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7/22/2019 Ejercicios de Estructuras Algebraicas I
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p Z+ p
p1 mod 4
p= 2
x= x0+ x1(10) + x2(10)
2 + . . . + xk(10)k
x0 x1+ x2 x3+. . .+ (1)kxk
518 39
270 + 370
17363
17283
298
p
n
p
1n < p
p
n
0 modp
p
n
pk=1
nmcd(k,p) 0 modp
x2 +y2 =z2
x
y
z
xyz
x
y
z
xyz0 mod 60
7/22/2019 Ejercicios de Estructuras Algebraicas I
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m
a
b
m am2 + am3b + am4b + + bm2 m
n N
An= 2n + 4n + 8n
nm mod(3)
AnAmmod 7
a
b
c Z
mZ+
acbc modm
ab modm/d
d= (c, m)
p
Z+
Zp
[a][b] = 0
[a] = 0
[b] = 0
p Z+
np n mod
p
a
b
p Z+
(a + b)p
ap +bp modp
p
p4 1
7/22/2019 Ejercicios de Estructuras Algebraicas I
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l.c.i.
Z Z
(a, b) (c, d) = (ac + 2bd,ad + bc)
Z
a b= a + b + 4
R+
a b= ab+1
a+b
1.b
G=
l.c.i.
G
a, bG
ab =a xb x
ab =x ax b
xG
A={1, 3, 5, 7, 9}
A
ab=
ab
l.c.i.
A
S
S
l.c.i.
f :SS
S
abf(a) =f(b)
S
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S
(b)
S
f(S)
R2 (x, y) (a, b) y = b
l.c.i.
R2
R2
(a, b) (c, d) = (3 + ac, 2b 4d2 + bd 1)
Q
Q
a
b= a +b+ab
Q
R
x y x2 x = y2 y
l.c.i.
ab = a2 +b2 (a+ b) + 1
S= Z+ Z+
(a, b) (a, b) = (aa, bb)
S
f : (S, ) (Q, )
f(a, b) = a
b
S
f
S
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Q
a b= a +b kab
k
Q
Q
(Q, )
(Q {}, )
S = QQ
(a, b)
(x, y) = (ax, ay+ b)
(3, 4) (1, 2)
(1, 3) (5, 2)
S
S
Z+
Z+
(Z+
, )
Z+
(b)
Z+
G ={X = (xn)nN xn =1 1, n Z+}
XY = (xnyn)nN
(G, )
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G
G
(ab)2 = a2b2
a, bG
G
e
a2 = e, a G
G
({a,b,c,d,p,q}, )
G
a
a2
G
G
e
x, yG
xn =e
(y1xy)n =e
(G, )
a
b
G
a x= b
y a= b
G
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l.c.i.
S
x S
xx= x
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7/22/2019 Ejercicios de Estructuras Algebraicas I
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C([0,1]) ={f : [0, 1] R f }
(C([0,1]),+) +
H={f C([0,1])
01
f(x)dx= 0}
H
C([0,1])
H={f C([0,1]) } HC([0,1])
S3 x, y
(xy)2
=x2y2
(G, )
H G
H
H
(G, )
H={S8 2 =I}
S8
(G, )
H
G
G
H
G
A
B(A) ={f : AA
f
}
(B(A), )
a A
Ha ={f B(A) f(a) = a} Ha
B(A)
CA
{f B(A)
f(a) =a
aC}
B(A)
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G
e
x G x2 = e
H G
H
x
xn =e
n1
G
e
(G, )
n Z+
H ={xn
x G}
G
(L(R2), +)
GLn(R) ={A Rnn x G}
n
(GLn(R), )
SLn(R) ={AGLn(R) det(A) = 1}
On(R) ={AGLn(R) AAt =I}
SOn(R) =On(R) SLn(R)
H={ a bb a a, b R} (H, ) (GL2(R), )
(G, )
H
G
H
ab modHab1 H
a, bG
Ha ={ha
h H}
H
Ha= [a]
(G, ) aG N(a) ={xG xa= ax}
N(a)
G
S
G
Hs ={x G xs = sx s S} G
G
Z
G
Z ={z G
zx = xz
x G}
Z
G
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ab a, b
R)
ab :xax + b G={ab a= 0}
G
abcd
H={abG a } H G
N={1bG}
N
G
aG
nN
ana1 N
(C, )
C = C {0}
C
S1 ={z C
|z|= 1}
Un={z C z }
H
K
(G, )
H K
(G, )
HK
KH
(G, ) a, bG a b b a
H
K
G
HK
HK={hk
hH
kK}
G
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7/22/2019 Ejercicios de Estructuras Algebraicas I
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G
H K G
[G : H]
[H : K]
[G: K]
[G: K] = [G: H][H :K]
G
H
K
G
(|H| , |K|) = 1
H K={e}
G
x
n
G
d
n
G
d
G
G
=e
() = 2
= 1
G
T(G)
G
T(G) ={ G
()
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(xk) =n/(k, n)
k
= 0
k
n
(xk) =n/k
G
a=e
n
aZ(G)
n= 2
Z(G) ={aG
ax= xa
xG}
nZ+
(n)
1k n
n
: Z+ Z+
G =x
n
G
(n)
xk
G(k, n) = 1
p, n Z+
p
(pn) =pn pn1
G
n
xn =e
xG
n Z+
(a, n) = 1
a(n) 1 modn
p
G
2p
G
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G
AG
C(A) ={gG
ga = ag
aA}
C(A)G
A
G=A C(A)
AG
B G=ABG
AB ={ab
aA bB}
G
e
N G
[G: N] =n
xG xm =e (m, n) = 1 xN
G
N G
Z(N)G
SOn(R) On(R)
G
HKG
H G
H K
G
Ln ={x G xn = e} G
n N
G
M
N
G
M N={e}
mn= nm
mm nN
G
S={xyx1y1
x, yG}
S G
HG
xG
xHx1 G
G
n
G
n
G
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G G
G
H
G
[G: H] = 2
H G
a /H
b /H=abH
{a2
aG} H
n
G
(xy)n =xnyn
x, yG
H=
{x
G xn =e
} K=
{xn
x
G}
H G
K G
N
G
N
N
G
Z(G)
H
G
H
G
N(H) ={gG gH g1 =H}
N(H)
G
H
N(H)
H
K
G
KN(H)
N(H)
G
H
H
GN(H) =G
H
K
G
H
N(K)
K
G
HK
G
H
G K
G
HK
G
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G
gG
Ig :GG Ig(x) =gxg1
I(G) ={Ig g G}
G
(I(G), )(Aut(G), )
f : (R, +)(GL2(R), ) f(x) =
cos x sin x
sin x cos x
f
: (Z, +) ({z C/zn = 1}, )
(k) =e2ki
n
(R+, )
(R, +) Z
3Z
(R, +) (R, +) (Z, +)
({(a, b)
a, b Q}, +}
f : G1 G2 x
f(x)
x G1
A
n
k
n
f : A A
f(a) = ak
G
n
p
n
aG
xp =a
G
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(Z, +)
(G,
)
g
G
f : Z G G
g
G = (R2, +)
M
fM : G G fM(x, y) = (x, y)M
fM
f
G
G
f1
G
G
C
G= G G G
f : G G
f(g) = g1
G
G
f : G1 G2 x y
G1 f(x) f(y) G2
G2 G1
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H
K
G
H K
H
K
G
HK
K G
L G
K L= G
K L={e}
G/L=K G/K=L
G
M G
N G
H G
H M =H N
HMM
= HNN
H
K
G
N
G
HN=K N HHN = KKN
f
G
Z n N G
n
G
Zn
G
H
K
G
K G
|H|
[G: K]
HK
G
N
G
G/N
= Z
n G
n
G
Z
G1 G2 f :G1G2
Nu(f)
J
G1
N u(f)
J
G1 f(J)
G2
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G1 G2 f Nu(f) J1 J2
G1 N u(f) J1contieneaJ2
[J1:J2] = [f(J1) :f(J2)]
H
K
G
|H||K|=|HK||H K|
H
K
L
G
H K
K L
H=K=L
A= G/H
B =K/H
C=L/H
B
C
A
B
G
(A/B)/(C/B)
N G
N
G/N
H
G
K
H
K
H/K
G
N
M
G
H
G
n
m
h
(n, h) = 1
(m, h) = 1
HMM
= HNN
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G
d Z+
G
d
G
d
k Z+
d
k
[d, k]
d
k
(d, k)
G
a G
G
a
f : G G
f(a)
a
f
f(a)) =
a
Z30 25
Z42 30
Z12 Z13 Z36 Z50
G
G
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G=x
n
(n, s) = 1G =xs
p
e
f
G
Im(f)
|H|
|Im(f)|
|G|
G
N
G
G/N
G
m
x
y
G
(m, s) = 1
xs =ys
x= y
G
H
G
K
H
K
G
n
m
nZ
mZ
(Z, +)
nZ mZ mZ/nZ
G
a
G
ax= xaxG
G
H
K
|H| = r
|K| = s
(r, s) = 1
G
n
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Z3 Z4
A
B
A B
A
B
G ={(a, b)
a, b
}
G
(a, b) (c, d) = (a+c(1)b, b+d)
(G, )
G
Z Z
A B A B B A
G
M=G G
D={(g, g)G G
gG}
G
D
M
G
M =x
N =y
m
n
(m, n) = 1
M N
M
N
m
n
M N
(m, n) = 1
A
G
B
H
A B
G H
GHAB
GA H
B
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H
G
K
G
|HK
|= |H||K|
|HK|
G
H
K
|H||K|=|G|
H K={e}
G
H K
HK=G
G
H K
Z2 S3
H
H
K
K
H=H K=K
H K=H K
(Z6, +) M N (Z6, +)
(M N, +)
Z2 Z4 (Z8, +)
Zm Zn= Zmn(m, n) = 1
(a1, a2, , an)
i=1n Gi ai ri
Gi (a1, a2, , an) i=1
n Gi [r1, r2, , rn]
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G
pn
p
Z(G)
pn1
G
G
GZ(G)
G
p2
p
G
|G| = p3
p
|Z(G)|= p
G |G| p
G
p
G
H
p
G
H
G
G
|G|= pnm
n1
p
m
P
p
G
H
p
P HG
H=P
G
pn
n 1
G
pn1
P
p2
p
p
P
Zp Zp
53 73
p
p
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G
G
p
q
p < q
G
pq
q
G
q
p
G
G
G
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X
P(X)
(P(X), , )
(P(X))
(Zp, +, ) (a + b)p =ap + bp
p
(G, +)
G
End(G) ={f :GG
f(a + b) =f(a) + f(b)}
(f+ g)(a) =f(a) + g(a)
(f g)(a) =f(g(a))
f g End(G)
(End(G), +, )
(End(G), +, )
Q[i] =
{a+bi
a, b
Q
}
C
Q[
2]
(a + b
2) + (c +d
2 ) = (a+c) + (b + d)
2
(a + b
2) (c + d2) = (ac + 2bd) + (ad+bc)2
(A)
A
(A)
A= C([0, 1])
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A= R22
A= Z[i]
A= Zn
A= Q[
2]
(A, +)
A
(A, +, )
(A, +)
(1 + 1)(a + b)
A=
{n/2k
n
Z
k
Z
} (A)
p
Ep ={a/b a, b Z p b} Ep
(Ep)
Q ={(qn) qn Q (qn) }
(qn) + (rn) = (qn+ rn)
(qn)(rn) = (qnrn)
>0
N N
|qn qm|< n, m > N
A
B
AB
(a, b) + (c, d) = (a+ c, b+ d)
(a, b)(c, d) = (ac, bd) A B
AB
A
B
AB
m
[2] (Zm)
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