntu
.
’
g

132
.
b) T`ım tˆa
´
tca
’
c´ac phˆa
`
ntu
.
’
cˆa
´
p 14 cu
’
a nh´om cˆo
.
ng Z
140
c´ac sˆo
´
nguyˆen mˆod¯ulˆo
140.
5
28.
Cho C l`a mˆo
.
t nh´om cyclic sinh bo
.
’

i phˆa
`
ntu
.
’
a.Ch´u
.
ng minh r˘a
`
ng:
a) Nˆe
´
u G l`a mˆo
.
t nh´om con cu
’
a C th`ı G c˜ung l`a mˆo
.
t nh´om cyclic.
b) Nˆe
´
u C l`a nh´om h˜u
.
uha
.
nv`am l`a sˆo
´
nguyˆen du
.
o

.
ng u
.
´o
.
ccu
’
a |C| th`ı tˆo
`
n
ta
.
i duy nhˆa
´
t nh´om con G cu
’
a C sao cho |G| = m.
c) Nˆe
´
u C l`a nh´om vˆo ha
.
nth`ıC c´o 2 phˆa
`
ntu
.
’
sinh l`a a v`a a
−1
.
d) Nˆe

´
u |C| = n th`ı a
m
l`a phˆa
`
ntu
.
’
sinh cu
’
a C khi v`a chı
’
khi m v`a n nguyˆen
tˆo
´
c`ung nhau.
29.
X´et nh´om cˆo
.
ng Q c´ac sˆo
´
h˜u
.
utı
’
.Ch´u
.
ng minh r˘a
`
ng ´anh xa

.
f : Q −→ Q l`a
d¯ ˆo
`
ng cˆa
´
u nh´om khi v`a chı
’
khi tˆo
`
nta
.
i duy nhˆa
´
tmˆo
.
tsˆo
´
a ∈ Q sao cho f(x)=
ax, ∀x ∈ Q.
30.
Cho m v`a n l`a hai sˆo
´
nguyˆen du
.
o
.
ng nguyˆen tˆo
´
c`ung nhau. Ch´u

.
ng minh
d¯ ˘a
’
ng cˆa
´
u nh´om Z
m
× Z
n
∼
=
Z
mn
.T`u
.
d¯´o suy ra Z
3
× Z
2
khˆong d¯˘a
’
ng cˆa
´
uv´o
.
i
nh´om d¯ˆo
´
ix´u

.
ng S
3
.
31.
Cho G l`a mˆo
.
t nh´om, M v`a N l`a hai nh´om con chuˆa
’
nt˘a
´
ccu
’
a G sao cho
G = MN.Ch´u
.
ng minh r˘a
`
ng
G/(M ∩ N )
∼
=
G/M × G/N.
32.
a) Trˆen tˆa
.
pho
.
.
p G = Z

3
,v´o
.
i Z l`a tˆa
.
p c´ac sˆo
´
nguyˆen, x´et ph´ep to´an hai
ngˆoi:
∀(a, b, c), (a

,b

,c

) ∈ G, (a, b, c) ∗ (a

,b

,c

)=(a + a

,b+ b

,c+ c

− ba

).

Ch´u
.
ng minh r˘a
`
ng (G,∗)l`amˆo
.
t nh´om khˆong aben.
b) Trˆen tˆa
.
pho
.
.
p R
2
c´ac c˘a
.
psˆo
´
thu
.
.
c, x´et ph´ep to´an ◦:
(x, y) ◦ (x

,y

)=(xx

− yy


,yx

+ xy

).
(R
2
,◦)c´ol`amˆo
.
t nh´om khˆong?
33.
X´et nh´om R c´ac sˆo
´
thu
.
.
cv´o
.
i ph´ep to´an hai ngˆoi:
∀x, y ∈ R,x∗ y = x

1+y
2
+ y

1+x
2
v`a ´anh xa
.
f : R −→ R x´ac d¯i

.
nh bo
.
’
i f(x)=
e
x
− e
−x
2
.Ch´u
.
ng minh r˘a
`
ng f l`a
mˆo
.
td¯˘a
’
ng cˆa
´
ut`u
.
nh´om (R, +) lˆen nh´om (R,∗).
34.
K´y hiˆe
.
u U
n
l`a nh´om nhˆan c´ac phˆa

`
ntu
.
’
kha
’
nghi
.
ch cu
’
a v`anh Z
n
c´ac sˆo
´
nguyˆen mˆod¯ulˆo n.
a) Lˆa
.
pba
’
ng nhˆan cu
’
a U
22
v`a ch´u
.
ng minh r˘a
`
ng U
22
l`a mˆo

.
t nh´om cyclic.
b) U
24
c´o l`a nh´om cyclic khˆong? V`ı sao?
6
35.
Cho R l`a mˆo
.
t v`anh c´o d¯o
.
nvi
.
1. Trˆen R, x´et ph´ep to´an ∗:
x ∗ y = x + y − xy.
K´y hiˆe
.
u R
∗
= {x ∈ R |∃y ∈ R, x ∗ y = y ∗ x =0}.Ch´u
.
ng minh r˘a
`
ng:
a) (R
∗
,∗)l`amˆo
.
t nh´om.
b) R

∗
∼
=
U(R), v´o
.
i U (R) l`a nh´om c´ac phˆa
`
ntu
.
’
kha
’
nghi
.
ch cu
’
a v`anh R.
36.
X´et nh´om thay phiˆen A
4
(nh´om con cu
’
a nh´om d¯ˆo
´
ix´u
.
ng S
4
gˆo
`

m c´ac ph´ep
thˆe
´
ch˘a
˜
nbˆa
.
c 4).
a) Ch´u
.
ng to
’
r˘a
`
ng nh´om A
4
khˆong c´o nh´om con cˆa
´
p6.
b) T`ım tˆa
´
tca
’
c´ac p-nh´om con Sylow cu
’
a A
4
v´o
.
i p = 2 v`a 3.

37.
Cho G l`a mˆo
.
t nh´om h˜u
.
uha
.
nc´ocˆa
´
pl`ap
r
m,v´o
.
i r ≥ 1v`ap |m.Ch´u
.
ng
minh r˘a
`
ng:
a) Nˆe
´
u P l`a mˆo
.
t p-nh´om con Sylow cu
’
a G v`a H l`a mˆo
.
t p-nh´om sao cho
P ⊂ H ⊂ G th`ı H = P .
b) Nˆe

´
u G chı
’
c´o p-nh´om con Sylow duy nhˆa
´
tl`aP th`ı PG.
38.
Cho G l`a mˆo
.
t nh´om h˜u
.
uha
.
nc´ocˆa
´
pl`apq, trong d¯´o p v`a q l`a hai sˆo
´
nguyˆen
tˆo
´
m`a p<q.Ch´u
.
ng minh r˘a
`
ng:
a) G c´o mˆo
.
t v`a chı
’
mˆo

.
t nh´om con cˆa
´
p q.
b) Nˆe
´
u q =1+kp v´o
.
isˆo
´
nguyˆen k t `u y ´y t h ` ı G l`a nh´om cyclic cˆa
´
p pq.
39.
Cho G l`a mˆo
.
t nh´om nhˆan h˜u
.
uha
.
n sao cho G c´o mˆo
.
ttu
.
.
d¯ ˘a
’
ng cˆa
´
u ϕ thoa

’
m˜an ϕ(a) = a, ∀a =1
G
.Ch´u
.
ng minh r˘a
`
ng:
a) V´o
.
imo
.
i α ∈ G,tˆo
`
nta
.
i g ∈ G sao cho α = g
−1
ϕ(g).
b) Nˆe
´
u ϕ c´o cˆa
´
pb˘a
`
ng 2, t´u
.
cl`aϕ = id v`a ϕ
2
= id th`ı ϕ(α)=α

−1
v´o
.
imo
.
i
α ∈ G v`a G l`a mˆo
.
t nh´om aben c´o cˆa
´
p l`a mˆo
.
tsˆo
´
le
’
.
7
TRA
’
L
`
O
.
IV
`
AHU
.
´
O

.
NG D
ˆ
A
˜
N GIA
’
IB
`
AI T
ˆ
A
.
P
CHU
.
O
.
NG I – NH
´
OM
1.
∀a, b, c ∈ Q, (a∗ b) ∗ c =(a + b + ab)∗ c = a + b + ab + c + ac + bc + abc =
a + b + c + bc + ab + ac + abc = a∗ (b + c + bc)=a ∗ (b ∗ c), hay ph´ep to´an ∗
c´o t´ınh kˆe
´
tho
.
.
p. ∀a ∈ Q,a∗ 0=0∗ a = a hay 0 l`a phˆa

`
ntu
.
’
d¯ o
.
nvi
.
cu
’
a Q d¯ ˆo
´
i
v´o
.
i ph´ep to´an ∗. Do d¯´o Q v´o
.
i ph´ep to´an ∗ l`a mˆo
.
tvi
.
nh´om, nhu
.
ng khˆong pha
’
i
l`a mˆo
.
t nh´om, v`ı phˆa
`

ntu
.
’
a = −1 khˆong c´o phˆa
`
ntu
.
’
nghi
.
ch d¯a
’
o.
T`u
.
a + b + ab +1 = (a + 1)(b + 1), ta c´o ∀a, b ∈ Q \{−1},a∗ b = −1hay
a ∗ b ∈ Q \{−1}. Do d¯´o Q \{−1} l`a mˆo
.
tvi
.
nh´om v´o
.
i ph´ep to´an ∗. Ngo`ai ra,
∀a ∈ Q \{−1},ac´o phˆa
`
ntu
.
’
nghi
.

ch d¯a
’
ol`a−
a
1+a
∈ Q \{−1}.Vˆa
.
y Q \{−1}
l`a mˆo
.
t nh´om v´o
.
i ph´ep to´an ∗.
2.
∀(a, b), (a

,b

), (a

,b

) ∈ G, ((a, b)(a

,b

))(a

,b


)=(ab

+ a

,bb

)(a

,b

)=
(ab

b

+ a

b

+ a

,bb

b

)=(a, b)(a

b

+ a


,b

b

)=(a, b)((a

,b

)(a

,b

)) hay
ph´ep to´an nhˆan c´o t´ınh kˆe
´
tho
.
.
p. ∀(a, b) ∈ G, (a, b)(0, 1) = (0, 1)(a, b)=
(a, b) hay (0, 1) l`a phˆa
`
ntu
.
’
d¯ o
.
nvi
.
cu

’
a G. ∀(a, b) ∈ G, (a, b)(−
a
b
,
1
b
)=
(−
a
b
,
1
b
)(a, b)=(0, 1) hay (a, b) c´o phˆa
`
ntu
.
’
nghi
.
ch d¯a
’
ol`a(−
a
b
,
1
b
). Vˆa

.
y G
l`a mˆo
.
t nh´om.
H = ∅ v`ı(0, 1) ∈ H. ∀(a, 1), (a

, 1) ∈ H, (a, 1)(a

, 1)
−1
=(a, 1)(−
a

1
,
1
1
)=
(a − a

, 1) ∈ H.Vˆa
.
y H l`a mˆo
.
t nh´om con cu
’
a G.
3. a)
∀(x, y), (x


,y

), (x

,y

) ∈ G,
((x, y)∗(x

,y

))∗(x

,y

)=(xx

,xy

+
y
x

)∗(x

,y

)=(xx


x

,xx

y

+
xy

x

+
y
x

x

)
=(x(x

x

),x(x

y

+
y

x


)+
y
x

x

)=(x, y) ∗ (x

x

,x

y

+
y

x

)=(x, y) ∗ ((x

,y

) ∗
(x

,y

)).

∀(x, y) ∈ G, (x, y) ∗ (1, 0) = (x, y)=(0, 1) ∗ (x, y).
∀(x, y) ∈ G, (x, y) ∗ (
1
x
,−y)=(1, 0) = (
1
x
,−y) ∗ (x, y).
Vˆa
.
y G l`a mˆo
.
t nh´om.
b) (1, 0) = (1,k(1−
1
1
)) ∈ H
k
nˆen H
k
= ∅.
∀(x, k(x−
1
x
)), (y, k(y −
1
y
)) ∈ H
k
,

(x, k(x−
1
x
))∗(y, k(y−
1
y
))
−1
=(x, k(x−
1
x
))∗(
1
y
,k(
1
y
−y)) = (
x
y
,k(
x
y
−
y
x
)) ∈
H
k
.

(x, k(x−
1
x
))∗ (y, k(y−
1
y
)) = (xy, k(xy−
1
xy
)) = (y, k(y−
1
y
))∗ (x, k(x−
1
x
)).
Vˆa
.
y H
k
l`a mˆo
.
t nh´om con giao ho´an cu
’
a G.
8
c) Z(G)={(x, y) | (x, y) ∗ (a, b)=(a, b) ∗ (x, y), ∀(a, b) ∈ G}
= {(x, y) | (xa, xb +
y
a

)=(ax, ay +
b
x
), ∀(a, b) ∈ G}
= {(x, y) | b(x −
1
x
)=y(a −
1
a
), ∀(a, b) ∈ G}
= {(x, y) | x −
1
x
=0,y=0}
= {(1, 0), (−1, 0)}.
4.
Tru
.
´o
.
chˆe
´
t ta c´o ∀x ∈ R, ∀n ∈ Z, [x + n]=[x]+n.
∀x, y ∈ G, (x⊕ y)=x + y − [x + y]=y + x − [y + x]=y ⊕ x.
∀x,y,z∈ G, (x⊕ y)⊕ z = x⊕ y + z − [x⊕ y + z]=x + y − [x + y]+z − [x +
y + z− [x + y]] = x + y + z− [x + y]− [x + y + z]+[x +y]=x +y + z−[x +y + z]=
x + y + z− [y + z]− [x + y + z]+[y + z]=x+ y + z− [y + z]− [x+ y + z− [y + z]] =
x + y ⊕ z − [x + y ⊕ z]=x ⊕ (y ⊕ z).
∀x ∈ G, [x]=0nˆenx ⊕ 0=x +0− [x +0]=x.

∀x ∈ G,nˆe
´
u x =0th`ı0⊕ 0=0,nˆe
´
u x =0th`ı1− x ∈ G v`a x⊕ (1− x)=0.
Vˆa
.
y(G,⊕)l`amˆo
.
t nh´om aben.
b) ∀x, y ∈ G, f(x ⊕ y) = cos 2π(x ⊕ y)+i sin 2π(x ⊕ y) = cos(2πx +2πy−
2π[x + y]) + i sin(2πx +2πy − 2π[x + y]) = cos(2πx +2πy)+i sin(2πx +2πy)=
(cos 2πx + i sin 2πx)(cos 2πy + i sin 2πy)=f (x)f (y).
Vˆa
.
y f l`a mˆo
.
td¯ˆo
`
ng cˆa
´
u nh´om.
5.
Gia
’
su
.
’
G l`a mˆo
.

t nh´om m`a khˆong c´o nh´om con thu
.
.
csu
.
.
n`ao. Nˆe
´
u G = {1}
th`ı tˆo
`
nta
.
i a ∈ G v´o
.
i a = 1. Nh´om con <a>cu
’
a G sinh ra bo
.
’
i a kh´ac {1}
nˆen pha
’
ib˘a
`
ng G hay G l`a nh´om cyclic. Nˆe
´
u G c´o cˆa
´
p p v`a gia

’
su
.
’
p = mn v´o
.
i
1 <m,n<pth`ı G c´o nh´om con cˆa
´
p m, d¯ˆay l`a nh´om con thu
.
.
csu
.
.
cu
’
a G.D
-
iˆe
`
u
mˆau thuˆa
’
n n`ay dˆa
˜
n d¯ ˆe
´
n p l`a sˆo
´

nguyˆen tˆo
´
.
6.
Gia
’
su
.
’
G/H = <aH>(a ∈ G). ∀x, y ∈ G, ∃m, n ∈ Z sao cho xH =
(aH)
m
= a
m
H, yH =(aH)
n
= a
n
H. Khi d¯´o x = a
m
u v`a y = a
n
v,v´o
.
i
u, v ∈ H (nˆen thuˆo
.
c Z(G)).
xy = a
m

ua
n
v = a
m
a
n
uv = a
m+n
vu = a
n
a
m
vu = a
n
va
m
u = yx.
Vˆa
.
y G l`a mˆo
.
t nh´om aben.
7.
a) G c´o d¯´ung hai l´o
.
pkˆe
`
l`a xH v`a H, trong d¯´o x/∈ H. Khi d¯´o, v´o
.
imo

.
i
g ∈ G = H ∪ xH,v´o
.
imo
.
i a ∈ H,tac´ogag
−1
∈ H khi g ∈ H v`a khi g/∈ H
ngh˜ıa l`a g = xh v´o
.
i h ∈ H th`ı gag
−1
= xhah
−1
x
−1
∈ H,v`ınˆe
´
u ngu
.
o
.
.
cla
.
i
xhah
−1
x

−1
= xh

v´o
.
i h

∈ H hay x = h

−1
hah
−1
∈ H,d¯iˆe
`
u n`ay vˆo l´y. Do d¯´o
HG.
b) Do HG nˆen ta c´o nh´om thu
.
o
.
ng G/H c´o cˆa
´
pl`a|G/H| =[G : H]=m.
Do d¯´o v´o
.
imo
.
i a ∈ G, aH ∈ G/H,tac´oa
m
H =(aH)

m
= H hay a
m
∈ H.
9
8.
– AB ≤ G:
x ∈ AB ⇒ x
−1
∈ AB (v`ı AB ≤ G) ⇒ x
−1
= ab, a ∈ A, b ∈ B ⇒ x =
b
−1
a
−1
∈ BA (v`ı b
−1
∈ B v`a a
−1
∈ A). Do d¯´o AB ⊂ BA.
x ∈ BA ⇒ x = ba, b ∈ B, a ∈ A ⇒ x
−1
= a
−1
b
−1
∈ AB (v`ı a
−1
∈ A v`a

b
−1
∈ B) ⇒ x ∈ AB (v`ı AB ≤ G). Do d¯´o BA ⊂ AB.
Vˆa
.
y AB = BA.
– AB = BA:
1=1.1 ∈ AB ⇒ AB = ∅.
x, x
1
∈ AB ⇒ x = ab, x
1
= a
1
b
1
,a,a
1
∈ A, b, b
1
∈ B ⇒ xx
−1
1
= abb
−1
1
a
−1
1
.

b

= bb
−1
1
∈ B ⇒ b

a
−1
1
∈ BA = AB ⇒ b

a
−1
1
= a
2
b
2
,a
2
∈ A, b
2
∈ B ⇒
xx
−1
1
= ab

a

−1
1
= aa
2
b
2
∈ AB (v`ı aa
2
∈ A v`a b
2
∈ B).
Vˆa
.
y AB l`a mˆo
.
t nh´om con cu
’
a G.
9.
a) K´y hiˆe
.
u1l`ad¯o
.
nvi
.
cu
’
a G.
1 ∈ A ∧ 1 ∈ B ⇒ 1 ∈ A ∩ B ⇒ A ∩ B = ∅.
∀x, y ∈ A ∩ B ⇒ x, y ∈ A ∧ x, y ∈ B ⇒ xy

−1
∈ A ∧ xy
−1
∈ B ⇒
xy
−1
∈ A ∩ B.
Vˆa
.
y A ∩ B l`a mˆo
.
t nh´om con cu
’
a G.
b) Gia
’
su
.
’
A ∪ B l`a mˆo
.
t nh´om con cu
’
a G v`a A ⊂ B. Khi d¯´o ∃a ∈ A, a /∈ B
v`a ∀b ∈ B, c = ab ∈ A ∪ B (v`ı A ∪ B l`a mˆo
.
t nh´om con cu
’
a G)hayc ∈ A ho˘a
.

c
c ∈ B.Nˆe
´
u c ∈ B th`ı a = cb
−1
∈ B, mˆau thuˆa
’
nv´o
.
i a/∈ B.Vˆa
.
y c ∈ A, suy ra
b = a
−1
c ∈ A. Do d¯´o B ⊂ A.
Ngu
.
o
.
.
cla
.
i, nˆe
´
u A ⊂ B th`ı A∪ B = B l`a mˆo
.
t nh´om con cu
’
a G v`a nˆe
´

u B ⊂ A
th`ı A ∪ B = A l`a mˆo
.
t nh´om con cu
’
a G.
c) Gia
’
su
.
’
C ⊂ A ∪ B v`a C ⊂ A. Khi d¯´o ∃c
0
∈ C, c
0
/∈ A nˆen c
0
∈ B (v`ı
c
0
∈ A ∪ B).
∀c ∈ C (⇒ c ∈ A∪B ⇒ c ∈ A ∨ c ∈ B) ⇒ b = cc
0
∈ C ⇒ b ∈ A ∨ b ∈ B.
–V´o
.
i c ∈ A,nˆe
´
u b ∈ A th`ı c
0

= c
−1
b ∈ A, mˆau thuˆa
’
nv´o
.
i c
0
/∈ A.Vˆa
.
y
b ∈ B,nˆenc = bc
−1
0
∈ B.
–V´o
.
i c ∈ B th`ı b`ai to´an d¯u
.
o
.
.
cch´u
.
ng minh; t´u
.
c l`a, C ⊂ B.
10.
a) ∀x, y ∈ G, x
2

y
2
=(xy)
2
(= 1) hay xxyy = xyxy, do d¯´o xy = yx.Vˆa
.
y G
l`a mˆo
.
t nh´om aben.
b) Xem ph´ep to´an trˆen G l`a ph´ep cˆo
.
ng, khi d¯´o ta c´o 2x =0, ∀x ∈ G.V`ı
vˆa
.
y c´o ph´ep nhˆan vˆo hu
.
´o
.
ng cu
’
a Z
2
lˆen G:
∀
a ∈ Z
2
, ∀x ∈ G, ax = ax.
Kiˆe
’

mch´u
.
ng dˆe
˜
d`ang G l`a mˆo
.
t Z
2
- khˆong gian vecto
.
,doG h˜u
.
uha
.
nnˆenG
l`a khˆong gian vecto
.
h˜u
.
uha
.
nchiˆe
`
u. Gia
’
su
.
’
dim G = n. Khi d¯´o G
∼

=
Z
n
2
hay
|G| =2
n
.
10
11.
a) Cho ac ∈ A(B ∩ C), trong d¯´o a ∈ A v`a c ∈ B ∩ C. Khi d¯´o ac ∈ AB
v`a ac ∈ aC = C.V`ıthˆe
´
A(B ∩ C) ⊂ AB ∩ C.M˘a
.
t kh´ac, nˆe
´
u ab ∈ AB ∩ C,
trong d¯´o a ∈ A v`a b ∈ B th`ı b ∈ a
−1
C = C v`a v`ı vˆa
.
y ab ∈ A(B ∩ C). Vˆa
.
y
AB ∩ C ⊂ A(B ∩ C).
b) Theo a) v`a c´ac gia
’
thiˆe
´

t, ta c´o
A = A(A ∩ K)=A(B ∩ K)=AK ∩ B = BK ∩ B = B.
12.
a) Ba
’
ng nhˆan cu
’
a Z
13
:
. 0 1 2 3 4 5 6 7 8 9 10 11 12
0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 10 11 12
2 0 2 4 6 8 10 12 1 3 5 7 9 11
3 0 3 6 9 12 2 5 8 11 1 4 7 10
4 0 4 8 12 3 7 11 2 6 10 1 5 9
5 0 5 10 2 7 12 4 9 1 6 11 3 8
6 0 6 12 5 11 4 10 3 9 2 8 1 7
7 0 7 1 8 2 9 3 10 4 11 5 12 6
8 0 8 3 11 6 1 9 4 12 7 2 10 5
9 0 9 5 1 10 6 2 11 7 3 12 8 4
10 0 10 7 4 1 11 8 5 2 12 9 6 3
11 0 11 9 7 5 3 1 12 10 8 6 4 2
12 0 12 11 10 9 8 7 6 5 4 3 2 1
2
1
= 2, 2
2
= 4, 2
3

= 8, 2
4
= 3, 2
5
= 6, 2
6
= 12,
2
7
= 11, 2
8
= 9, 2
9
= 5, 2
10
= 10, 2
11
= 7, 2
12
= 1.
Nhu
.
vˆa
.
y, Z
∗
13
l`a mˆo
.
t nh´om cyclic v´o

.
i phˆa
`
ntu
.
’
sinh l`a
2.
b) Gia
’
su
.
’
R
∗
l`a mˆo
.
t nh´om cyclic sinh bo
.
’
i x, ngh˜ıa l`a
R
∗
= {x
n
| n ∈ Z}.
Khi d¯´o ´anh xa
.
f : Z −→ R
∗

cho bo
.
’
i f (n)=x
n
l`a mˆo
.
t to`an ´anh, nˆen R
∗
l`a khˆong
qu´a d¯ˆe
´
md¯u
.
o
.
.
c. D
-
iˆe
`
u n`ay vˆo l´y v`ı R
∗
l`a tˆa
.
pho
.
.
pvˆoha
.

n khˆong d¯ˆe
´
md¯u
.
o
.
.
c. Vˆa
.
y
R
∗
khˆong l`a nh´om cyclic.
11
13.
a)
x = −
√
2
2
+
√
2
2
i, x
2
= −i, x
3
=
√

2
2
+
√
2
2
i, x
4
= −1,
x
5
=
√
2
2
−
√
2
2
i, x
6
= i, x
7
= −
√
2
2
−
√
2

2
i, x
8
=1.
Vˆa
.
y <x>= {1,x,x
2
,x
3
,x
4
,x
5
,x
6
,x
7
}.
b)
x = cos
4π
7
+ i sin
4π
7
,x
2
= cos
8π

7
+ i sin
8π
7
,
x
3
= cos
12π
7
+ i sin
12π
7
,x
4
= cos
2π
7
+ i sin
2π
7
,
x
5
= cos
6π
7
+ i sin
6π
7

,x
6
= cos
10π
7
+ i sin
10π
7
,x
7
=1.
Vˆa
.
y <x>= {1,x,x
2
,x
3
,x
4
,x
5
,x
6
}.
14.
a) Mˆo
˜
i phˆa
`
ntu

.
’
cu
’
a S
3
l`a mˆo
.
t ho´an vi
.
cu
’
a {1, 2, 3},t´u
.
cl`amˆo
.
t song ´anh
{1, 2, 3}−→{1, 2, 3}. Ph´ep to´an t´ıch trˆen S
3
ch´ınh l`a ph´ep ho
.
.
p th`anh ´anh xa
.
.
C´ac phˆa
`
ntu
.
’

cu
’
a S
3
l`a:

123
123

k.h.
= (1) ,

123
213

k.h.
= (1 2),

123
321

k.h.
= (1 3) ,

123
132

k.h.
= (2 3),


123
231

k.h.
= (1 2 3) ,

123
312

k.h.
= (1 3 2).
(1) (1 2) (1 3) (2 3) (1 2 3) (1 3 2)
(1) (1) (1 2) (1 3) (2 3) (1 2 3) (1 3 2)
(1 2) (1 2) (1) (1 3 2) (1 2 3) (2 3) (1 3)
(1 3) (1 3) (1 2 3) (1) (1 3 2) (1 2) (2 3)
(2 3) (2 3) (1 3 2) (1 2 3) (1) (1 3) (1 2)
(1 2 3) (1 2 3) (1 3) (2 3) (1 2) (1 3 2) (1)
(1 3 2) (1 3 2) (2 3) (1 2) (1 3) (1) (1 2 3)
12
V`ı ph´ep ho
.
.
p th`anh c´o t´ınh kˆe
´
tho
.
.
p nˆen ph´ep to´an trˆen S
3
c´o t´ınh kˆe

´
tho
.
.
p.
S
3
c´o phˆa
`
ntu
.
’
d¯ o
.
nvi
.
l`a (1). C˘an c´u
.
v`ao ba
’
ng nhˆan, ta thˆa
´
ymo
.
i phˆa
`
ntu
.
’
cu

’
a
S
3
d¯ ˆe
`
u kha
’
nghi
.
ch. Cu
.
thˆe
’
,
(1)
−1
= (1), (1 2)
−1
= (1 2), (1 3)
−1
= (1 3),
(2 3)
−1
= (2 3), (1 2 3)
−1
= (1 3 2), (1 3 2)
−1
= (1 2 3).
Vˆa

.
y S
3
l`a mˆo
.
t nh´om.
b) D
-
˘a
.
t X = {(1 2), (1 3), (2 3)} v`a Y = {(1 2 3), (1 3 2). C˘an c´u
.
v`ao
ba
’
ng nhˆan ta thˆa
´
ynˆe
´
u nh´om con H cu
’
a S
3
ch´u
.
a2phˆa
`
ntu
.
’

cu
’
a X ho˘a
.
c 1 phˆa
`
n
tu
.
’
cu
’
a X v`a 1 phˆa
`
ntu
.
’
cu
’
a Y th`ı H = S
3
.Vˆa
.
y c´ac nh´om con cu
’
a S
3
l`a:
{(1)}, {(1), (1 2)}, {(1), (1 3)}, {(1), (2 3)}, {(1), (1 2 3), (1 3 2)},S
3

,
trong d¯´o c´ac nh´om con chuˆa
’
n t´ac l`a {(1)}, {(1), (1 2 3), (1 3 2)},S
3
.
c) V`ı c ˆa
´
pcu
’
a G
3
pha
’
i l`a mˆo
.
tu
.
´o
.
c chung cu
’
a 24 v`a 30 cho nˆen n´o pha
’
il`a
mˆo
.
tu
.
´o

.
ccu
’
a6.
Ta biˆe
´
tr˘a
`
ng nh´om c´o cˆa
´
p nho
’
ho
.
n6d¯ˆe
`
u l`a aben v`a nh´om cˆa
´
p6chı
’
c´o
hai loa
.
i (sai kh´ac d¯˘a
’
ng cˆa
´
u): aben (khi d¯´o l`a nh´om cyclic) v`a khˆong aben. Vˆa
.
y

G
3
∼
=
S
3
.
15.
a) Gia
’
su
.
’
Q l`a nh´om cyclic sinh ra bo
.
’
i
m
n
trong d¯´o m v`a n l`a c´ac sˆo
´
nguyˆen
nguyˆen tˆo
´
c`ung nhau. 1 ∈ Q nˆen tˆo
`
nta
.
isˆo
´

nguyˆen k sao cho 1 = k.
m
n
, d¯ i ˆe
`
u n`ay
dˆa
˜
n d¯ ˆe
´
nsu
.
.
vˆo l´y l`a n = km.
b) Mˆo
˜
i phˆa
`
ntu
.
’
cu
’
a Q/Z c´o cˆa
´
ph˜u
.
uha
.
nv`ıv´o

.
i m, n ∈ Z,n>0, ta c´o
n

m
n
+ Z

= Z; trong khi mo
.
i phˆa
`
ntu
.
’
kh´ac khˆong cu
’
a Q d¯ ˆe
`
uc´ocˆa
´
p vˆo ha
.
n.
Do d¯´o Q/Z khˆong thˆe
’
d¯ ˘a
’
ng cˆa
´

uv´o
.
i Q.
16.
a) R˜o r`ang H = ∅ v`a c´o 14 phˆa
`
ntu
.
’
v`ı m c´o 2 c´ach cho
.
nv`ab c´o 7 c´ach
cho
.
n.

±
1 b
0 1

±
1 c
0 1

=

±
1 b ± c
0 1


∈ H

±
1 b
0 1

±
1 ∓b
0 1

=

1 0
0 1

hay

±
1 b
0 1

−1
=

±
1 ∓b
0 1

∈ H.
Vˆa

.
y H l`a mˆo
.
t nh´om con cu
’
a GL(2, Z
7
) c´o 14 phˆa
`
ntu
.
’
.
b) Ta c´o:

1 b
0 1

1 c
0 1

=

1 b + c
0 1

,

−
1 0

0 1

2
=

1 0
0 1

= I
2
,

1 b
0 1

−
1 0
0 1

=

−
1 b
0 1

. Do d¯´o 14 phˆa
`
ntu
.
’

cu
’
a H l`a:
I
2
,

1 1
0 1

= A,

1 2
0 1

= A
2
,

1 3
0 1

= A
3
,

1 4
0 1

= A

4
,

1 5
0 1

= A
5
,
13

1 6
0 1

= A
6
,

−
1 0
0 1

= B,

−
1 1
0 1

= AB,


−
1 2
0 1

= A
2
B,

−
1 3
0 1

= A
3
B,

−
1 4
0 1

= A
4
B,

−
1 5
0 1

= A
5

B,

−
1 6
0 1

= A
6
B.
17.
a) Do G =<x,y>v`a x
−1
= x
2
,y
−1
= y,mˆo
˜
i phˆa
`
ntu
.
’
cu
’
a G c´o da
.
ng:
x
k

1
y
l
1
...x
k
n
y
l
n
, trong d¯´o k
i
,l
i
,v´o
.
i1≤ i ≤ n, l`a c´ac sˆo
´
tu
.
.
nhiˆen. T`u
.
c´ac quan
hˆe
.
cu
’
a G, ta c´o:
yx

3
y = yy =1=(xy)
2
= xyxy ⇒ xy = yx
2
.
Do d¯´o c´ac phˆa
`
ntu
.
’
cu
’
a G l`a y
k
x
l
,v´o
.
i k =0, 1v`al =0, 1, 2. C´ac phˆa
`
ntu
.
’
n`ay
d¯ˆoi mˆo
.
t kh´ac nhau nˆen ta c´o:
G = {1,x,x
2

, y, yx, yx
2
}.
Ba
’
ng nhˆan cu
’
a G:
. 1 x x
2
y yx yx
2
1 1 x x
2
y yx yx
2
x x x
2
1 yx
2
y yx
x
2
x
2
1 x yx yx
2
y
y y yx yx
2

1 x x
2
yx yx yx
2
y x
2
1 x
yx
2
yx
2
y yx x x
2
1
2) G c´o c´ac phˆa
`
ntu
.
’
bˆa
.
c3l`ax, x
2
v`a c´ac phˆa
`
ntu
.
’
bˆa
.

c2l`ay, yx, yx
2
. C˘an
c´u
.
v`ao ba
’
ng nhˆan, nˆe
´
u H l`a mˆo
.
t nh´om con cu
’
a G ch´u
.
a 1 phˆa
`
ntu
.
’
trong {x, x
2
}
v`a 1 phˆa
`
ntu
.
’
trong {y, yx, yx
2

} th`ı H = G.Vˆa
.
y c´ac nh´om con cu
’
a G l`a:
{1}, {1,x,x
2
}, {1,y}, {1,yx}, {1,yx
2
},G.
18.
a) A
2
=

−10
0 −1

,A
3
=

0 −1
10

,A
4
=

10

01

= I
2
,B
2
= I
2
.
BA =

−10
01

= A
3
B. Do d¯´o ta c´o:
G =<A,B| A
4
= B
2
= I
2
,BA= A
3
B>.
= {I
2
,A,A
2

,A
3
,B, AB,A
2
B, A
3
B}.
b) C´ac nh´om con cu
’
a G l`a:
{I
2
}, {I
2
,A,A
2
,A
3
}, {I
2
,A
2
},
{I
2
,B}, {I
2
,AB}, {I
2
,A

2
B}, {I
2
,A
3
B},{I
2
,A
2
,B,A
2
B},G.
14
19.
a) V`ı f
n
l`a mˆo
.
t to`an ´anh nˆen tˆo
`
nta
.
i z ∈ G sao cho y = z
n
.
V´o
.
i x ∈ G,v`ı f
n
l`a mˆo

.
td¯ˆo
`
ng cˆa
´
u, ta c´o: (xzx
−1
)
n
= x
n
z
n
x
−n
.T`u
.
d¯´o:
xyx
−1
= xz
n
x
−1
=(xzx
−1
)
n
= x
n

z
n
x
−n
= x
n
yx
−n
.
Vˆa
.
y x
n−1
y = yx
n−1
.
b) V´o
.
i n = 3, ta c´o: x
2
y = yx
2
. Ngo`ai ra,
x(yx)
2
y =(xy)
3
= x
3
y

3
= x(x
2
y
2
)y.
Vˆa
.
y, (yx)
2
= x
2
y
2
=(x
2
y)y =(yx
2
)y =(yx)(xy). T`u
.
d¯ ´o yx = xy.
20.
a) 1=1
n
v`a 1
n
=1nˆen1∈ G
(n)
v`a 1 ∈ G
(n)

, ngh˜ıa l`a G
(n)
= ∅ v`a
G
(n)
= ∅. ∀x
n
,y
n
∈ G
(n)
,x
n
(y
n
)
−1
= x
n
(y
−1
)
n
=(xy
−1
)
n
∈ G
(n)
. ∀x, y ∈

G
(n)
,x
n
= y
n
=1,nˆen(xy
−1
)
n
= x
n
(y
n
)
−1
=1hayxy
−1
∈ G
(n)
. Ngo`ai ra,
∀y ∈ G, ∀x ∈ G, yx
n
y
−1
=(yxy
−1
)
n
∈ G

(n)
; ∀z ∈ G
(n)
, (yzy
−1
)
n
= yz
n
y
−1
=
yy
−1
=1hayyzy
−1
∈ G
(n)
.Vˆa
.
y G
(n)
v`a G
(n)
l`a c´ac nh´om con chuˆa
’
nt˘a
´
ccu
’

a
G.
b)X´et ´anh xa
.
f : G −→ G
(n)
cho bo
.
’
i f (x)=x
n
. R˜o r`ang f l`a mˆo
.
t to`an
cˆa
´
u. Kerf = {x ∈ G | x
n
=1} = G
(n)
. Do d¯´o ta c´o G/Kerf
∼
=
Imf hay
G/G
(n)
∼
=
G
(n)

.
21.
a) X´et ´anh xa
.
ϕ : C
∗
−→ R
∗
+
cho bo
.
’
i ϕ(z)=|z|. Khi d¯´o ϕ l`a mˆo
.
t to`an cˆa
´
u
v`a kerϕ = H,nˆenC
∗
/H
∼
=
R
∗
+
.
b) T`u
.
d¯ i
.

nh ngh˜ıa vˆe
`
nh´om con chuˆa
’
nt˘a
´
c suy ra dˆe
˜
d`ang N = f
−1
(M) G
khi MH. X´et ´anh xa
.
ϕ : G −→ H/M cho bo
.
’
i ϕ(x)=f(x)M. Khi d¯´o ϕ l`a mˆo
.
t
to`an cˆa
´
u v`a kerϕ = N ,nˆenG/N
∼
=
H/M.
22.
a) Gia
’
su
.

’
G =<a>v`a f,g ∈ Aut(G)v´o
.
i f(a)=a
r
v`a g(a)=a
s
. Khi d¯´o
(g ◦ f)(a)=g(f (a)) = g(a
r
)=g(a)
r
= a
sr
= a
rs
= f(a)
s
= f(a
s
)=f (g(a)) =
(f ◦ g)(a). Do d¯´o g ◦ f = f ◦ g hay Aut(G) l`a nh´om aben.
b) Nˆe
´
u G =<a>c´o cˆa
´
p p nguyˆen tˆo
´
th`ı v´o
.

imˆo
˜
itu
.
.
d¯ ˆo
`
ng cˆa
´
u nh´om cu
’
a G
cho bo
.
’
i f (a)=a
r
, trong d¯´o r l`a sˆo
´
nguyˆen khˆong ˆam, ta c´o f ∈ Aut(G) khi v`a
chı
’
khi a
r
l`a phˆa
`
ntu
.
’
sinh cu

’
a G t´u
.
c l`a khi v`a chı
’
khi r nguyˆen tˆo
´
c`ung nhau v´o
.
i
p hay r =1, 2, ..., p − 1. Do d¯´o Aut(G) l`a nh´om c´o cˆa
´
p p − 1. Ngo`ai ra, Aut(G)
l`a nh´om d¯˘a
’
ng cˆa
´
uv´o
.
i Z
∗
p
= Z
p
\{0}, nˆen Aut(G) l`a nh´om cyclic.
23.
a) V`ı G/kerf
∼
=
f(G)nˆen|f(G)| = |G/kerf| chia hˆe

´
t |G|.
b) Gia
’
su
.
’
G c´o n l´o
.
pkˆe
`
phˆan biˆe
.
tl`ax
1
H, x
2
H, ..., x
n
H.V´o
.
imˆo
˜
i y ∈ K
tˆo
`
nta
.
i x ∈ G sao cho y = f(x). Khi d¯´o x ∈ x
i

H v´o
.
i i n`ao d¯´o v`a y ∈ f (x
i
)f(H).
Nˆe
´
u f(x
i
)f(H)=f (x
j
)f(H)th`ıf(x
j
)
−1
f(x
i
)=f (x

)v´o
.
i x

∈ H, nˆen ta c´o
x
−1
j
x
i
x


−1
∈ Kerf , m`a Kerf ⊂ H, do d¯´o x
−1
j
x
i
∈ H hay x
i
H = x
j
H,t`u
.
d¯ ´o i = j.
N´oi c´ach kh´ac K c´o n l´o
.
pkˆe
`
phˆan biˆe
.
tl`af (x
1
)f(H),f(x
2
)f(H), ..., f(x
n
)f(H)
hay [K : f (H)] = n.
15
24.

a) V´o
.
imo
.
i y ∈ G,tˆo
`
nta
.
i duy nhˆa
´
t x = g
−1
yg sao cho C
g
(x)=y,nˆenC
g
l`a mˆo
.
t song ´anh. Ngo`ai ra, C
g
(xx

)=gxx

g
−1
= gxg
−1
.gx


g
−1
= C
g
(x)C
g
(x

).
Do d¯´o C
g
l`a mˆo
.
ttu
.
.
d¯ ˘a
’
ng cˆa
´
ucu
’
a G. Aut(G)l`amˆo
.
t nh´om v´o
.
i ph´ep to´an ho
.
.
p

th`anh, d¯o
.
nvi
.
l`a ´anh xa
.
d¯ ˆo
`
ng nhˆa
´
t id
G
, nghi
.
ch d¯a
’
ocu
’
a f ∈ Aut(G) l`a ´anh xa
.
ngu
.
o
.
.
c f
−1
.V´o
.
imo

.
i f ∈ Aut(G), v´o
.
imo
.
i g ∈ G,
(f
−1
C
g
f)(x)=f
−1
(gf(x)g
−1
)=(f
−1
(g))x(f
−1
(g))
−1
= C
f
−1
(g)
(x)
v´o
.
imo
.
i x ∈ G nˆen f

−1
C
g
f = C
f
−1
(g)
∈ Inn(G). Do d¯´o Inn(G)  Aut(G).
b) Dˆe
˜
d`ang c´o d¯u
.
o
.
.
c Z(G)l`amˆo
.
t nh´om con chuˆa
’
nt˘a
´
ccu
’
a G. X´et ´anh xa
.
ϕ : G −→ Inn(G)
x´ac d¯i
.
nh bo
.

’
i ϕ(g)=C
g
th`ı ϕ l`a mˆo
.
t to`an ´anh v`a do C
gg

= C
g
C
g

nˆen ϕ l`a mˆo
.
t
to`an cˆa
´
u. Ngo`ai ra, C
g
= id
G
khi v`a chı
’
khi g ∈ Z(G)nˆenZ(G)=kerf.V`ıvˆa
.
y,
G/Z(G)
∼
=

Inn(G).
25.
a) ∀x ∈ G, ∀a ∈ [G, G], ta c´o x
−1
ax = a(a
−1
x
−1
ax) ∈ [G, G]. Vˆa
.
y
[G, G] G.
V´o
.
i H  G, G/H l`a aben ⇐⇒ ∀ x, y ∈ G, (xH)(yH)=(yH)(xH) ⇐⇒
∀x, y ∈ G, xyH = yxH ⇐⇒ ∀ x, y ∈ G, (yx)
−1
xy ∈ H ⇐⇒ ∀ x, y ∈ G, x
−1
y
−1
xy
∈ H ⇐⇒ [G, G] ⊂ H. Do d¯´o [G, G] l`a nh´om con chuˆa
’
nt˘a
´
c nho
’
nhˆa
´

tcu
’
a G sao
cho G/[G, G] l`a aben.
b) [xy, z]=(xy)
−1
z
−1
xyz = y
−1
x
−1
z
−1
xyz
= y
−1
(x
−1
z
−1
xz)y(y
−1
z
−1
yz)=y
−1
[x, z]y[y, z].
c) Nˆe
´

u[G, G] ⊂ Z(G)th`ıtac´o
[xy, a]=y
−1
[x, a]y[y, a]=[x, a]y
−1
y[y, a]=[x, a][y, a]
hay f (xy)=f (x)f (y), ∀x, y ∈ G.Vˆa
.
y f l`a mˆo
.
td¯ˆo
`
ng cˆa
´
u nh´om.
Kerf = {x ∈ G | [x, a]=1} = {x ∈ G | xa = ax}.
d) Nh´om S
3
gˆo
`
m 6 phˆa
`
ntu
.
’
:

123
123


k.h.
= (1),

123
231

k.h.
= (1 2 3),

123
312

k.h.
= (1 3 2),

123
213

k.h.
= (1 2),

123
321

k.h.
= (1 3),

123
132


k.h.
= (2 3).
Kiˆe
’
m tra d¯u
.
o
.
.
cr˘a
`
ng: ∀x, y ∈ S
3
, [x, y] = (1) ho˘a
.
c (1 2 3) ho˘a
.
c (1 3 2),
t´u
.
cl`a[S
3
,S
3
]={(1), (1 2 3), (1 3 2)} =< (1 2 3) >.
Do [S
3
,S
3
]S

3
nˆen c´o nh´om thu
.
o
.
ng S
3
/[S
3
,S
3
] v`a nh´om n`ay c´o cˆa
´
p6/3=2.
Vˆa
.
y S
3
/[S
3
,S
3
]
∼
=
Z
2
.
16
26.

Go
.
i l =ord(a
m
)v`al

=
n
(m, n)
.Tac´o
a
ml
=1 ⇒ n | ml ⇒
n
(m, n)



m
(m, n)
l.
Do d¯´o l

|l.M˘a
.
t kh´ac n | ml

nˆen a
ml


=1t´u
.
c l`a ord(a
m
)|l

hay l | l

.Vˆa
.
y l = l

.
27.
a) Do (168, 132) = 12 nˆen cˆa
´
pcu
’
a g
132
l`a
168
12
= 14.
b) V´o
.
i Z
140
=< 1 > v`a m l`a sˆo
´

nguyˆen thoa
’
m˜an 0 ≤ m ≤ 139,
ord(m
1) = 14 ⇔
140
(140,m)
=14⇔ (140,m)=10
⇔ m =10, 30, 50, 90, 110, 130.
Vˆa
.
y c´ac phˆa
`
ntu
.
’
cˆa
´
p 14 cu
’
a nh´om cˆo
.
ng Z
140
l`a 10, 30, 50, 90, 110, 130.
28.
a) Nˆe
´
u G = {1} th`ı G l`a cyclic. Nˆe
´

u G = {1},go
.
i a l`a phˆa
`
ntu
.
’
sinh cu
’
a
G,th`ı a
k
∈ G v´o
.
i k l`a sˆo
´
nguyˆen du
.
o
.
ng n`ao d¯´o. Go
.
i m l`a sˆo
´
nguyˆen du
.
o
.
ng nho
’

nhˆa
´
t sao cho a
m
∈ G.V´o
.
imo
.
i b ∈ G, ta c´o b = a
n
v´o
.
isˆo
´
nguyˆen n n`ao d¯´o.
Theo thuˆa
.
t to´an chia, n = qm + r v´o
.
i0≤ r<m. Khi d¯´o a
r
=(a
m
)
−q
a
n
∈ G.
Do t´ınh nho
’

nhˆa
´
tcu
’
a m suy ra r =0. V`ıvˆa
.
y n = qm v`a b =(a
m
)
q
,t´u
.
cl`aG l`a
nh´om cyclic sinh ra bo
.
’
i a
m
.
b) Gia
’
su
.
’
|C| = n v`a m|n.Tac´oG =<a
n
m
> l`a nh´om con cu
’
a C c´o cˆa

´
p
m. Ngo`ai ra, nˆe
´
u H l`a nh´om con cu
’
a C c´o cˆa
´
p m. Gia
’
su
.
’
H =<a
s
>. Khi d¯´o
a
sm
=(a
s
)
m
=1nˆenn | sm do d¯´o
n
m
| s.V`ıvˆa
.
y a
s
∈ G hay H ⊂ G v`a suy ra

H = G v`ı |H| = |G|.
c) Nˆe
´
u |C| = ∞ v`a C =<a>=<a
i
> th`ı tˆo
`
nta
.
isˆo
´
nguyˆen j sao cho
a =(a
i
)
j
= a
ij
. Khi d¯´o ij = 1, nˆen i =1hayi = −1.
d) Nˆe
´
u |C| = n<∞ th`ı a
m
l`a phˆa
`
ntu
.
’
sinh cu
’

a C khi v`a chı
’
khi ord(a
m
)=n
t´u
.
c l`a khi v`a chı
’
khi
n
(n, m)
= n hay (n, m)=1.
29.
Nˆe
´
u f : Q −→ Q x´ac d¯i
.
nh bo
.
’
i f(x)=ax v´o
.
i a ∈ Q th`ı f l`a mˆo
.
td¯ˆo
`
ng cˆa
´
u

nh´om. Thˆa
.
tvˆa
.
y, ∀x, y ∈ Q,f(x + y)=a(x + y)=ax + ay = f (x)+f (y).
D
-
a
’
ola
.
i, nˆe
´
u f : Q −→ Q l`a mˆo
.
td¯ˆo
`
ng cˆa
´
u nh´om th`ıd¯˘a
.
t a = f(1), ta c´o
a = f(1) = f(n.
1
n
)=nf(
1
n
)hayf (
1

n
)=
a
n
v´o
.
imo
.
isˆo
´
nguyˆen du
.
o
.
ng n. Khi d¯´o
∀x ∈ Q,x=
m
n
,m∈ Z, n l`a sˆo
´
nguyˆen du
.
o
.
ng, ta c´o f(x)=f(
m
n
)=f(m.
1
n

)=
mf(
1
n
)=m.
a
n
= a.
m
n
= ax. R˜o r`ang a duy nhˆa
´
t sao cho f(x)=ax, ∀x ∈ Q.
30.
X´et ph´ep tu
.
o
.
ng ´u
.
ng
f : Z
mn
−→ Z
m
× Z
n
: x(mod mn) → (x(mod m),x(mod n)).
17
x(mod mn)=y(mod mn) ⇔ x − y ≡ 0(mod mn) ⇔ x − y ≡ 0(mod m)v`a

x − y ≡ 0(mod n) ⇔ x(mod m)=y(mod m)v`ax(mod n)=y(mod n) ⇔
(x(mod m),x(mod n)) = (y(mod m),y(mod n)). Do d¯´o f l`a mˆo
.
td¯o
.
n ´anh. V`ı
|Z
mn
| = |Z
m
× Z
n
| = mn nˆen f l`a mˆo
.
t song ´anh. Ngo`ai ra,
f(x(mod mn)+y(mod mn)) = f (x + y(mod mn))
=(x + y(mod m),x+ y(mod n))
=(x(mod m),x(mod n)) + (y(mod m),y(mod n))
= f (x(mod mn)) + f (y(mod mn)).
Vˆa
.
y f l`a mˆo
.
td¯˘a
’
ng cˆa
´
u.
Do Z
3

× Z
2
∼
=
Z
6
nˆen Z
3
× Z
2
l`a mˆo
.
t nh´om aben, trong khi nh´om d¯ˆo
´
ix´u
.
ng
S
3
khˆong aben. V`ıvˆa
.
y Z
3
× Z
2

∼
=
S
3

.
31.
X´et ´anh xa
.
f : G −→ G/M × G/N, x → (xM, xN ).
∀x, y ∈ G, f(xy)=(xyM, xyN)=(xM.yM, xN.yN)
=(xM, xN )(yM, yN)=f (x)f(y).
Vˆa
.
y f l`a mˆo
.
td¯ˆo
`
ng cˆa
´
u nh´om.
∀(aM, bN ) ∈ G/M × G/N, a = uv, b = zt, u, z ∈ M, v,t ∈ N (v`ı
G = MN). D
-
˘a
.
t x = zv th`ı do MG,u
−1
z ∈ M, t
−1
v ∈ N,tac´o

a
−1
x = v

−1
u
−1
zv = v
−1
(u
−1
z)v ∈ M
b
−1
x = t
−1
z
−1
zv = t
−1
v ∈ N
⇒

xM = aM
xN = bN
.
T´u
.
cl`a∃x = zv ∈ G sao cho f(x)=(xM, xN). Do d¯´o f l`a mˆo
.
t to`an cˆa
´
u nh´om.
Kerf = {x ∈ G | f(x)=(xM, xN )=(M,N)}

= {x ∈ G | x ∈ M ∧ x ∈ N} = M ∩ N.
Vˆa
.
y G/Kerf
∼
=
Imf hay G/(M ∩ N )
∼
=
G/M × G/N .
32.
a) Ph´ep to´an ∗ thoa
’
m˜an t´ınh kˆe
´
tho
.
.
p. G c´o phˆa
`
ntu
.
’
trung ho`a l`a (0, 0, 0).
Nghi
.
ch d¯a
’
ocu
’

a(a, b, c) ∈ G l`a (−a,−b,−c − ba). Do d¯´o (G,∗)l`amˆo
.
t nh´om.
Nh´om n`ay khˆong aben v`ı
(1, 0, 0) ∗ (0, 1, 0) = (1, 1, 0) =(1, 1, 1) = (0, 1, 0) ∗ (1, 0, 0).
b) R
2
c´o phˆa
`
ntu
.
’
trung ho`a l`a (1, 0). V´o
.
i x, y ∈ R cho tru
.
´o
.
c, hˆe
.
phu
.
o
.
ng
tr`ınh

xx

− yy


=1
yx

+ xy

=0
chı
’
c´o nghiˆe
.
m duy nhˆa
´
t khi




x −y
yx




= x
2
+ y
2
=0.
Do d¯´o (R

2
,◦) khˆong l`a mˆo
.
t nh´om v`ı phˆa
`
ntu
.
’
(0, 0) khˆong kha
’
nghi
.
ch.
18
33.
f l`a mˆo
.
td¯ˆo
`
ng cˆa
´
u nh´om v`ı ∀x, y ∈ R
f(x) ∗ f(y)=
e
x
− e
−x
2

1+


e
y
− e
−y
2

2
+
e
y
− e
−y
2

1+

e
x
− e
−x
2

2
=
e
x
− e
−x
2

.
e
y
+ e
−y
2
+
e
y
− e
−y
2
.
e
x
+ e
−x
2
=
e
x+y
− e
−(x+y)
2
= f (x + y).
f c`on l`a mˆo
.
t song ´anh v`ı f c´o ´anh xa
.
ngu

.
o
.
.
cl`af
−1
(x)=ln(x +
√
x
2
+ 1).
Do d¯´o f l`a mˆo
.
td¯˘a
’
ng cˆa
´
u nh´om.
34.
a) Ba
’
ng nhˆan cu
’
a U
22
= {1, 3, 5, 7, 9, 13, 15, 17, 19, 21}:
. 1 3 5 7 9 13 15 17 19 21
1 1 3 5 7 9 13 15 17 19 21
3 3 9 15 21 5 17 1 7 13 19
5 5 15 3 13 1 21 9 19 7 17

7 7 21 13 5 19 3 17 9 1 15
9 9 5 1 19 15 7 3 21 17 13
13 13 17 21 3 7 15 19 1 5 9
15 15 1 9 17 3 19 5 13 21 7
17 17 7 19 9 21 1 13 3 15 5
19 19 13 7 1 17 5 21 15 9 3
21 21 19 17 15 13 9 7 5 3 1
7
1
= 7, 7
2
= 5, 7
3
= 13, 7
4
= 3, 7
5
= 21,
7
6
= 15, 7
7
= 17, 7
8
= 9, 7
9
= 19, 7
10
= 1,
Vˆa

.
y U
22
l`a mˆo
.
t nh´om cyclic sinh bo
.
’
i
7.
b) U
24
= {1, 5, 7, 11, 13, 17, 19, 23}.
5
2
= 25, 5
4
= 1.
7
2
= 1.
11
2
= 1.
13
2
= 1.
17
2
= 1.

19
2
= 1.
23
2
= 1.
19
Vˆa
.
y U
24
khˆong l`a nh´om cyclic.
35.
a) R˜o r`ang 0 ∈ R
∗
, ph´ep to´an ∗ c´o t´ınh kˆe
´
tho
.
.
p, 0 l`a phˆa
`
ntu
.
’
d¯ o
.
nvi
.
(do

x ∗ 0=0∗ x = x) v`a mo
.
i x ∈ R
∗
d¯ ˆe
`
u kha
’
nghi
.
ch (do d¯i
.
nh ngh˜ıa cu
’
a R
∗
). Do
d¯´o, R
∗
l`a mˆo
.
t nh´om.
b) V´o
.
i x ∈ U(R), (1− x)∗ (1− x
−1
)=(1− x
−1
)∗ (1− x) = 0, nˆen ta c´o ´anh
xa

.
f : U(R) −→ R
∗
cho bo
.
’
i f (x)=1− x. R˜o r`ang f l`a mˆo
.
t song ´anh. Ngo`ai ra,
f(x)∗ f (y)=(1−x)∗ (1− y)=(1−x)+(1−y)−(1−x)(1− y)=1−xy = f (xy).
Do d¯´o f l`a mˆo
.
td¯˘a
’
ng cˆa
´
u nh´om.
36.
a) 12 phˆa
`
ntu
.
’
cu
’
a A
4
v´o
.
id¯o

.
nvi
.
ι l`a:
ι =

1234
1234

,τ
1
=

1234
1342

,τ
2
=

1234
1423

,
τ
3
=

1234
3241


,τ
4
=

1234
4213

,τ
5
=

1234
2431

,
τ
6
=

1234
4132

,τ
7
=

1234
2314


,τ
8
=

1234
3124

,
σ
1
=

1234
2143

,σ
2
=

1234
3412

,σ
3
=

1234
4321

.

Gia
’
su
.
’
A
4
c´o mˆo
.
t nh´om con H cˆa
´
p 6. Do σ
2
i
= ι v`a τ
3
j
= ι nˆen σ
i
c´o cˆa
´
p
2v`aτ
j
c´o cˆa
´
p3v´o
.
i i =1, 2, 3v`aj =1, 2,... ,8. Do H c´o cˆa
´

p6nˆenH ch´u
.
a
mˆo
.
t 3-nh´om con Sylow v`a mˆo
.
t 2-nh´om con Sylow. Do d¯´o τ
j
∈ H v`a σ
i
∈ H
v´o
.
i j v`a i n`ao d¯´o, ch˘a
’
ng ha
.
n τ
1
∈ H v`a σ
2
∈ H. Khi d¯´o H ch´u
.
a c´ac phˆa
`
ntu
.
’
ι, τ

1
,τ
2
1
= τ
2
,σ
1
,σ
1
τ
1
= τ
8
,τ
1
σ
1
= τ
5
,τ
2
8
= τ
7
,τ
2
5
= τ
6

.D
-
iˆe
`
u n`ay cho biˆe
´
t
H c´o ´ıt nhˆa
´
t 8 phˆa
`
ntu
.
’
, mˆau thuˆa
˜
nv´o
.
i |H| = 6. Vˆa
.
y A
4
khˆong ch´u
.
amˆo
.
t nh´om
con cˆa
´
p 6 n`ao.

b) Cˆa
´
pcu
’
amˆo
.
t 2-nh´om con Sylow l`a 4, v`ı 2
2
l`a l˜uy th`u
.
a cao nhˆa
´
tcu
’
a2
chia hˆe
´
t12=|A
4
|. Do khˆong c´o τ
j
n`ao c´o thˆe
’
l`a phˆa
`
ntu
.
’
cu
’

amˆo
.
t 2-nh´om con
Sylow (v`ı ch´ung d¯ˆe
`
u c´o cˆa
´
p l`a 3), σ
i
σ
k
= σ
l
v´o
.
i i, k, l ∈{1, 2, 3},σ
2
i
= ι v´o
.
i
i =1, 2, 3v`aι, σ
1
,σ
2
,σ
3
l`a bˆo
´
n phˆa

`
ntu
.
’
duy nhˆa
´
t trong A
4
c´o cˆa
´
pu
.
´o
.
ccu
’
a4,
ta c´o 2-nh´om con Sylow duy nhˆa
´
tl`aP = {ι, σ
1
,σ
2
,σ
3
}.
Cˆa
´
pcu
’

amˆo
.
t 3-nh´om con Sylow l`a 3. C´ac tˆa
.
p
{ι, τ
1
,τ
2
1
}, {ι, τ
3
,τ
2
3
}, {ι, τ
5
,τ
2
5
}, {ι, τ
7
,τ
2
7
}
l`a c´ac nh´om con cˆa
´
p 3. Sˆo
´

c´ac 3-nh´om con Sylow l`a s
3
=1+3k,v´o
.
i k ∈ Z, pha
’
i
chia hˆe
´
t cho 12. R˜o r`ang k = 0, v`a nˆe
´
u k>1th`ıs
3
khˆong chia hˆe
´
t 12. Do d¯´o
k = 1 v`a c´o d¯´ung bˆo
´
n 3-nh´om con Sylow nhu
.
trˆen.
37.
a) Gia
’
su
.
’
|H| = p
t
,t≥ 0. Theo d¯i

.
nh l´y Lagrange, p
t
| p
r
m.V`ıp |m nˆen
t ≤ r.DoP ⊂ H v`a |P| = p
r
,tac´ot = r v`a v`ıvˆa
.
y P = H.
20
b) V´o
.
imˆo
˜
i g ∈ G, ´anh xa
.
P −→ g
−1
Pg : x → g
−1
xg l`a mˆo
.
t song ´anh, nˆen
|g
−1
Pg| = |P| = p
r
. Do d¯´o g

−1
Pg l`a p-nh´om con Sylow, v´o
.
imˆo
˜
i g ∈ G.DoG
chı
’
c´o p-nh´om con Sylow duy nhˆa
´
tl`aP nˆen g
−1
Pg = P, ∀g ∈ G hay PG.
38.
a) Sˆo
´
q-nh´om con Sylow cˆa
´
p q cu
’
a G l`a s
q
=1+kq,v´o
.
i k ≥ 0 n`ao d¯´o. Ngo`ai
ra, 1 + kq chia hˆe
´
t pq, nˆen c´o bˆo
´
n kha

’
n˘ang xa
’
y ra: 1 + kq = q ho˘a
.
c1+kq = p
ho˘a
.
c1+kq = pq ho˘a
.
c1+kq =1. V`ıq khˆong chia hˆe
´
t1+kq,chı
’
c`on hai kha
’
n˘ang 1 + kq = p ho˘a
.
c1+kq =1. V`ıq>pnˆen 1 + kq = p v`a do d¯´o 1 + kq =1.
Vˆa
.
y c´o d¯´ung mˆo
.
t nh´om con cˆa
´
p q.
b) Sˆo
´
p-nh´om con Sylow cˆa
´

p p cu
’
a G l`a s
p
=1+kp,v´o
.
i k ≥ 0 n`ao d¯´o. Lˆa
.
p
luˆa
.
nnhu
.
trˆen, ta c´o hai kha
’
n˘ang xa
’
y ra: 1 + kp = 1 ho˘a
.
c1+kp = q.Tru
.
`o
.
ng
ho
.
.
p cuˆo
´
i l`a khˆong d¯´ung theo gia

’
thiˆe
´
t, nˆen chı
’
c´o mˆo
.
t nh´om con cˆa
´
p p cu
’
a G.
Go
.
i H l`a nh´om con cˆa
´
p q v`a K l`a nh´om con cˆa
´
p p cu
’
a G. Khi d¯´o v´o
.
i
h ∈ H, k ∈ K, h =1,k= 1, ta c´o H =<h>Gv`a K =<k>G. Ngo`ai ra,
H ∩ K = {1} v`ı c´ac phˆa
`
ntu
.
’
kh´ac d¯o

.
nvi
.
cu
’
a H c´o cˆa
´
p q v`a cu
’
a K c´o cˆa
´
p p.Do
h
−1
k
−1
hk = h
−1
(k
−1
hk) ∈ H v`ı HG
=(h
−1
k
−1
h)k ∈ K v`ı KG
nˆen h
−1
k
−1

hk =1hayhk = kh. Theo d¯i
.
nh l´y Lagrange, cˆa
´
pcu
’
a hk l`a p, q ho˘a
.
c
pq.Nhu
.
ng (hk)
p
= h
p
k
p
v`ı h giao ho´an v´o
.
i k,v`ıvˆa
.
y(hk)
p
= h
p
=1. Tu
.
o
.
ng tu

.
.
(hk)
q
= k
q
= 1. Vˆa
.
ycˆa
´
pcu
’
a hk l`a pq hay G l`a nh´om cyclic sinh bo
.
’
i hk.
39.
a) V´o
.
i g,h ∈ G,nˆe
´
u g
−1
ϕ(g)=h
−1
ϕ(h)th`ıϕ(g)ϕ(h)
−1
= gh
−1
hay

ϕ(gh
−1
)=gh
−1
.T`u
.
gia
’
thiˆe
´
tvˆe
`
ϕ ta pha
’
ic´ogh
−1
=1
G
hay g = h. Do d¯´o
{g
−1
ϕ(g) | g ∈ G} = G.
b) ∀α ∈ G, ∃g ∈ G sao cho α = g
−1
ϕ(g) v`a ta c´o
ϕ(α)=ϕ(g
−1
ϕ(g)) = ϕ(g
−1
).ϕ(ϕ(g)) = ϕ(g)

−1
g =(g
−1
ϕ(g))
−1
= α
−1
.
∀α, β ∈ G, ϕ(αβ)=(αβ)
−1
= β
−1
α
−1
= ϕ(β)ϕ(α)=ϕ(βα). Do d¯´o
αβ = βα, ∀α, β ∈ G hay G l`a aben.
V´o
.
imo
.
i α ∈ G, α =1
G
,tac´oα = ϕ(α)=α
−1
.Nhu
.
vˆa
.
y G khˆong c´o phˆa
`

n
tu
.
’
cˆa
´
p2. T`u
.
d¯´o suy ra G c´o cˆa
´
ple
’
.
21
B
`
AI T
ˆ
A
.
P CHU
.
O
.
NG II – V
`
ANH
1.
Cho S l`a mˆo
.

ttˆa
.
pho
.
.
p, k´yhiˆe
.
u P(S) l`a tˆa
.
pgˆo
`
m c´ac tˆa
.
p con cu
’
a S.Ch´u
.
ng
to
’
r˘a
`
ng P(S)v´o
.
i 2 ph´ep to´an cˆo
.
ng v`a nhˆan nhu
.
sau:
A + B =(A ∪ B) \ (A ∩ B) ,AB= A ∩ B,∀A, B ∈P(S)

l`a mˆo
.
t v`anh giao ho´an c´o d¯o
.
nvi
.
.
2.
Cho R l`a mˆo
.
t v`anh, Z l`a v`anh c´ac sˆo
´
nguyˆen, trˆen tˆa
.
p Z × R ta d¯i
.
nh ngh˜ıa
2 ph´ep to´an cˆo
.
ng v`a nhˆan nhu
.
sau:
(m, x)+(n, y)=(m + n, x + y) , (m, x)(n, y)=(mn, my + nx + xy) .
Ch´u
.
ng minh r˘a
`
ng Z × R v´o
.
i 2 ph´ep to´an n`ay l`a mˆo

.
t v`anh c´o d¯o
.
nvi
.
v`a R d¯ ˘a
’
ng
cˆa
´
uv´o
.
imˆo
.
t id¯ˆean cu
’
a v`anh n`ay.
3.
Cho S l`a mˆo
.
ttˆa
.
pho
.
.
p, R l`a mˆo
.
t v`anh v`a f l`a mˆo
.
t song ´anh t`u

.
R lˆen S.
Ch´u
.
ng minh r˘a
`
ng S v´o
.
i 2 ph´ep to´an:
a + b = f (f
−1
(a)+f
−1
(b)) ,ab= f(f
−1
(a)f
−1
(b)) ,∀a, b ∈ S
l`a mˆo
.
t v`anh v`a f l`a mˆo
.
td¯˘a
’
ng cˆa
´
u v`anh. D`ung d¯iˆe
`
u n`ay d¯ˆe
’

ch´u
.
ng minh r˘a
`
ng
mˆo
.
t v`anh bˆa
´
tk`y c´o d¯o
.
nvi
.
1c˜ung c`on l`a mˆo
.
t v`anh d¯ˆo
´
iv´o
.
i 2 ph´ep to´an:
a ⊕ b = a + b − 1 ,a∗ b = a + b − ab.
4.
a) Cho R l`a mˆo
.
t v`anh. Ch´u
.
ng minh r˘a
`
ng
Z(R)={a ∈ R | ax = xa, ∀x ∈ R}

l`a mˆo
.
t v`anh con giao ho´an cu
’
a R go
.
i l`a tˆam cu
’
a R.Nˆe
´
u R l`a mˆo
.
tthˆe
’
th`ı Z(R)
c´o cˆa
´
u tr´uc g`ı?
b) X´ac d¯i
.
nh tˆam cu
’
a v`anh M(3, R) c´ac ma trˆa
.
n vuˆong cˆa
´
p3hˆe
.
sˆo
´

thu
.
.
c.
5.
Mˆo
.
t v`anh R d¯ u
.
o
.
.
cgo
.
il`amˆo
.
t v`anh Boole nˆe
´
uv´o
.
imˆo
˜
i a ∈ R, a
2
= a.
Cho R l`a mˆo
.
t v`anh Boole. Ch´u
.
ng minh r˘a

`
ng:
a) R c´o d¯˘a
.
csˆo
´
2.
b) R l`a mˆo
.
t v`anh giao ho´an.
c) Nˆe
´
u R khˆong c´o u
.
´o
.
ccu
’
a 0 th`ı ho˘a
.
c R = {0} ho˘a
.
c R chı
’
c´o hai phˆa
`
ntu
.
’
.

6.
Cho R l`a v`anh c´o d¯o
.
nvi
.
1 =0v`ax, y ∈ R.Ch´u
.
ng minh r˘a
`
ng:
a) Nˆe
´
u xy v`a yx kha
’
nghi
.
ch th`ı x v`a y kha
’
nghi
.
ch.
22
b) Nˆe
´
u R khˆong c´o u
.
´o
.
ccu
’

a khˆong v`a xy kha
’
nghi
.
ch th`ı x v`a y kha
’
nghi
.
ch.
7.
Cho R l`a v`anh c´o d¯o
.
nvi
.
1 =0.
a) Ch´u
.
ng minh r˘a
`
ng nˆe
´
u a ∈ R, a =0, c´o nghi
.
ch d¯a
’
o tr´ai th`ı a khˆong l`a
u
.
´o
.

ccu
’
a 0 bˆen tr´ai v`a d¯iˆe
`
u ngu
.
o
.
.
cla
.
ivˆa
˜
n d¯ ´u n g n ˆe
´
u a ∈ aRa.
b) V´o
.
i a, b ∈ R,ch´u
.
ng minh r˘a
`
ng nˆe
´
u1− ba kha
’
nghi
.
ch tr´ai th`ı1− ab c˜ung
kha

’
nghi
.
ch tr´ai.
8.
Cho R l`a v`anh h˜u
.
uha
.
n. Ch´u
.
ng minh r˘a
`
ng
a) Nˆe
´
u R khˆong c´o u
.
´o
.
ccu
’
a khˆong th`ı n´o c´o d¯o
.
nvi
.
v`a mo
.
i phˆa
`

ntu
.
’
kh´ac
khˆong cu
’
a R d¯ ˆe
`
u kha
’
nghi
.
ch.
b) Nˆe
´
u R c´o d¯o
.
nvi
.
th`ı mo
.
i phˆa
`
ntu
.
’
kha
’
nghi
.

ch mˆo
.
tph´ıa trong R d¯ ˆe
`
u kha
’
nghi
.
ch.
9.
Cho R l`a mˆo
.
t v`anh. Mˆo
.
t phˆa
`
ntu
.
’
x cu
’
a R d¯ u
.
o
.
.
cgo
.
il`al˜uy linh nˆe
´

utˆo
`
nta
.
i
n ∈ N
∗
sao cho x
n
= 0. Ch´u
.
ng minh r˘a
`
ng:
a) Nˆe
´
u x, y l˜uy linh v`a giao ho´an th`ı x + y c˜ung l`a l˜uy linh.
b) Nˆe
´
u x l˜uy linh v`a xy = yx th`ı xy c˜ung l`a l˜uy linh.
c) Nˆe
´
u x l˜uy linh th`ı1− x kha
’
nghi
.
ch v`a t´ınh (1 − x)
−1
.
10.

Cho p l`a mˆo
.
tsˆo
´
nguyˆen tˆo
´
.Ch´u
.
ng minh r˘a
`
ng tˆa
.
pho
.
.
p c´ac sˆo
´
h˜u
.
utı
’
c´o da
.
ng
m/n, trong d¯´o n nguyˆen tˆo
´
v´o
.
i p, l`a mˆo
.

tmiˆe
`
n nguyˆen. T`ım tru
.
`o
.
ng c´ac thu
.
o
.
ng
cu
’
amiˆe
`
n nguyˆen n`ay.
11.
Ch´u
.
ng minh r˘a
`
ng mo
.
imiˆe
`
n nguyˆen h˜u
.
uha
.
n d¯ ˆe

`
u l`a tru
.
`o
.
ng.
12.
Cho R l`a mˆo
.
t v`anh giao ho´an, kh´ac khˆong v`a c´o d¯o
.
nvi
.
.Ch´u
.
ng minh r˘a
`
ng
c´ac d¯iˆe
`
u sau l`a tu
.
o
.
ng d¯u
.
o
.
ng:
a) R l`a mˆo

.
t tru
.
`o
.
ng.
b) R chı
’
c´o hai id¯ˆean l`a {0} v`a R.
c) Mo
.
id¯ˆo
`
ng cˆa
´
u kh´ac khˆong t`u
.
v`anh R v`ao mˆo
.
t v`anh kh´ac khˆong d¯ˆe
`
ul`a
d¯ o
.
ncˆa
´
u.
13.
Ch´u
.

ng minh r˘a
`
ng tˆa
.
pho
.
.
p c´ac ma trˆa
.
n c´o da
.
ng

ab
3ba

,v´o
.
i a, b l`a nh˜u
.
ng
sˆo
´
h˜u
.
utı
’
t`uy ´y, l`a mˆo
.
t tru

.
`o
.
ng d¯ˆo
´
iv´o
.
i ph´ep cˆo
.
ng v`a ph´ep nhˆan ma trˆa
.
n, tru
.
`o
.
ng
n`ay d¯˘a
’
ng cˆa
´
uv´o
.
i tru
.
`o
.
ng A = {a + b
√
3 | a, b ∈ Q}, Q l`a tru
.

`o
.
ng c´ac sˆo
´
h˜u
.
utı
’
.
14.
Ch´u
.
ng minh r˘a
`
ng tˆa
.
pho
.
.
p c´ac ma trˆa
.
n c´o da
.
ng

ab
−ba

,v´o
.

i a, b l`a nh˜u
.
ng
sˆo
´
thu
.
.
ct`uy ´y, l`a mˆo
.
t tru
.
`o
.
ng d¯ˆo
´
iv´o
.
i ph´ep cˆo
.
ng v`a ph´ep nhˆan ma trˆa
.
n, tru
.
`o
.
ng
n`ay d¯˘a
’
ng cˆa

´
uv´o
.
i tru
.
`o
.
ng C c´ac sˆo
´
ph´u
.
c.
23
15.
a) Cho p l`a mˆo
.
tsˆo
´
nguyˆen tˆo
´
.K´yhiˆe
.
u
Q(
√
p)={a + b
√
p | a, b ∈ Q},
trong d¯´o Q l`a tru
.

`o
.
ng c´ac sˆo
´
h˜u
.
utı
’
.Ch´u
.
ng minh r˘a
`
ng Q(
√
p) l`a mˆo
.
t tru
.
`o
.
ng
(tru
.
`o
.
ng con cu
’
a tru
.
`o

.
ng R c´ac sˆo
´
thu
.
.
c).
b) Ch´u
.
ng minh r˘a
`
ng tru
.
`o
.
ng Q(
√
7) khˆong d¯˘a
’
ng cˆa
´
uv´o
.
i tru
.
`o
.
ng Q(
√
11).

16.
H˜ay t`ım c´ac tu
.
.
d¯ ˆo
`
ng cˆa
´
ucu
’
a tru
.
`o
.
ng F:
a) F l`a tru
.
`o
.
ng Q c´ac sˆo
´
h˜u
.
utı
’
.
b) F l`a tru
.
`o
.

ng R c´ac sˆo
´
thu
.
.
c.
c) F l`a tru
.
`o
.
ng Z
p
c´ac sˆo
´
nguyˆen mˆod¯ulˆo p,v´o
.
i p l`a mˆo
.
tsˆo
´
nguyˆen tˆo
´
.
d) F l`a tru
.
`o
.
ng C c´ac sˆo
´
ph´u

.
c m`a gi˜u
.
nguyˆen c´ac sˆo
´
thu
.
.
c.
17.
Trˆen v`anh M(2, C) c´ac ma trˆa
.
n vuˆong cˆa
´
p2hˆe
.
sˆo
´
trˆen tru
.
`o
.
ng c´ac sˆo
´
ph´u
.
c
C, x´et tˆa
.
p con

Q =


ab
−
b a

| a, b ∈ C

.
a) Ch´u
.
ng minh r˘a
`
ng Q l`a mˆo
.
tthˆe
’
con cu
’
a v`anh M(2, C).
b) D
-
˘a
.
t I =

i 0
0 −i


,J=

01
−10

,K=

0 i
i 0

.D
-
ˆo
`
ng nhˆa
´
tsˆo
´
thu
.
.
c
a v´o
.
i

a 0
0 a

∈Q.Ch´u

.
ng minh r˘a
`
ng I
2
= J
2
= K
2
= −1,IJ= −JI =
K, JK = −KJ = I, KI = −IK = J v`a mo
.
i phˆa
`
ntu
.
’
cu
’
a Q d¯ ˆe
`
u c´o da
.
ng:
a
1
+ a
2
I + b
1

J + b
2
K, a
1
,a
2
,b
1
,b
2
∈ R.
(Thˆe
’
Q d¯ u
.
o
.
.
cgo
.
il`athˆe
’
quaternion.)
18.
Cho K l`a mˆo
.
tthˆe
’
v`a x, y ∈ K \{0} sao cho x + y = −1v`ax
−1

+ y
−1
=1.
Ch´u
.
ng minh r˘a
`
ng:
xy = −1,x
4
+ y
4
=7.
(O
.
’
d¯ˆay ta k´y hiˆe
.
u n thay cho n1
K
,v´o
.
i n ∈ Z v`a 1
K
l`a d¯o
.
nvi
.
cu
’

a K.)
19.
Tˆo
`
nta
.
i hay khˆong mˆo
.
tthˆe
’
K sao cho c´ac nh´om K v´o
.
i ph´ep cˆo
.
ng v`a K\{0}
v´o
.
i ph´ep nhˆan d¯˘a
’
ng cˆa
´
uv´o
.
i nhau?
20.
K´y hiˆe
.
u T l`a v`anh tˆa
´
tca

’
c´ac ma trˆa
.
n tam gi´ac du
.
´o
.
icˆa
´
p 3 trˆen v`anh Z c´ac
sˆo
´
nguyˆen. D
-
˘a
.
t
I =



000
a 00
b 2c 0








a, b, c ∈ Z

,J =



000
l 00
2m 2n 0







l, m, n ∈ Z

.
24
Ch´u
.
ng minh r˘a
`
ng I l`a id¯ˆean 2 ph´ıa cu
’
a T , J l`a id¯ˆean 2 ph´ıa cu
’
a I v`a J l`a id¯ˆean

pha
’
icu
’
a T nhu
.
ng khˆong l`a id¯ˆean tr´ai.
21.
X´et v`anh Z c´ac sˆo
´
nguyˆen.
a) H˜ay t`ım tˆa
´
tca
’
c´ac id¯ˆean cu
’
a v`anh Z.
b) Ch´u
.
ng to
’
r˘a
`
ng mo
.
i d˜ay t˘ang c´ac id¯ˆean cu
’
a Z d¯ ˆe
`

ud`u
.
ng, t´u
.
cl`anˆe
´
u
I
1
⊂ I
2
⊂···⊂I
n
⊂··· l`a d˜ay c´ac id¯ˆean cu
’
a Z th`ı tˆo
`
nta
.
isˆo
´
nguyˆen i sao cho
v´o
.
imo
.
i j l´o
.
nho
.

n i th`ı I
j
= I
i
.
D
-
ˆo
´
iv´o
.
i d˜ay gia
’
m c´ac id¯ˆean cu
’
a Z th`ı thˆe
´
n`ao?
22.
Cho R v`a S l`a c´ac v`anh c´o d¯o
.
nvi
.
.Ch´u
.
ng minh r˘a
`
ng M l`a mˆo
.
t id¯ˆean cu

’
a
v`anh t´ıch R× S khi v`a chı
’
khi M = I × J, trong d¯´o I v`a J lˆa
`
nlu
.
o
.
.
t l`a c´ac id¯ˆean
cu
’
a R v`a S.
T`ım c´ac id¯ˆean cu
’
a c´ac v`anh t´ıch Z
2
, R
2
, trong d¯´o Z v`a R lˆa
`
nlu
.
o
.
.
t l`a v`anh
c´ac sˆo

´
nguyˆen v`a c´ac sˆo
´
thu
.
.
c.
23.
Cho R l`a mˆo
.
t v`anh giao ho´an c´o d¯o
.
nvi
.
.
a) Ch´u
.
ng to
’
mo
.
i id¯ˆean cu
.
.
cd¯a
.
icu
’
a R d¯ ˆe
`

u l`a id¯ˆean nguyˆen tˆo
´
.
b) Gia
’
su
.
’
R c´o t´ınh chˆa
´
t: ∀x ∈ R tˆo
`
nta
.
isˆo
´
tu
.
.
nhiˆen n>1 sao cho x
n
= x.
Ch´u
.
ng to
’
mo
.
i id¯ˆean nguyˆen tˆo
´

c˜ung l`a id¯ˆean cu
.
.
cd¯a
.
i.
24.
K´y hiˆe
.
u Z[ i ]={a + ib ∈ C | a, b ∈ Z}, trong d¯´o C l`a tru
.
`o
.
ng c´ac sˆo
´
ph´u
.
c
v`a Z l`a v`anh c´ac sˆo
´
nguyˆen. V´o
.
i u ∈ Z[ i ], k´yhiˆe
.
u(u)={ux | x ∈ Z[ i ]}.
Ch´u
.
ng minh:
a) Z[ i ] l`a v`anh con cu
’

a C v`a (u) l`a id¯ˆean cu
’
a Z[ i ].
b) V`anh thu
.
o
.
ng Z[ i ]/(2) khˆong pha
’
i l`a tru
.
`o
.
ng.
c) V`anh thu
.
o
.
ng Z[ i ]/(3) l`a tru
.
`o
.
ng c´o 9 phˆa
`
ntu
.
’
.
25.
Cho R l`a mˆo

.
t v`anh giao ho´an v`a I l`a mˆo
.
t id¯ˆean sinh ra bo
.
’
i phˆa
`
ntu
.
’
a ∈ R.
Ch´u
.
ng minh r˘a
`
ng:
I =

{ra | r ∈ R} nˆe
´
u R c´o d¯o
.
nvi
.
{ra + na | r ∈ R v`a n ∈ Z} nˆe
´
u R khˆong c´o d¯o
.
nvi

.
.
26.
K´y hiˆe
.
u M (2, F) l`a v`anh c´ac ma trˆa
.
n vuˆong cˆa
´
p2hˆe
.
sˆo
´
trˆen tru
.
`o
.
ng F.
Ch´u
.
ng to
’
r˘a
`
ng:
a) Tˆa
.
pho
.
.

p


ab
00

∈ M (2, F) | a, b ∈ F

l`a 1 id¯ˆean pha
’
i m`a khˆong l`a
id¯ˆean tr´ai cu
’
a M (2, F).
b) V`anh M (2, Z
2
) l`a mˆo
.
t v`anh d¯o
.
n ngh˜ıa l`a M (2, Z
2
) khˆong c´o id¯ˆean n`ao
kh´ac ngo`ai id¯ˆean khˆong v`a ch´ınh n´o (Z
2
l`a tru
.
`o
.
ng c´ac sˆo

´
nguyˆen mod 2).
25