Ejercicios de Estructuras Algebraicas I

7/22/2019 Ejercicios de Estructuras Algebraicas I

1/31

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


2/31

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


3/31

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


4/31

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


5/31

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, )


6/31

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


7/31

l.c.i.

S

x S

xx= x


8/31


9/31

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)


10/31

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


11/31

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


12/31


13/31

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

()


14/31

(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


15/31

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


16/31

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


17/31

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


18/31

(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


19/31

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


20/31

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


21/31

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


22/31

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


23/31

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


24/31

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]


25/31

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


26/31

G

G

p

q

p < q

G

pq

q

G

q

p

G

G

G


27/31

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])


28/31

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)


29/31


30/31


31/31

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Ejercicios de Estructuras Algebraicas I