[PDF] aata-20200730pdf - Abstract Algebra: Theory and Applications

[PDF] math 113: abstract algebra solutions to practice problems for

27 sept 2007 · MATH 113: ABSTRACT ALGEBRA SOLUTIONS TO PRACTICE PROBLEMS FOR MIDTERM 1 1 Show that if (G, ·) is a group of order 9, then G is abelian

[PDF] EXERCISES AND SOLUTIONS IN GROUPS RINGS AND FIELDS

18 avr 2012 · These notes are prepared in 1991 when we gave the abstract al- gebra course Our intention was to help the students by giving them

[PDF] aata-20200730pdf - Abstract Algebra: Theory and Applications

30 juil 2020 · Often in the solutions a proof is only sketched, and it is up to the student to provide the details The exercises range in difficulty from very

[PDF] Examples in abstract algebra - Penn Math

13 nov 2020 · We know that (a9)2 = e, so a9 is its own inverse Solution 22 Let H be the cyclic subgroup generated by a We have that Why is order(a)

[PDF] aata-20200730pdf - Abstract Algebra: Theory and Applications

1374_6aata_20200730.pdf #bi`+i H;2#` h?2Q`v M/ TTHB+iBQMb #bi`+i H;2#` h?2Q`v M/ TTHB+iBQMb h?QKb qX Cm/bQM ai2T?2M 6X mbiBM aii2 lMBp2`bBiv a;2 1t2`+Bb2b 7Q` #bi`+i H;2#` h`/m++B¦M H 2bT¢QH

CmHv jy- kyky

Ĝ

ǳ Ǵ Ĝ Ĝ Ĝ Ĵ Ƕ Ƕ

Ƕ Ƕ

Ƕ

1 S S S

3 + 5613 + 8/2

2 + 3 = 5

2x= 6 x= 4

ax2+bx+c= 0a̸= 0 x=bp b 24ac
2a x

34x2+ 5x6

ǳ2x= 6 x= 4Ǵ

24 6̸= 8

Ƕ

ǳ10/5 =

2Ǵ ǳ

p qǴ pq p q ax

2+bx+c= 0a̸= 0

x=bp b 24ac
2a ax2+bx+c= 0a̸= 0 x=bp b 24ac
2a ax2+bx+c= 0a̸= 0 ax

2+bx+c= 0

x 2+b a x=c a x 2+b a x+(b 2a) 2 =(b 2a) 2 c a ( x+b 2a) 2 =b24ac 4a2 x+b 2a=p b 24ac
2a x =bp b 24ac
2a rs r=s ǳp qǴ ǳ q pǴ ě x x AX a A a2A x

X=fx1;x2;:::;xng

x1;x2;:::;xn

X=fx:xPg

xX P E E

E=f2;4;6;:::gE=fx:x x >0g

22E E 3 /2E 3
E

N=fn:n g=f1;2;3;:::g;

Z=fn:n g=f:::;1;0;1;2;:::g;

Q=fr:r g=fp/q:p;q2Zq̸= 0g;

R=fx:x g;

C=fz:z g

A B ABBA A B

f4;5;8g f2;3;4;5;6;7;8;9g

NZQRC

B ABA B̸=A A B A̸B f4;7;9g ̸ f2;4;5;8;9g A=B ABBA ∅

A[B AB

A[B=fx:x2Ax2Bg;

A\B=fx:x2Ax2Bg

A=f1;3;5gB=f1;2;3;9g

A[B=f1;2;3;5;9gA\B=f1;3g

n∪ i=1A i=A1[:::[An n∩ i=1A i=A1\:::\An A1;:::;An

E O EO

AB A\B=∅ U

AU A A′

A ′=fx:x2Ux/2Ag AB

AnB=A\B′=fx:x2Ax/2Bg

A=fx2R: 0< x3gB=fx2R: 2x <4g

A\B=fx2R: 2x3g

A[B=fx2R: 0< x <4g

AnB=fx2R: 0< x <2g

A ′=fx2R:x0x >3g □

AB C

A[A=AA\A=A AnA=∅

A[ ∅=AA\ ∅=∅

A[(B[C) = (A[B)[CA\(B\C) = (A\B)\C

A[B=B[AA\B=B\A

A[(B\C) = (A[B)\(A[C)

A\(B[C) = (A\B)[(A\C)

A[A=fx:x2Ax2Ag

=fx:x2Ag =A

A\A=fx:x2Ax2Ag

=fx:x2Ag =A

AnA=A\A′=∅

AB C

A[(B[C) =A[ fx:x2Bx2Cg

=fx:x2Ax2B;x2Cg =fx:x2Ax2Bg [C = (A[B)[C A\(B\C) = (A\B)\C■ Ƕ AB (A[B)′=A′\B′ (A\B)′=A′[B′ A[B=∅ AB (A[B)′A′\B′(A[B)′A′\B′ x2(A[B)′ x/2A[B x A B x2A′x2B′ x2A′\B′ (A[B)′A′\B′ x2A′\B′ x2A′x2B′ x/2Ax/2B x/2A[B x2(A[B)′ (A[B)′A′\B′ (A[B)′=A′\B′ ■ (AnB)\(BnA) =∅ (AnB)\(BnA) = (A\B′)\(B\A′) =A\A′\B\B′ =∅ □ AB AB A

B

AB=f(a;b) :a2Ab2Bg

A=fx;ygB=f1;2;3g C=∅ AB

f(x;1);(x;2);(x;3);(y;1);(y;2);(y;3)g A C=∅ □ n A

1 An=f(a1;:::;an) :ai2Aii= 1;:::;ng

A=A1=A2==An AnA AA

n R3 AB fAB A B a2A b2B (a;b)2f Af B f:A!BAf!B (a;b)2AB f(a) =bf:a7!b A f f(A) =ff(a) :a2Ag B f Ƕ Ƕ A=f1;2;3gB=fa;b;cg f gAB f g 12A B g(1) =ag(1) =b a b c a b c A B A B g f □ f:A!B f:R!R f(x) =x3f:x7!x3 f:Q!Z f(p/q) =p 1/2 = 2/4 f(1/2) = 12 f:A!B fB f(A) =B f a2A b2B f(a) =b f a1̸=a2f(a1)̸=f(a2) f(a1) =f(a2)a1=a2 f:Z!Q f(n) =n/1 f g:Q!Zg(p/q) =pp/q g □ f:A!Bg:B!C fgAC (g◦f)(x) =g(f(x)) A B C a b c X Y Z f g A C X Y Z g◦f f:A!Bg:B!C g◦f:A!C □ f(x) =x2g(x) = 2x+ 5 (f◦g)(x) =f(g(x)) = (2x+ 5)2= 4x2+ 20x+ 25 ( g◦f)(x) =g(f(x)) = 2x2+ 5 f◦g̸=g◦f□ f◦g=g◦f f(x) =x3g(x) =3p x (f◦g)(x) =f(g(x)) =f(3p x) = (3p x)3=x (g◦f)(x) =g(f(x)) =g(x3) =3p x 3=x □ 22

A=(a b

c d) TA:R2!R2 T

A(x;y) = (ax+by;cx+dy)

(x;y)R2 (a b c d)( x y) =(ax+by cx+dy) RnRm □ S=f1;2;3g :S!S (1) = 2; (2) = 1; (3) = 3 (1 2 3 (1)(2)(3)) =(1 2 3

2 1 3)

S :S!S S□ f:A!Bg:B!C h:C!D (h◦g)◦f=h◦(g◦f) fg g◦f fg g◦f fg g◦f h◦(g◦f) = (h◦g)◦f a2A (h◦(g◦f))(a) =h((g◦f)(a)) =h(g(f(a))) = (h◦g)(f(a)) = ((h◦g)◦f)(a) fg c2C a2A (g◦f)(a) =g(f(a)) =c g b2B g(b) =c a2A f(a) =b (g◦f)(a) =g(f(a)) =g(b) =c ■ S idSid S id(s) =s s2S g:B!A f:A!Bg◦f=idAf◦g=idB ǳǴ f1 f f(x) =x3 f1(x) =3p x □ f(x) =x f

1(x) =ex

f(f1(x)) =f(ex) =ex=x f

1(f(x)) =f1(x) =ex=x

□

A=(3 1

5 2)

A R2R2

A(x;y) = (3x+y;5x+ 2y)

TA A T1 A=TA1 A 1=(21 5 3) ; T 1

A(x;y) = (2xy;5x+ 3y)

T 1

A◦TA(x;y) =TA◦T1

A(x;y) = (x;y)

B(x;y) = (3x;0)

B=(3 0

0 0) T 1

B(x;y) = (ax+by;cx+dy)

(x;y) =TB◦T1

B(x;y) = (3ax+ 3by;0)

xy y 0□ =(1 2 3

2 3 1)

S=f1;2;3g

1=(1 2 3

3 1 2)

□ f:A!B g:B!A g◦f=idA g(f(a)) =a a1;a22Af(a1) =f(a2) a

1=g(f(a1)) =g(f(a2)) =a2 f b2B

f a2A f(a) =b f(g(b)) =b g(b)2A a=g(b) f b2B f a2A f(a) =b f a g g(b) =a f■ X

RXX

(x;x)2R x2X (x;y)2R(y;x)2R (x;y)(y;z)2R(x;z)2R R X xy (x;y)2R = = pqr s qs p/qr/s ps=qr p/qr/sr/st/u qs u ps=qrru=st psu=qru=qst s̸= 0pu=qt p/qt/u□ fg R f(x)g(x)f′(x) =g′(x) f(x)g(x) g(x)h(x) f(x)g(x) =c1g(x)h(x) =c2 c

1c2

f(x)h(x) = (f(x)g(x)) + (g(x)h(x)) =c1+c2 f′(x)h′(x) = 0 f(x)h(x)□ (x1;y1)(x2;y2)R2 (x1;y1)(x2;y2)x21+y21=x22+y22 R2□

AB22

22 AB P PAP1=B

A=(1 2

1 1)

B=(18 33

11 20)

ABPAP1=B

P=(2 5

1 3)

I 22

I=(1 0

0 1)

IAI1=IAI=A

AB P PAP1=B

A=P1BP=P1B(P1)1

ABBC PQ PAP1=BQBQ1=C

C=QBQ1=QPAP1Q1= (QP)A(QP)1

□

P X X1;X2;::: Xi\Xj=

∅i̸=j∪ kXk=X X x2X [x] =fy2X:yxg x X X X P=fXig X X Xi X x2X x2[x] [x] X=∪ x2X[x] x;y2X [x] = [y][x]\[y] =∅ [x][y] z2[x]\[y] zxzy xy [x][y] [y][x] [x] = [y] P=fXig X x y y x xyyx x yy z x z ■ (p;q) (r;s) □ f(x)g(x) □ R2(x1;y1)(x2;y2)x21+y21= x

22+y22

□ rs n2N r sn r sn rs n rs=nk k2Z rs(n)

4117 (8)4117 = 24 8

n Z r rr= 0 n rs (n) rs=(sr) n sr nsr (n) rs(n)st(n) kl rs=knst=ln rt n rt=rs+st=kn+ln= (k+l)n rt n 3 [0] =f:::;3;0;3;6;:::g; [1] =f:::;2;1;4;7;:::g; [2] =f:::;1;2;5;8;:::g [0][[1][[2] =Z [0][1] [2] n n □

A=fx:x2Nx g;

B=fx:x2Nx g;

C=fx:x2Nx 5g

A\B B\C A[B

A\(B[C)

A=fa;b;cgB=f1;2;3gC=fxg D=∅

AB BA ABC AD AB AB=BA

A[ ∅=AA\ ∅=∅

A[B=B[AA\B=B\A

A[(B\C) = (A[B)\(A[C)

A\(B[C) = (A\B)[(A\C)

AB A\B=A

(A\B)′=A′[B′

A[B= (A\B)[(AnB)[(BnA)

(A[B)C= (AC)[(BC) (A\B)nB=∅ (A[B)nB=AnB

An(B[C) = (AnB)\(AnC)

A\(BnC) = (A\B)n(A\C)

(AnB)[(BnA) = (A[B)n(A\B) f:Q!Q f f(p/q) =p+ 1 p 2 f(p/q) =3p 3q f(p/q) =p+q q 2 f(p/q) =3p2 7q2p q f:R!R f(x) =ex f:Z!Z f(n) =n2+ 3 f:R!R f(x) =x f:Z!Z f(x) =x2 f:A!Bg:B!C f

1g1 (g◦f)1=f1◦g1

f:N!N f:N!N R2(x1;y1)(x2;y2)x21+y21=x22+y22 f:A!Bg:B!C fg g◦f g◦f g g◦f f g◦f f g g◦f g f f(x) =x+ 1 x1 f f f◦f1 f

1◦f

f:X!Y A1;A2XB1;B2Y f(A1[A2) =f(A1)[f(A2) f(A1\A2)f(A1)\f(A2) f1(B1[B2) =f1(B1)[f1(B2) f

1(B) =fx2X:f(x)2Bg

f1(B1\B2) =f1(B1)\f1(B2) f1(YnB1) =Xnf1(B1) xyRxy mnZmn >0 xyRjxyj 4 mnZmn(6) R2 (a;b)(c;d) a2+b2c2+d2 mn RnRm ǳ xy yx xxǴ R2nf(0;0)g (x1;y1)(x2;y2) (x1;y1) = (x2;y2) R2n(0;0) P(R) 2

1 + 2 ++n=n(n+ 1)

2 n n= 1

23 4

n (n+ 1) n= 1

1 =1(1 + 1)

2 n

1 + 2 ++n+ (n+ 1) =n(n+ 1)

2 +n+ 1 = n2+ 3n+ 2 2 = (n+ 1)[(n+ 1) + 1] 2 (n+ 1) S N S

S(n)

n2N S(n0) n0 k kn0S(k) S(k+ 1) S(n) n n0 n32n> n+ 4 8 = 2

3>3 + 4 = 7

n0= 3 2k> k+ 4k3 2k+1= 22k>

2(k+ 4)

2(k+ 4) = 2k+ 8> k+ 5 = (k+ 1) + 4

k n3□

10n+1+ 310n+ 5 9n2N n= 1

1+1+ 310 + 5 = 135 = 915

9 10k+1+ 310k+ 5 9k1 10 (k+1)+1+ 310k+1+ 5 = 10k+2+ 310k+1+ 5045 = 10(10 k+1+ 310k+ 5)45 9□ (a+b)n=n∑ k=0( n k) a kbnk ab n2N (n k) =n! k!(nk)! (n+ 1 k) =(n k) +(n k1) (n k) +(n k1) =n! k!(nk)!+n! (k1)!(nk+ 1)! = (n+ 1)! k!(n+ 1k)! =(n+ 1 k) n= 1 n 1 (a+b)n+1= (a+b)(a+b)n = (a+b)( n∑ k=0( n k) a kbnk) =n∑ k=0( n k) a k+1bnk+n∑ k=0( n k) a kbn+1k =an+1+n∑ k=1( n k1) a kbn+1k+n∑ k=1( n k) a kbn+1k+bn+1 =an+1+n∑ k=1[( n k1) +(n k)] a kbn+1k+bn+1 = n+1∑ k=0( n+ 1 k) a kbn+1k □

S(n)

n2N S(n0) n0 S(n0);S(n0+

1);:::;S(k) S(k+1)kn0 S(n)

nn0 SZS Z 1

S=fn2N:n1g 12S n2S 0<1

n=n+0< n+1 1n < n+1 n2S n+ 1 S S=N■ N S S S S k 1kn S S n+ 1 S S n+1 n+1

S S S n

S ■ n! n n! = 123(n1)n

1! = 1n! =n(n1)!n >1

ab b >0 qr a=bq+r

0r < b

qr q′r′ q=q′r=r′ qr

S=fabk:k2Zabk0g

02S ba q=a/br= 0 0 /2S

S a >0 ab02S a <0 ab(2a) =a(12b)2S S̸=∅

S r=abq a=bq+rr0

r < b r > b ab(q+ 1) =abqb=rb >0 ab(q+ 1) S ab(q+ 1)< abq r=abq S rb

0 /2Sr̸=b r < b

qr rr′q q′ a=bq+r;0r < ba=bq′+r′;0r′< b bq+r=bq′+r′ r′r b(qq′) =r′r b r′r0r′rr′< b r′r= 0 r=r′q=q′■ ab b=ak k ajb d abdjadjb ab d d ab d′ ab d′jd d=(a;b) (24;36) = 12 (120;102) = 6 ab (a;b) = 1 ab rs (a;b) =ar+bs ab

S=fam+bn:m;n2Zam+bn >0g

S S d=ar+bs d=(a;b) a=dq+r′0r′< d r′>0 r ′=adq =a(ar+bs)q =aarqbsq =a(1rq) +b(sq) S d S r ′= 0da db d ab d′ ab d′jd a=d′hb=d′k d=ar+bs=d′hr+d′ks=d′(hr+ks) d′ d d ab■ ab rs ar+bs= 1 9452415

2415 = 9452 + 525

945 = 5251 + 420

525 = 4201 + 105

420 = 1054 + 0

105420105525105945 1052415

105 9452415 d 9452415

d 105 (945;2415) = 105 rs 945r+ 2415s= 105

105 = 525 + (1)420

= 525 + (1)[945 + (1)525] = 2525 + (1)945 = 2[2415 + (2)945] + (1)945 = 22415 + (5)945 r=5s= 2 rs r= 41s=16 □ (a;b) =d r1> r2>> rn=d b=aq1+r1 a=r1q2+r2 r

1=r2q3+r3

r n2=rn1qn+rn r n1=rnqn+1 rs ar+bs=d d=rn =rn2rn1qn =rn2qn(rn3qn1rn2) =qnrn3+ (1 +qnqn1)rn2 =ra+sb d ab d ab p p >1 p p p1p n >1 ab p pjab pjapjb p a pjb (a;p) = 1 rs ar+ps= 1 b=b(ar+ps) = (ab)r+p(bs) p ab p b= (ab)r+p(bs)■ p1;p2;:::;pn P=p1p2pn+ 1 P pi1in pi Pp1p2pn= 1 P p̸=pi P■ n n >1 n=p1p2pk p1;:::;pk n=q1q2ql k=l qiǶ piǶ n n= 2 n m 1m < n n=p1p2pk=q1q2ql p1p2 pkq1q2 ql p1jqi i= 1;:::;lq1jpj j= 1;:::;k piǶ qiǶ p1=qi q1=pj p1=q1p1pj=q1qi=p1 n ′=p2pk=q2ql k=lqi=pii= 1;:::;k S

S a aa1

a a=a1a21< a1< a1< a2< a a12Sa22S a S a

1=p1pr

2=q1qs

a=a1a2=p1prq1qs a/2S ■ Ƕ n f f(n) n Ĝ 22n+ 1 n Ĝ 2

25+ 1 = 4;294;967;297

Ƕ

4 = 2 + 26 = 3 + 38 = 3 + 5:::

41018

Ƕ

123456792
8452
rs r(84) +s(52) =(84;52) ǳ Ǵ Ƕ ǳǴ 1

2+ 22++n2=n(n+ 1)(2n+ 1)

6 n2N 1

3+ 23++n3=n2(n+ 1)2

4 n2N n!>2nn4 x+ 4x+ 7x++ (3n2)x=n(3n1)x 2 n2N 10n+1+ 10n+ 1 3n2N 4102n+ 9102n1+ 5 99n2N n p a

1a2an1

n n ∑ k=1a k f(n)(x) f(n) n f (fg)(n)(x) =n∑ k=0( n k) f (k)(x)g(nk)(x) 1 + 2 + 22++ 2n= 2n+11n2N 1 2 +1 6 ++1 n(n+ 1)=n n+ 1 n2N x (1+x)n1nxn= 0;1;2;:::

X X P(X)

P(fa;bg) =f∅;fag;fbg;fa;bgg

n n 2n SN

12Sn+ 12Sn2S S=N

ab (a;b) rs (a;b) =ra+sb 1439

234165

17399923

471562

2377119945

43573754

ab rs ar+bs= 1 ab

1;1;2;3;5;8;13;21;:::

f1= 1f2= 1 fn+2=fn+1+fnn2N fn<2n fn+1fn1=f2n+ (1)nn2 fn= [(1 +p 5) n(1p 5) n]/2np 5 n!1fn/fn+1= (p 51)/2
fnfn+1 ab (a;b) = 1 rs ar+bs= 1 (a;s) =(r;b) =(r;s) = 1 x;y2N xy xy 4k4k+1 k a;b;r;s a

2+b2=r2

2b2=s2

ar s b n2N n 0;1;:::;n1 r sZ 0s < n[r] = [s] n ab (a;b) m abm a b n m n ab d=(a;b)m=(a;b) dm=jabj (a;b) =ab (a;b) = 1 (a;c) =(b;c) = 1 (ab;c) = 1 ab c a;b;c2Z (a;b) = 1ajbc ajc p2 2p1 p 6n+ 5 4n1 2 pq p

2= 2q2 p

2 N n

1< n < N 2

3 5 4

N p

N

N= 250 N N

N= 120

N0=N[ f0g Ƕ A:N0N0!N0

A(0;y) =y+ 1;

A(x+ 1;0) =A(x;1);

A(x+ 1;y+ 1) =A(x;A(x+ 1;y))

A(3;1) Ƕ Ƕ A(4;1) A(5;1) ab (a;b) rs (a;b) =ra+sb 3 Z 22 22 n n ab nn ab n Zn Zn 12 [0] =f:::;12;0;12;24;:::g; [1] =f:::;11;1;13;25;:::g; [11] = f:::;1;11;23;35;:::g 0;1;:::;11 [0];[1];:::;[11] Zn ab n (a+b) (n) a+b n n (ab) (n) ab n n

7 + 41 (5)731 (5)

3 + 50 (8)357 (8)

3 + 47 (12)340 (12)

n 0n□ Zn Z8 24 6
n= 24 6 k kn1 (8)

0 1 2 3 4 5 6 7

0 0 0 0 0 0 0 0

0 1 2 3 4 5 6 7

0 2 4 6 0 2 4 6

0 3 6 1 4 7 2 5

0 4 0 4 0 4 0 4

0 5 2 7 4 1 6 3

0 6 4 2 0 6 4 2

0 7 6 5 4 3 2 1

Z8 □

Zn na;b;c2

Z n a+bb+a(n) ab ba(n) (a+b) +ca+ (b+c) (n) (ab)ca(bc) (n) a+ 0a(n) a1a(n) a(b+c)ab+ac(n) a a a+ (a)0 (n) a (a;n) = 1 ba(n) b ab1 (n) n a+b n b+a n (a;n) = 1 rs ar+ns= 1 ns= 1ar ar1 (n) b rab1 (n) b ab1 (n) n ab1 k abnk= 1 d=(a;n) d abnkd 1 d= 1■ A D B C C B D A A D B C A D B C 180
◦ A D B C D A C B A D B C B C A D 180◦360◦
90◦
A B C B C A

3=(A B C

B A C)

A B C C A B

2=(A B C

C B A)

A B C A B C

1=(A B C

A C B)

A B C B A C

2=(A B C

C A B)

A B C C B A

1=(A B C

B C A)

A B C A C B id=(A B C

A B C)

△ABC △ABC AB C

S :S!S 3! = 6

321 = 3! = 6 ABBC CA (A B C

B C A)

120◦
△ABC 11 1 1 (11)(A) =1(1(A)) =1(B) =C (11)(B) =1(1(B)) =1(C) =B (11)(C) =1(1(C)) =1(A) =A 2

11 3

11̸=11 △ABC

= ◦ 12123 12123 1

12312

21231

12312

23121

31212

n G GG!G (a;b)2GG a◦b abG ab (G;◦) G (a;b)7!a◦b (a◦b)◦c=a◦(b◦c) a;b;c2G e2G a2G e◦a=a◦e=a a2G a1 a◦a1=a1◦a=e G a◦b=b◦a a;b2G Z=f:::;1;0;1;2;:::g m;n2Z + ◦ m+n m◦n 0 n2Z n n1 m+n=n+m □ ab a◦b m+nn mn m+ (n) n n Z

5 0123 4

Z5 m+n Z5 2+3 = 3+2 = 0 Z5

Zn=f0;1;:::;n1g

n +

0 1 2 3 4

1 2 3 4 0

2 3 4 0 1

3 4 0 1 2

4 0 1 2 3

(Z5;+) □ Zn Zn 1k=k1 =k k2Zn 0 0k=k0 = 0 kZn Znnf0g 22Z6

02 = 0 12 = 2

22 = 4 32 = 0

42 = 2 52 = 4

k Znk n ZnU(n) U(n) Zn U(8)

1 3 5 7

3 1 7 5

5 7 1 3

7 5 3 1

U(8) □ = S3D3 □ M2(R) 22 GL2(R) M2(R)

A=(a b

c d) GL2(R) A1 AA1=A1A=I I 22 A

A A=adbc̸= 0

I=(1 0

0 1) A2GL2(R) A 1=1 adbc( db c a) AB=BA

GL2(R) □

1 = (1 0 0 1)

I=(0 1

1 0) J=(0i i0) K=(i0 0i) i2=1 I2=J2=K2=1IJ=KJK=IKI=J

JI=KKJ=I IK=J Q8=f1;I;J;Kg

Q8 □ C

C 1 z=a+bi

z 1=abi a 2+b2 z □ G n jGj=n Z5 5 Z jZj=1 G e2G eg=ge=g g2G ee′ G eg=ge=ge′g=ge′=g g2G e=e′ e ee′=e′ e′ ee′=e e=ee′=e′ ■ g′g′′ g G gg′=g′g=egg′′=g′′g=e g′=g′′

g′=g′e=g′(gg′′) = (g′g)g′′=eg′′=g′′

g G g g1

G a;b2G (ab)1=b1a1

a;b2G abb1a1=aea1=aa1=e b1a1ab=e (ab)1=b1a1■

G a2G(a1)1=a

a1(a1)1=e a (a1)1=e(a1)1=aa1(a1)1=ae=a ■ ab G x2G ax=b x

G ab G

ax=bxa=b G ax=b x ax=ba1 x=ex=a1ax=a1b x1x2 ax=b ax1= b=ax2 x1=a1ax1=a1ax2=x2 xa=b ■ G a;b;c2G ba=cab=cab=ac b=c G g2G g0=e n2N g n=ggg| {z } n g n=g1g1g1 | {z } n g;h2G g mgn=gm+n m;n2Z (gm)n=gmn m;n2Z (gh)n= (h1g1)n n2Z G (gh)n=gnhn (gh)n̸=gnhn ZZn ng gn mg+ng= (m+n)g m;n2Z m(ng) = (mn)g m;n2Z m(g+h) =mg+mh n2Z ZZn Ĝ Ĝ Ƕ Ƕ 2Z=f:::;2;0;2;4;:::g

H G HG

G HH G

H=feg G G R

1 a2R

1/a

Q =fp/q:pq g R R1 1 = 1/1 R Q Q p/qr/s pr/qs Q p/q2Q Q ( p/q)1=q/p R Q □ C

H=f1;1;i;ig H C H

HC□

SL2(R) GL2(R)

A=(a b

c d) SL2(R) adbc= 1 SL2(R) 22 SL2(R) A A 1=(db c a)

SL2(R) □

H G G H G G 22M2(R) 22 M2(R) M2(R) (1 0 0 1) +(1 0 01) =(0 0 0 0) GL2(R)□ Z4 02 Z2 Z2Z2 (a;b) + (c;d) = (a+c;b+d) Z2Z2 Z2Z2 H

1=f(0;0);(0;1)gH2=f(0;0);(1;0)g H3=f(0;0);(1;1)gZ4Z2Z2

+ (0;0) (0;1) (1;0) (1;1) (0;0) (0;0) (0;1) (1;0) (1;1) (0;1) (0;1) (0;0) (1;1) (1;0) (1;0) (1;0) (1;1) (0;0) (0;1) (1;1) (1;1) (1;0) (0;1) (0;0) Z2Z2 □ HG eG H h1;h22H h1h22H h2H h12H H G H eH eH=e e G eHeH=eH eeH=eHe=eH eeH=eHeH e=eH H h2H H h ′2H hh′=h′h=e Gh′=h1 H G ■

H G H G

H̸=∅ g;h2Hgh1 H

H G gh12H gh H h H h1 H gh12H HG H̸=∅gh12Hg;h2H g2H gg1=e H g2H eg1=g1 H h1;h22H H h1(h12)1=h1h22H H G■ Ƕ Z8 6 + 7 21 U(16) 57 31 ǳǴ x2Z

3x2 (7)

5x+ 113 (23)

5x+ 113 (26)

9x3 (5)

5x1 (6)

3x1 (6)

G=fa;b;c;dg ◦ a b c d a a c d a b b b c d c c d a b d d a b c ◦ a b c d a a b c d b b a d c c c d a b d d c b a ◦ a b c d a a b c d b b c d a c c d a b d d a b c ◦ a b c d a a b c d b b a c d c c b a d d d d b c (Z4;+) D4 U(12)

S=Rnf1g Sab=a+b+ab

(S;) ABGL2(R)AB̸=BA SL2(R) 0 @1x y 0 1z

0 0 11

A 0 @1x y 0 1z

0 0 11

A0 @1x′y′

0 1z′

0 0 11

A =0 @1x+x′y+y′+xz′

0 1z+z′

0 011 A (AB) =(A)(B)GL2(R) GL2(R) AB GL2(R)

AB2GL2(R)

Zn2=f(a1;a2;:::;an) :ai2Z2g Zn2

(a1;a2;:::;an) + (b1;b2;:::;bn) = (a1+b1;a2+b2;:::;an+bn) Zn2 R=Rn f0g RZ G=RZ ◦G (a;m)◦(b;n) = (ab;m+n) G G g;h2G(gh)n̸=gnhn n! n

0 +aa+ 0a(n)

a2Zn n a1a(n) a2Zn b2Zn a+bb+a0 (n) n n n n a(b+c)ab+ac(n) ab G abna1= (aba1)nn2Z

U(n) Zn n >2 k2U(n)

k2= 1k̸= 1 g1g2gng1ng1n1g11 G a;b2G xa=b G G

Gba=cab=cab=acb=c a;b;c2G

a2=e a G G G a2G a a2=e

G (ab)2=a2b2 abG G

Z3Z3 Z3Z3 Z9

H=f2k:k2Zg H Q

n= 0;1;2;:::nZ=fnk:k2Zg nZ Z Z

T=fz2C:jzj= 1g T C

G 22

( ) 2R G SL2(R)

G=fa+bp

2 :a;b2Qab g

G 22

H={(a b

c d) :a+d= 0} H G SL2(Z) 22 SL2(R) Q8 G G HK G H[K G HK G HK=fhk:h2

Hk2Kg G G

G g2G

Z(G) =fx2G:gx=xg g2Gg

G G ab G a4b=baa3=e ab=ba xy=x1y1 xyG G H G

C(H) =fg2G:gh=hg h2Hg

C(H) G HG

H G g2G gHg1=fghg1:h2Hg

G d

1d2d12

3d1+ 1d2+ 3d3++ 3d11+ 1d120 (10)

d12 (d1;d2;:::;dk)(w1;w2;:::;wk)0 (n) d

1w1+d2w2++dkwk0 (n)

(d1;d2;:::;dk)(w1;w2;:::;wk)0 (n) k d1d2dk 0di< n (wi;n) = 11ik (d1;d2;:::;dk)(w1;w2;:::;wk)0 (n) k d1d2dk 0di< n didj (wiwj;n) = 1i j1k Ƕ (d1;d2;:::;d10)(10;9;:::;1)0 (11) d10 ǳ Ǵ Ĝ 4 ZZn 32Z
3

3Z=f:::;3;0;3;6;:::g

3Z 3 3 ǳǴ 3□

H=f2n:n2Zg H

Q a= 2mb= 2n H ab1= 2m2n= 2mn H

H Q 2□

G a G

⟨a⟩=fak:k2Zg G ⟨a⟩ G a ⟨a⟩a0=e gh ⟨a⟩ ⟨a⟩ g=amh=an mn gh=aman=am+n ⟨a⟩ g=an⟨a⟩ g1=an ⟨a⟩ HGa a H⟨a⟩ ⟨a⟩ Ga ■ ǳǴ ⟨a⟩=fna:n2Zg a2G ⟨a⟩ a G a G=⟨a⟩ G a G a G a n an=e jaj=n n a jaj=1 a 1

5Z6 Z6

22Z63 2
⟨2⟩=f0;2;4g□ ZZn 11 Z Zn Zn Z6 U(9) Z9 U(9) f1;2;4;5;7;8g U(9) 2

1= 2 22= 4

3= 8 24= 7

5= 5 26= 1

□ S3 S3 f;1;2g f;1g f;2g f;3g S 3 fg S3 □

G a2G G gh G

a g=arh=as gh=aras=ar+s=as+r=asar=hg

G ■

G G G G G a H

G H=feg H H

g g an n H g1=an H nn

H an >0 m

am2H m h=am H h′2H h h′2HH Gh′=ak k qr k=mq+r

0r < m

a k=amq+r= (am)qar=hqar ar=akhq akhq Har H m am H r= 0 k=mq h ′=ak=amq=hq

H h■

Z nZn= 0;1;2;:::

G n a

G ak=e nk

ak=e k=nq+r0r < n e=ak=anq+r=anqar=ear=ar m am=enr= 0 nk k=ns s a k=ans= (an)s=es=e ■

G n a2G

b=ak bn/d d=(k;n) m e=bm=akm m nkm n/d m(k/d) d nkn/dk/d n/d m(k/d) m mn/d■ Zn r 1r < n(r;n) = 1 Z16 135791113 15 Z16 16 Z 16

19 = 929 = 239 = 11

49 = 459 = 1369 = 6

79 = 1589 = 899 = 1

109 = 10119 = 3129 = 12

139 = 5149 = 14159 = 7

□

C=fa+bi:a;b2Rg

i2=1 z=a+bi a zb z z=a+biw=c+di z+w= (a+bi) + (c+di) = (a+c) + (b+d)i i2=1 zw (a+bi)(c+di) =ac+bdi2+adi+bci= (acbd) + (ad+bc)i z=a+bi z12C zz1=z1z= 1 z=a+bi z 1=abi a 2+b2 z=a+bi z=abi z=a+bijzj=p a 2+b2 z= 2 + 3iw= 12i z+w= (2 + 3i) + (12i) = 3 +i zw= (2 + 3i)(12i) = 8i z 1=2 13 3 13 i jzj=p 13 z= 23i □ y x 0 z

1= 2 + 3i

3=3 + 2i

2= 12i

z=a+bi xy a x b y z1= 2+3iz2= 12i z

3=3 + 2i

y x 0 a+bi r x r z =a+bi=r(+i) r=jzj=√ a 2+b2 a=r b=r r(+i)r z 0◦ <360◦

0 <2

z= 260◦ a= 260◦= 1 b= 260◦=p 3 z= 1 +p 3i Ƕ z= 3p 23p
2i r=√ a

2+b2=p

36 = 6

=(b a ) =(1) = 315◦ 3p 23p

2i= 6315◦□

z=rw=sϕ zw=rs(+ϕ) z= 3(/3)w= 2(/6) zw= 6(/2) = 6i□ z=r [r]n=rn(n) n= 1;2;::: n n= 1 k 1kn z n+1=znz =rn(n+in)r(+i) =rn+1[(nn) +i(n+n)] =rn+1[(n+) +i(n+)] =rn+1[(n+ 1)+i(n+ 1)] ■ z= 1+i z10 (1 +i)10 z10 Ƕ z

10= (1 +i)10

= (p 2( 4 )) 10 = ( p 2) 10(5 2 ) = 32( 2 ) = 32i □ C

QR C

T=fz2C:jzj= 1g

C H=f1;1;i;ig H 11i i z4= 1 zn= 1 n zn= 1 n z=(2k n ) k= 0;1;:::;n1 n T n Ƕ z n=( n2k n ) =(2k) = 1 zǶ 2k/n 2 zn= 1 n n n T ■ n n !=p 2 2 +p 2 2 i ! 3=p 2 2 +p 2 2 i ! 5=p 2 2 p 2 2 i ! 7=p 2 2 p 2 2 i y x 0 ! i ! 3 1 ! 5 i ! 7 □ 22
2

8 21;000;000

37;398;332(46;389)

046;388
n a

2

a= 2k1+ 2k2++ 2kn k1< k2<< kn a 57 = 20+ 23+ 24+ 25
Zn bax(n)cay(n) bcax+y(n) a2k(n)k a 20(n) a 21(n)
a 2k(n) n 271321(481)

321 = 2

0+ 26+ 28;

271321(481)
271

20+26+28271202712627128(481)

2712i(481)i= 0;6;8
271

21= 73;441329 (481)

27122(481)
271

22(27121)2(481)

(329)2(481) 108;241 (481) 16 (481) (a2n)2a22na2n+1(n) 271

26419 (481)

271

2816 (481)

271

32127120+26+28(481)

271202712627128(481) 27141916 (481) 1;816;784 (481) 47 (481) □ n ě 3U(20) 5U(23) Z8 5 1540(23)
Z60 U(8) Q G G 52Z12
p 32R
p 32R
i2C

722Z240

3122Z471

Z 7 Z24 15 Z12 Z60 Z13 Z48 U(20) U(18) R 7 C ii2=1 C 2i C (1 +i)/p 2 C (1 +p 3i)/2 GL2(R) ( 0 1 1 0) ( 0 1/3 3 0) ( 11 1 0) ( 11 0 1) ( 11 1 0) ( p

3/2 1/2

1/2p 3/2 ) Z18 D4 Q8 U(30) Z32 ǳǴ Z Q R a24=e G a n n20 U(n)

A=(0 1

1 0) B=(01 11) GL2(R) AB AB (32i) + (5i6) (45i) (4i4) (54i)(7 + 2i) (9i) (9i) i 45
(1 +i) + (1 +i) a+bi

2(/6)

5(9/4)

3() (7/4)/2 1i 5

2 + 2i

p 3 +i 3i

2i+ 2p

3 (1 +i)1 (1i)6 ( p

3 +i)5

(i)10((1i)/2)4 (p 2p 2i)12 (2 + 2i)5 jzj=j zj z z=jzj2 z 1= z/jzj2 jz+wj jzj+jwj jzwj jjzj jwjj jzwj=jzjjwj 292

3171(582)

2557

341(5681)

2071

9521(4724)

971

321(765)

a;b2G a a1 g2Gjaj=jg1agj ab ba pq Zpq p r Zpr Zp p gh 1516 G ⟨g⟩\⟨h⟩ a G ⟨am⟩ \ ⟨an⟩ Zn n >2 G ab2G jaj=mjbj=n (m;n) = 1 ⟨a⟩ \ ⟨b⟩=feg

G G

G n x y=xk

(k;n) = 1 y G

G 2

G 4

G pq (p;q) = 1 G a

b pq G Z nZn= 0;1;2;::: Zn r 1r < n (r;n) = 1 G G G G mdjm G d n1n z=r(+i)w=s(ϕ+iϕ) zw=rs[(+ϕ) +i(+ϕ)] C n T n 2T m= 1n= 1 d= 1d=(m;n) z2C jzj ̸= 1 z z=+i T2Q z 2 ax(n) nx ǳ ě Ǵ 5 △ABC S=fA;B;Cg :S!S (A B C

A B C) (

A B C

C A B) (

A B C

B C A)

( A B C

A C B) (

A B C

C B A) (

A B C

B A C)

(A B C

B C A)

ABBC CA A7!B B7!C C7!A X SX X

X=f1;2;:::;ng Sn SX

S n n nSn n! Sn 1122:::nn f:Sn!Sn f1 f jSnj=n! ■ Sn GS5 =(1 2 3 4 5

1 2 3 5 4)

=(1 2 3 4 5

3 2 1 4 5)

=(1 2 3 4 5

3 2 1 5 4)

G ◦ □ X (◦)(x) =((x)) (x) ((x)) (x) (x) =(1 2 3 4

4 1 2 3)

=(1 2 3 4

2 1 4 3)

=(1 2 3 4

1 4 3 2)

=(1 2 3 4

3 2 1 4)

□ 2SX k a1;a2;:::;ak2X (a1) =a2 (a2) =a3 (ak) =a1 (x) =x x2X (a1;a2;:::;ak) =(1 2 3 4 5 6 7

6 3 5 1 4 2 7)

= (162354) 6 =(1 2 3 4 5 6

1 4 2 3 5 6)

= (243) 3 (1 2 3 4 5 6

2 4 1 3 6 5)

= (1243)(56) 4□ = (1352)= (256)

17!3;37!5;57!2;27!1

27!5;57!6;67!2

17!3;37!5;57!6;67!27!1

= (1356) = (1634) = (1652)(34)□ SX= (a1;a2;:::;ak)= (b1;b2;:::;bl) ai̸=bj ij (135)(27) (135)(347) (135)(27) = (135)(27) (135)(347) = (13475) □ SX == (a1;a2;:::;ak)= (b1;b2;:::;bl) (x) =(x) x2X x fa1;a2;:::;akgfb1;b2;:::;blg x (x) =x(x) =x (x) =((x)) =(x) =x=(x) =((x)) =(x) x2 fa1;a2;:::;akg (ai) =a(ik)+1 a 17!a2 a 27!a3
a k17!ak a k7!a1 (ai) =ai (ai) =((ai)) =(ai) =a(ik)+1 =(a(ik)+1) =((ai)) =(ai) x2 fb1;b2;:::;blg ■ Sn X=f1;2;:::;ng 2Sn X1 f(1);2(1);:::g X1 X i X X1 X2f(i);2(i);:::g X2 X3;X4;::: X r i i(x) ={ (x)x2Xi x x/2Xi =12r X1;X2;:::;Xr 1;2;:::;r ■ =(1 2 3 4 5 6

6 4 3 1 5 2)

=(1 2 3 4 5 6

3 2 1 5 6 4)

= (1624) = (13)(456) = (136)(245) = (143)(256) □ (1) 2 (a1;a2;:::;an) = (a1an)(a1an1)(a1a3)(a1a2) (16)(253) = (16)(23)(25) = (16)(45)(23)(45)(25) (12)(12) (13)(24)(13)(24) (16) (23)(16)(23) (35)(16)(13)(16)(13)(35)(56) (16) □ r =12r r r r >1 r= 2 r >2 r1r (ab)(ab) = (bc)(ab) = (ac)(bc) (cd)(ab) = (ab)(cd) (ac)(ab) = (ab)(bc) abc d r1r =12r3r2 r2 r r1r r a r2r1 r2 r a r2 r2 r3r2 a a r2 ■ =12m=12n m n m1 =m1=1nm1 n ■ Sn An

An n

An Sn A n An =12r i r

1=rr11

An■ Snn2 Ann!/2

An SnBn

Sn n2 :An!Bn () = () =() = =1=1= ■ A4 S4 A4 (1)(12)(34)(13)(24)(14)(23) (123)(132)(124)(142) (134)(143)(234)(243) A4 □ Ĝ Ƕ Ĝ n n n= 3;4;::: n Dn n 1;2;:::;n n k k+1 k1 2n n 1 n1 3 2 n 4 n Dn Sn 2n Dnn3 rs r n= 1 s 2= 1 srs=r1 n n ;360◦ n ;2360◦ n ;:::;(n1)360◦ n 360◦/nr r
r k=k360◦ n n ns1;s2;:::;sn sk k n s1=sn/2+1;s2= s n/2+2;:::;sn/2=sn s1;s2;:::;sn sk s=s1 s2= 1rn= 1 t n k k+1 k1 k+ 1 t=rk k1 t=rksrsDn Dn rs D n=f1;r;r2;:::;rn1;s;rs;r2s;:::;rn1sg srs=r1 n ■ D4 1234
r= (1234) r

2= (13)(24)

3= (1432)

4= (1)

1= (24)

2= (13)

D48 rs

1= (12)(34)

3s1= (14)(23)

D4 □ n ǳ Ǵ 6 64 = 24 24
S4 24
S4 123 4
S4 180◦

S4 S4

S4 ■ ě (134)(354) A3 (

1 2 3 4 5

2 4 1 5 3)

(

1 2 3 4 5

4 2 5 1 3)

(

1 2 3 4 5

3 5 1 4 2)

(

1 2 3 4 5

1 4 3 2 5)

(1345)(234) (12)(1253) (143)(23)(24) (1423)(34)(56)(1324) (1254)(13)(25) (1254)(13)(25) 2 (1254)

1(123)(45)(1254)

(1254)

2(123)(45)

(123)(45)(1254) 2 (1254) 100
j(1254)j j(1254)2j (12) 1 (12537) 1 [(12)(34)(12)(47)] 1 [(1235)(467)] 1 (14356) (156)(234) (1426)(142) (17254)(1423)(154632) (142637) (a1;a2;:::;an)1 S4 f2S4:(1) = 3g f2S4:(2) = 2g f2S4:(1) = 3(2) = 2g S4 A4 S7A7 A10 15

A8 26

Snn= 3;:::;10 A5 A6 2Sn n iji=j ij (n) =1m2Sn 1;:::;m D5 rs rs (12)(34) A 4 Sn n3 An n4 Dn n3 2Sn n1 2Sn n2 2 3 Ann3 3 Sn (12);(13);:::;(1n) (12);(23);:::;(n1;n) (12);(12:::n)

G g:G!Gg(a) =ga g

G n! n G

Z(G) =fg2G:gx=xg x2Gg

D8 D10 Dn = (a1;a2;:::;ak) k

1= ((a1);(a2);:::;(ak))

k k 1=Sn 2Sn 1= Sn 2SX n(x) =y n2Z xy X x2X2SX O x;=fy:xyg f1;2;3;4;5g S5 = (1254) = (123)(45) = (13)(25)

Ox;\ Oy;̸=∅ Ox;=Oy;

HSX x;y2X 2H (x) =y ⟨⟩ Ox;=X x2X 2Snn3 = 2Sn Sn 1 2An2Sn 12An 11 ;2Sn rs Dn srs=r1 rks=srkDn rk2Dnn/(k;n) 6 Ƕ

Ƕ

G H G H

g2G gH=fgh:h2Hg

Hg=fhg:h2Hg

H Z6 03

0 +H= 3 +H=f0;3g

1 +H= 4 +H=f1;4g

2 +H= 5 +H=f2;5g

ZZn □

H S3 f(1);(123);(132)g

H (1)H= (123)H= (132)H=f(1);(123);(132)g (12)H= (13)H= (23)H=f(12);(13);(23)g H

H(1) =H(123) =H(132) =f(1);(123);(132)g

H(12) =H(13) =H(23) =f(12);(13);(23)g

Ƕ K S3 f(1);(12)g K (1)K= (12)K=f(1);(12)g (13)K= (123)K=f(13);(123)g (23)K= (132)K=f(23);(132)g; K

K(1) =K(12) =f(1);(12)g

K(13) =K(132) =f(13);(132)g

K(23) =K(123) =f(23);(123)g

□

H G g1;g22G

1H=g2H

11=Hg12

1Hg2H

g 22g1H
g

11g22H

H G

H G HG

G G HG

g1Hg2H HG g1H\g2H=∅ g1H=g2H g1H\g2H̸=∅a2g1H\g2H a=g1h1=g2h2 h1h2H g1=g2h2h11 g

12g2H

g1H=g2H■

G

G H G HG

HG [G:H]

G=Z6H=f0;3g [G:H] = 3□

G=S3H=f(1);(123);(132)g K=f(1);(12)g [G:H] = 2[G:K] = 3□

H G HG

LHRH HG

ϕ:LH! RH gH2 LH

ϕ(gH) =Hg1

ϕ g1H=g2H Ƕ Hg11=Hg12 ϕ Hg

11=ϕ(g1H) =ϕ(g2H) =Hg12

g1H=g2H ϕ ϕ(g1H) =Hg■ Ƕ

H Gg2G ϕ:H!gH

ϕ(h) =gh ϕ H

gH ϕ ϕ(h1) =ϕ(h2) h1;h22H h1=h2 ϕ(h1) =gh1ϕ(h2) =gh2 gh

1=gh2 h1=h2 ϕ

gH gh h2Hϕ(h) =gh■

G H G

jGj/jHj= [G:H] HG H G G [G:H] jHj jGj= [G:H]jHj■ G g2G g G jGj=pp G g2G g̸=e g G g̸=e g j⟨g⟩j>1 p gG■ p Zp

HK G GHK

[

G:K] = [G:H][H:K]

[G:K] =jGj jKj=jGj jHjjHj jKj= [G:H][H:K] ■ Ƕ A4

12 6

Ƕ 12 123

4 6

A4 6

H A4 3

H 3 H 3

6 A4 6 [A4:H] = 2 HA4 H gH=HggHg1=H g2A4 3 A4 3 H (123) H (123)1= (132) Ƕ H ghg12H g2A4 h2H (124)(123)(124)

1= (124)(123)(142) = (243)

(243)(123)(243)

1= (243)(123)(234) = (142)

H (1);(123);(132);(243);(243)1= (234);(142);(142)1= (124)

A4 6■

Sn 2Sn =1 = (a1;a2;:::;ak) = (b1;b2;:::;bk) (a1) =b1 (a2) =b2 (ak) =bk =1 = (a1;a2;:::;ak) k 2Sn (ai) =b (a(ik)+1) =b′ (b) =b′ = ((a1);(a2);:::;(ak)) ■

Ƕ Ƕ

ϕ ϕ:N!N ϕ(n) = 1n= 1 n >1

ϕ(n) m1m < n (m;n) = 1

U(n) Zn ϕ(n) jU(12)j=ϕ(12) = 4 pϕ(p) =p1

U(n) Zn jU(n)j=ϕ(n)

Ƕ an n >0(a;n) =

1 aϕ(n)1 (n)

U(n)ϕ(n) aϕ(n)= 1 a2U(n) aϕ(n)1 n aϕ(n)1 (n)■ Ƕ n=p

ϕ(p) =p1

Ƕ Ƕ p p∤ap a a p11 (p) bbpb(p) Ĝ

Ƕ

ǳ Ǵ ǳ Ǵ Ƕ ⟨3⟩Z9 f();(12)(34);(13)(24);(14)(23)g S4 S 4

G G

p= 137909 57137909(137909) G g 5 h

7 jGj 35

G 60 G Ƕ ⟨8⟩Z24 ⟨3⟩U(8) 3ZZ A

4S4AnSn

D 4S4 TC

H=f(1);(123);(132)gS4

SL2(R)GL2(R) SL2(R)GL2(R) Ƕ n= 15a= 4 Ƕ p= 4n+ 3 x2 1 (p)

H G g1;g22G

1H=g2H

11=Hg12

1Hg2H

g 22g1H
g

11g22H

ghg12H g2Gh2H gH=Hg g2G

ϕ:LH! RH ϕ(gH) =Hg

gn=e gn = (12)(345)(78)(9) (2;3;2;1) (1;2;2;3) ;2Sn = 1 = 1

2Sn

jGj= 2n 2 G [G:H] = 2 ab H ab2H [G:H] = 2 gH=Hg

HK G gH\gK H\K

HK G Gab

h2H k2K hak=b H=f(1);(123);(132)gA4

G n ϕ(n) G

Ƕ n=pe11pe22pekk p1;p2;:::;pk

ϕ(n) =n(

11 p 1)( 11 p 2) ( 11 p k) n=∑ djnϕ(d) n 7 ::: :::

A

BC BCǶ A

B C AB C C C f f1 f f1 f = 00;= 01;:::;= 25 f(p) =p+ 326; A7!D;B7!E;:::;Z7!C f

1(p) =p326 =p+ 2326

3;14;9;7;4;20;3

0;11;6;4;1;17;0

26 □

26
b f(p) =p+b26 = 04 = 18 18 = 4 +b26 b= 14 f(p) =p+ 1426 f

1(p) =p+ 1226

□ 26
f(p) =ap+b26 f1 c=ap+b26 p a (a;26) = 1 f

1(p) =a1pa1b26

f(p) =ap+b26 a2Z26 (a;26) = 1 a= 5 (5;26) = 1 a1= 21 f(p) = 5p+326 3;6;7;23;8;10;3 f

1(p) = 21p21326 = 21p+ 1526

□ p1p2 =(p1 p 2)

A 22 Z26

f() =A+ Z26 f

1() =A1A1

7;4;11;15

A=(3 5

1 2) A

1=(2 21

25 3)
= (2;2) □ f f1 ǳǴ pq n=pq

ϕ(n) =m= (p1)(q1) ϕ ϕ

E m E

(E;m) = 1 D DE1 (m) nE En = 00;= 02;:::;= 25 n x y=xEn y x x=yDn D 25 p= 23
q= 29 n=pq= 667

ϕ(n) =m= (p1)(q1) = 616

E= 487 (616;487) = 1 25

487667 = 169

191E= 1+151m
(n;D) = (667;191) 169

191667 = 25

□ DE1 (m) k

DE=km+ 1 =kϕ(n) + 1

(x;n) = 1 y

D= (xE)D=xDE=xkm+1= (xϕ(n))kx= (1)kx=xn

x yDn (x;n)̸= 1 n=pqx < n x p q r r < qx=rp (x;q) = 1 m=ϕ(n) = (p1)(q1) =ϕ(p)ϕ(q) q x km=xkϕ(p)ϕ(q)= (xϕ(q))kϕ(p)= (1)kϕ(p)= 1q t xkm= 1 +tq y

D=xkm+1=xkmx= (1 +tq)x=x+tq(rp) =x+trn=xn

D nE n D 667 = 2329
D Ƕ (n′;E′) (n′;D′) Ƕ (n;E) (n;D) x x x ′=xD′n′ x′ x x ′ x′ Ƕ y ′=x′En Ƕ Ƕ Ƕ

ǳ Ƕ Ǵ

Ƕ

Ĵ ϕ(893456123) 7324(895)
Ƕ ǳǴ 22A Z26 ((A);26) = 1

A=(3 4

2 3) f() =A+ = (2;5) x x 2 x= 142528 1425 28 n= 3551;E= 629;x= 31 n= 2257;E= 47;x= 23 n = 120979;E= 13251;x= 142371 n= 45629;E= 781;x= 231561 D y n= 3551;D= 1997;y= 2791 n= 5893;D= 81;y= 34 n= 120979;D= 27331;y= 112135 n= 79403;D= 671;y= 129381 (n;E) D (n;E) = (451;231) (n;E) = (3053;1921) (n;E) = (37986733;12371) (n;E) = (16394854313;34578451) n n nE X X

EX(n)

1015 (n;E) D

Ƕ n n n d= 2;3;:::;p n n dn n Ƕ n=ab n n=x2y2= (xy)(x+y) xy n=x2y2 Ĵ n Ĵ Ĵ Ĵ Ƕ p (a;p) = 1 ap11 (p) Ƕ 15 2

1512144 (15)

17 1712161 (17)

n 2 n11 (n) 342
811
561
771
631
n b (b;n) = 1 bn11 (n) n b 341

2 3

2000
2000
2000
561 = 31117
25109
ǳ Ǵ Ĝ ǳ Ǵ Ĝ ǳ Ǵ Ĝ ǳ Ǵ Ĝ ǳ Ǵ ǳ Ǵ Ĝ 8 ǳǴ m n n m m n n n n n(x1;x2;:::;xn) 3n (x1;x2;:::;xn)7!(x1;x2;:::;xn;x1;x2;:::;xn;x1;x2;:::;xn) i i (0110) (0110 0110 0110) (0110 1110 0110) (0110) n2n n m m 3n□ 8 28= 2568 27= 128
= 6510= 010000012; = 6610= 010000102; = 6710= 010000112 128

00000000

2= 010;

01111111

2= 12710

01 1 = 010000012; = 010000102; = 110000112 (0100 0101) 1 m 816 32
01 1 1 1□ n

01

(1001 1000) 0 1 0 (000)1 (111) (101) 1 0 (111)

000 001 010 011 100 101 110 111

0000 1 1 2 1 2 2 3

1113 2 2 1 2 1 1 0

(000)(111)

3 □

p q pq n p p q q 0 1 p q= 1p 1 1 p 0 q npn p= 0:999 (0:999)10;0000:00005 n(x1;:::;xn) p k (n k) q kpnk k k q nk p n q kpnk k (n k) =n! k!(nk)! n k qkpnk (n k) q kpnk ■ p= 0:995 500 p n= (0:995)5000:082 (n 1) qp n1= 500(0:005)(0:995)4990:204 (n 2) q

2pn2=500499

2 (0:005)2(0:995)4980:257

10:0820:2040:257 = 0:457

□ (n;m) m n (n;m)

E:Zm2!Zn2

D:Zn2!Zm2

E E D (8;7)

E(x7;x6;:::;x1) = (x8;x7;:::;x1)

x8=x7+x6++x1 Z2□ = (x1;:::;xn)= (y1;:::;yn) n d(;) d d(;) w() 1 w() =d(;) = (000) = (10101)= (11010) = (00011) C d(;) = 4; d(;) = 3; d(;) = 3 w() = 3; w() = 3; w() = 2 □ n w() =d(;) d(;)0 d(;) = 0 = d(;) =d(;) d(;)d(;) +d(;) = (1101)= (1100) (1101) (1100) (1100) d(;) = 1 = (1100) = (1010) d(;) = 2 2 0000 0011 0101
0110
1001
1010
1100
1111
0000 0 2 2 2 2 2 2 4 0011 2 0 2 2 2 2 4 2 0101
2 2 0 2 2 4 2 2 0110
2 2 2 0 4 2 2 2 1001
2 2 2 4 0 2 2 2 1010
2 2 4 2 2 0 2 2 1100
2 4 2 2 2 2 0 2 1111
4 2 2 2 2 2 2 0 d(;) = 1 d(;) = 2 d= 2 d(;) = 2 d(;) =d(;) = 1 d3 d(;) = 1 d(;)2 ̸= 3

C d= 2n+ 1 C n

2n C n d(;)n

2n+ 1d(;)d(;) +d(;)n+d(;)

d(;)n+ 1

2n 1d(;)2n 2n+1

12n■ 1= (00000)2= (00111)3= (11100)

4= (11011)

00000 00111
11100
11011
00000 0 3 3 4 00111
3 0 4 3 11100
3 4 0 3 11011
4 3 3 0 □ Ƕ ǳ Ǵ Ƕ (32;6) Ƕ Zn2 n n (11000101) + (11000101) = (00000000) (0000000)(0001111)(0010101)(0011010) (0100110)(0101001)(0110011)(0111100) (1000011)(1001100)(1010110)(1011001) (1100101)(1101010)(1110000)(1111111) Z72 d= 3 3 □ n w(+) =d(;) n 1

1 + 1 = 0

0 + 0 = 0

1 + 0 = 1

0 + 1 = 1

■ d C d C d =fw() :̸=g d =fd(;) :̸=g =fd(;) :+̸=g =fw(+) :+̸=g =fw() :̸=g ■ 3 ǳǴ n =x1y1++xnyn = (x1;x2;:::;xn)= (y1;y2;:::;yn) = (011001)= (110101) = 0 = n =(x1x2xn)0 B BB@y 1 y 2 y n1 C CCA =x1y1+x2y2++xnyn 3 3 1 n= (x1;x2;:::;xn) 1 x1+x2++xn= 0 4= (x1;x2;x3;x4) 1 x1+x2+x3+x4= 0 ==(x1x2x3x4)0 B BB@1 1 1 11 C

CCA= 0

□

Mmn(Z2) mn Z2

Z2 H2Mmn(Z2) n H= H (H) H=0 @0 1 0 1 0

1 1 1 1 0

0 0 1 1 11

A 5= (x1;x2;x3;x4;x5) HH= x

2+x4= 0

1+x2+x3+x4= 0

3+x4+x5= 0

5 (00000) (11110) (10101) (01011) □

H Mmn(Z2) H

Zn2 ;2(H) HMmn(Z2) H= H=

H(+) =H+H=+=

+ H ■ H2 M mn(Z2) C H=0 @0 0 0 1 1 1

0 1 1 0 1 1

1 0 1 0 0 11

A 6= (010011) H=0 @0 1 11 A □ H H H mn Z2n > m m mm Im H= (AjIm) A m(nm) 0 B BB@a

11a12a1;nm

21a22a2;nm

a m1am2am;nm1 C CCA

Im mm

0 B

BB@1 00

0 10

0 011

C CCA n(nm)

G=(Inm

A ) G= H= G (000);(001);(010);:::;(111) A=0 @0 1 1 1 1 0

1 0 11

A G=0 B

BBBBBB@1 0 0

0 1 0 0 0 1 0 1 1 1 1 0

1 0 11

CCCCCCA

H=0 @0 1 1 1 0 0

1 1 0 0 1 0

1 0 1 0 0 11

A H

6 1 1

= (x1;x2;x3;x4;x5;x6) =H=0 @x

2+x3+x4

1+x2+x5

1+x3+x61

A x

2+x3+x4= 0

1+x2+x5= 0

1+x3+x6= 0

x4 x2x3x5 x1x2 x6 x1x3 x4x5 x6 x1x2 x3 x4x5 x6 H (000000) (001101) (010110) (011011) (100011) (101110) (110101) (111000): G □ G

000000000

001001101

010010110

011011011

100100011

101101110

110110101

111111000

H2Mmn(Z2) (H)

2Zn2 nm m

H= m

nm H (n;nm) n m m G nk C={:G=2Zk2} (n;k) C

G1=1G2=2 1+2 C

G(1+2) =G1+G2=1+2

G=G = G=G

GG=G() =

k G() x1y1;:::;xkyk Ik G G() = = ■

H= (AjIm) mn G=(Inm

A ) n(nm) HG=

C=HG ij C

c ij=n∑ k=1h ikgkj = nm∑ k=1h ikgkj+n∑ k=nm+1h ikgkj = nm∑ k=1a ikkj+n∑ k=nm+1 i(mn);kakj =aij+aij = 0 ij={ 1i=j

0i̸=j

■

H= (AjIm) mn

G=(Inm

A ) n(nm) H C G C H= C H 2C G= 2Zm2 H=HG= = (y1;:::;yn) H

Znm2 G= H=

11y1+a12y2++a1;nmynm+ynm+1= 0

21y1+a22y2++a2;nmynm+ynm+1= 0

a m1y1+am2y2++am;nmynm+ynm+1= 0 ynm+1;:::;yn y1;:::;ynm y nm+1=a11y1+a12y2++a1;nmynm y nm+1=a21y1+a22y2++a2;nmynm y nm+1=am1y1+am2y2++am;nmynm xi=yii= 1;:::;nm■ H

1= (10000)

2= (01000)

n= (00001) n Zn2 1 mn HHi i H 0 @1 1 1 0 0

1 0 0 1 0

1 1 0 0 11

A0 B

BBBB@0

1 0 0 01 C

CCCCA=0

@1 0 11 A □ i n 1 i 0Ƕ H2Mmn(Z2) Hi i H

H mn H

H (H) 2 2 i i= 1;:::;n Hi i H i H i H

Hi̸=■

H 1=0 @1 1 1 0 0

1 0 0 1 0

1 1 0 0 11

A H 2=0 @1 1 1 0 0

1 0 0 0 0

1 1 0 0 11

A H1 H2 □

H H2

3 H=0 @1 1 1 0

1 0 0 1

1 1 0 01

A H (H) 4 2 (1100)(1010)(1001)(0110)(0101) (0011) (H)

H H

H □

H H

H H ni+j1 i j w(i+j) = 2i̸=j =H(i+j) =Hi+Hj i j H ■ H 23= 8
0 @0 0 01 A ;0 @1 0 01 A ;0 @0 1 01 A ;0 @0 0 11 A 3 H mn nm m 2m ;1;:::;m 2m(1 +m) n n H=0 @1 1 1 0 0

0 1 0 1 0

1 0 0 0 11

A 5= (11011)= (01011) H=0 @0 0 01 A H=0 @1 0 11 A

H H

01 □

H mn 2Zn2 H

mn H n =+ H

H=H(+) =H+H=+H=H

■ H i H

H2Mmn(Z2) H

n H i i H=0 @1 0 1 1 0 0

0 1 1 0 1 0

1 1 1 0 0 11

A 6= (111110)= (111111) = (010111) H=0 @1 1 11 A ;H=0 @1 1 01 A ;H=0 @1 0 01 A (110110)(010011) H □ C Zn2 CZn2 C (n;m) C Z n2 +C 2Zn2 Ƕ 2nm CZn2

C (5;3)

H=0 @0 1 1 0 0

1 0 0 1 0

1 1 0 0 11

A (00000) (01101) (10011) (11110) 252= 23 CZ52 22= 4
□ C

C(00000)(01101)(10011)(11110)

(10000) +C(10000)(11101)(00011)(01110) (01000) +C(01000)(00101)(11011)(10110) (00100) +C(00100)(01001)(10111)(11010) (00010) +C(00010)(01111)(10001)(11100) (00001) +C(00001)(01100)(10010)(11111) (10100) +

C(00111)(01010)(10100)(11001)

(00110) +C(00110)(01011)(10101)(11000) n =+ =+ +C n + = (01111) (00010) +C (01101) = (01111) + (00010) □ (000)(00000) (001)(00001) (010)(00010) (011)(10000) (100)(00100) (101)(01000) (110)(00110) (111)(10100)

C (n;k) H

Zn2 C H=H n n C 2C H() = 0H=H■ C = (01111) H=0 @0 1 01 A (00010) □ (n;k) 2nk C (32;24) 224 23224= 28= 256
d= 6 5610
5610
ǵǶ C H=2

40 1 0 1 0

1 1 1 1 0

0 0 1 1 13

5 x= 11100 x C

Hx x

0 1 2 3 4 5 6 7 8

000 001 010 011 101 110 111 000 001

4 Z42 (0110) (1001) (1010) (1100) n (011010);(011100) (11110101);(01010100) (00110);(01111) (1001);(0111) n (011010) (11110101) (01111) (1011) C 7 C (011010) (011100) (110111) (110000) (011100) (011011) (111011) (100011) (000000) (010101) (110100) (110011) (000000) (011100) (110101) (110001) (0110110) (0111100) (1110000) (1111111) (1001001) (1000011) (0001111) (0000000) (n;k) 0 @0 1 0 0 0

1 0 1 0 1

1 0 0 1 01

A 0 B

BB@1 0 1 0 0 0

1 1 0 1 0 0

0 1 0 0 1 0

1 1 0 0 0 11

C CCA (

1 0 0 1 1

0 1 0 1 1)

0 B

BB@0 0 0 1 1 1 1

0 1 1 0 0 1 1

1 0 1 0 1 0 1

0 1 1 0 0 1 11

C CCA (5;2)

C

H=0 @0 1 0 0 1

1 0 1 0 1

0 0 1 1 11

01111 10101 01110 00011

1000
p p= 0:01 p= 0:0001 0 B

BB@1 1 0 0 0

0 0 1 0 0

0 0 0 1 0

1 0 0 0 11

C CCA 0 B

BB@0 1 1 0 0 0

1 1 0 1 0 0

0 1 0 0 1 0

1 1 0 0 0 11

C CCA (

1 1 1 0

1 0 0 1)

0 B

BB@0 0 0 1 0 0 0

0 1 1 0 1 0 0

1 0 1 0 0 1 0

0 1 1 0 0 0 11

C CCA H=0 @0 1 1 1 1

0 0 0 1 1

1 0 1 0 11

C Z32 (000)(111)

CZ32 C 0 @0 1 0 0 0

1 0 1 0 1

1 0 0 1 01

A 0 B

BB@0 0 1 0 0

1 1 0 1 0

0 1 0 1 0

1 1 0 0 11

C CCA (

1 0 0 1 1

0 1 0 1 1)

0 B

BB@1 0 0 1 1 1 1

1 1 1 0 0 1 1

1 0 1 0 1 0 1

1 1 1 0 0 1 01

C CCA n w() =d(;) d(;) =d(+;+) d(;) =w() X d:XX!R d(;)0 ;2X d(;) = 0 = d(;) =d(;) d(;)d(;) +d(;) Zn2

[PDF] aata-20200730pdf - Abstract Algebra: Theory and Applications

[PDF] math 113: abstract algebra solutions to practice problems for

[PDF] EXERCISES AND SOLUTIONS IN GROUPS RINGS AND FIELDS

[PDF] aata-20200730pdf - Abstract Algebra: Theory and Applications

[PDF] Examples in abstract algebra - Penn Math

CmHv jy- kyky

Ĝ

Ƕ Ƕ

Ƕ

3 + 5613 + 8/2

2 + 3 = 5

2x= 6 x= 4

34x2+ 5x6

24 6̸= 8

Ƕ

2Ǵ ǳ

2+bx+c= 0a̸= 0

2+bx+c= 0

X=fx1;x2;:::;xng

X=fx:xPg

E=f2;4;6;:::gE=fx:x x >0g

N=fn:n g=f1;2;3;:::g;

Z=fn:n g=f:::;1;0;1;2;:::g;

Q=fr:r g=fp/q:p;q2Zq̸= 0g;

R=fx:x g;

C=fz:z g

A B ABBA A B

NZQRC

A[B AB

A[B=fx:x2Ax2Bg;

A\B=fx:x2Ax2Bg

A=f1;3;5gB=f1;2;3;9g

A[B=f1;2;3;5;9gA\B=f1;3g

E O EO

AU A A′

AnB=A\B′=fx:x2Ax/2Bg

A=fx2R: 0< x3gB=fx2R: 2x <4g

A\B=fx2R: 2x3g

A[B=fx2R: 0< x <4g

AnB=fx2R: 0< x <2g

AB C

A[A=AA\A=A AnA=∅

A[ ∅=AA\ ∅=∅

A[(B[C) = (A[B)[CA\(B\C) = (A\B)\C

A[B=B[AA\B=B\A

A[(B\C) = (A[B)\(A[C)

A\(B[C) = (A\B)[(A\C)

A[A=fx:x2Ax2Ag

A\A=fx:x2Ax2Ag

AnA=A\A′=∅

A[(B[C) =A[ fx:x2Bx2Cg

B

AB=f(a;b) :a2Ab2Bg

A=fx;ygB=f1;2;3g C=∅ AB

1 An=f(a1;:::;an) :ai2Aii= 1;:::;ng

A=A1=A2==An AnA AA

A=(a b

A(x;y) = (ax+by;cx+dy)

2 1 3)

1(x) =ex

1(f(x)) =f1(x) =ex=x

A=(3 1

A R2R2

A(x;y) = (3x+y;5x+ 2y)

A(x;y) = (2xy;5x+ 3y)

A◦TA(x;y) =TA◦T1

A(x;y) = (x;y)

B(x;y) = (3x;0)

B=(3 0

B(x;y) = (ax+by;cx+dy)

B(x;y) = (3ax+ 3by;0)

2 3 1)

S=f1;2;3g

1=(1 2 3

3 1 2)

1=g(f(a1)) =g(f(a2)) =a2 f b2B

RXX

1c2

AB22

A=(1 2

A B ABBA A B

NZQRC

AU A A′

A=fx2R: 0< x3gB=fx2R: 2x <4g

A\B=fx2R: 2x3g

RXX

ABPAP1=B

AB P PAP1=B

4117 (8)4117 = 24 8

AB A\B=A

1);:::;S(k) S(k+1)kn0 S(n)

S=fn2N:n1g 12S n2S 0<1

0r < b

S=fabk:k2Zabk0g

S r=abq a=bq+rr0

1a2an1