4

:

A 2002

1. Rotman, J.J.: Galois, xii+185 , 2000.

2. Rudin, W.: , xvi+524 , 2000.

3. Fine, B. & Rosenberger, G.: , xix+264 ,

2001.

Mark Antony Armstrong

, ,

,

:

.

A 2002

: Groups and Symmetry: .. Armstrong: 1988, Springer -Verlag,

New York, Berlin, HeidelbergCopyright c1988: Springer-Verlag New York, Inc.Copyright c2002 : Leader Books A.E.

, . :

. / / e-mail: ddais@ucy.ac.cy

: , e-mail: klagou@cc.uoi.gr

:

, e-mail: apolimer@panafonet.gr

:

: e-mail: nmarmar@cc.uoi.gr

1 : 2002

ISBN 960-7901-28-X

LEADER BOOKS A.E.

17, ,

11521

.: 010/64.52.825-64.50.048, Fax: 010/64.49.924

web-page: http://www.leaderbooks.com, e-mail: info@leaderbooks.com

: Cosmoware

. 53,

.: 010/60.13.922, Fax: 010/60.01.642

.

.

.

( ) -

. -

L. Euler (1707-1783), C.-F. Gauss (1777-1855) ..

18 19 , , , -

1 J.-L. Lagrange (1736-

1813), E. Galois (1811-1832) .. , ,

, 2

A. Cayley (1821-1895) 1854, -

3 1870. (

R.Dedekind (1831-1916), C. Jordan

(1838-1922) W. van Dyck (1856-1934) .)

, , -

- -

.

. -

.

.

1. Rotmann J.J.: Galois, ..., 1, . . , Leader Books, 2000.

2

Cayley A.: On the theory of groups, as depending in the symbolic equation n = 1, Phil. Mag. 7, (1854).

3. Scholz E. (Hrsg.): Geschichte der Algebra, B.I., Mannheim, 1990, . 309.

viii

-

,

, , , -

( -

) 4 1] Hermann Weyl5.

-

X X ,

X ( ). -

(. . 9).

( ) -

, , -

1872, Erlanger

Programm Felix Klein (1849-1925), -

-

6.

, ,

, -

,

(. . 24)

... ,

(, , ) .

.

;

, ,

, -

. ,

,

-

4 .

5 H. Weyl (. . 145): , . .

6 - H.Wussing: Die Genesis des abstrakten Gruppenbegriffes, VEBDeutscher Verlag der Wissenschaften, Berlin, 1969.

ix

-

- - (

,

), -

, ,

,

.

Felix Klein,

19 ,

, ,

-

. ,

7:

, . -

, -

] -

] .

] -

,

] ,

( ) .

, , -

, ,

, , -

,

( ) . , -

] , ]

. 8.

-

,

, -

- - .

,

( ,

7: Klein F.: Vorlesungen uber die Entwicklung der Mathematik im 19 Jahrhundert, Teil I,Springer, 1926, . 335.

8 : ( ) , , .

x ) ; -

(

)

( ) . -

(. . 1, 4, 5, 7, 8 ..)

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()

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() 300 ( ) -

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()

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() .

:

(i)

(. 8), (. 12), -

(. 15), (. 19),

xi

(. 20) , Nielsen Schreier (. 28),

(ii) ,

(iii) , -

.

. :

() . -

-

( ). -

. (). (A B), (A B) (AB) A B, A B, . -

.

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9. .

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, ,

( ).

() 10. -

. () ()

. . .

, a = b(mod n), .

() . f X Y

(X,Y,f ) :

(i) f XY X Y

Dom(f) = Dom(f ) = X, Rg(f) = Y,

Im (f) = Im(f ) Y,

(ii) x X y Y, (x, y) f .

f f . x X,

y Y, (x, y) f ,

f x x f , y = f(x).

, f x y.

9 ( , -) 12 .

10. . 4 (. 135-184) :

( ), ..., . , 2001. (: . , . , . , . , . .)

xii

f X Y

:

f : X Y, x f(x) .

A X B Y , f(A) = {f(x) | x A}

A, f1(B) = {x X | f(x) B } -

B f . (, A = X, f(X) = Im(f).)

f : X Y , A,B X

C,D Y. :

(i) f() = = f1 () , f (X) Y.

(ii) A B = f(A) f (B) . , C D = f1(C) f1 (D) .

(iii) A f1 (f (A)) f(f1 (C)

) C.

(iv) f (A B) = f(A) f(B) f1 (C D) = f1(C) f1(D).

(v) f (A B) f(A) f(B), f1 (C D) = f1(C) f1(D).

(vi) B A = f(A)f(B) f(AB).

(vii)D C = f1 (CD) = f1(C)f1(D).

f : X Y . U X, f |U : U Y,

f |U (x) = f(x), x U, f U.

g : U Y , U

X. f : X Y , g = f |U , -

g X. ( g

X .)

f : X Y

(i) ( - (1-1)), ,

x, y X, f (x) = f (y) = x = y,

(ii) ( -), , f(X) = Y,

( y Y, x X : y = f(x)) ,

(iii) ( -), ,

.

, 1X : X X, 1X(x) = x, x,

x X, , A X

1X |A : A X .

xiii

f : X Y . :

(i) H f (f(A B) = f (A) f (B) A,B X).

(ii) H f (A = f1 (f (A)) A X).

(iii) H f (f(f1 (C)

)= C C Y ).

(iv) H f (f(XA) = Yf (A) A X).

f : X Y g : Y Z . H

gf g f ( g f)

gf : X Z ,

(gf)(x) = g(f(x)), x, x X,

Dom(gf) = X, Rg(gf) = Z

Im(gf) = Im(gIm(f)

). , A X B Z,

(gf) (A) = g(f(A)) (gf)1 (B) = f1(g1(B)

), , h : Z W

,

, h (gf) = (hg) f.

f : X Y g : Y Z .

:

(i) f, g = gf

(ii) f, g = gf

(iii) gf = f

(iv) gf = g

f : X Y .

(i) f g : Y X, gf = 1X .

(ii) f h : Y X, fh = 1Y .

(iii) f

{ : Y X,

f = 1X f = 1Y .

, ( ),

f1, f .

X Y

: X Y . X

n, X

{1, 2, . . . , n}. , n X (-

|X|) X n. (-

)

xiv

11.

. ( X

).

: X Y

. X Y

f : X Y X Y . :

( f ) ( f ) .

() . -

R. . -

. . 12.

C. -

. . -

( C)13.

() 14. ,

, , , , , -

, . . . , , -

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, , ..,

. . . 19.)

() 15. 16

,

11 . P. R.Halmos 7], . 24 25.12. . 1, 2 5 :

( ), ..., . , 2001. ( : . , . -, . , . , . ). , . . 1 ( ).13. . 2 (. 85-126) :

( ), ..., . , 2001. (: . , . , . , . , . , . .)14. . 2, 3, 4, 7, 8, 9, 12 13 :

( ), ..., 1999. ( : . , . ,. . . .: . , . .)

( ), ..., 2001. ( : . , ., . , . , . . . .: . .)15 . 1 (. 11-84) :

( ), ..., . , 2001. (: . , . , . , . , . , . ). , , .. .16 entry (. Eitrag) , ( ) ]. -- ( ), , . ( .)

xv

A = [ai j ]1im, 1jn, B = [bi j ]1im, 1jn :

A+B = [ai j + bi j ]1im, 1jn ,

A = [ai k]1im, 1kp, B = [bk j ]1kp, 1jn :

AB =

[p

k=1

ai kbk j

]1im, 1jn

,

n n

In =

1 0 0 0

0 1 0 0...

.... . .

......

0 0 1 0

0 0 0 1

( In, I, n -

),

AIn = InA = A,

n n A.

At = [aj i]1im, 1jn m n A = [ai j ]1im, 1jn( A A).

2 2

det

[a b

c d

]= ad cb,

3 3

det

a b cd e fg h j

= aej + dhc+ gbf ahf gec dbj

Sarrus , , n n A -

--

det (A) =Sn

sign() 1(1) 2(2) n(n),

Sn 17 n sign()

18 Sn. A1 n n A

17 Sn

. . 36.18 sign() . . 40.

xvi

det (A) = 0,

AA1 = A1A = In.

. . .

. . -

. .

() . . -

.

fA : Rn Rn, fA(x) = xA

t

A GLn (R). -

. . , n = 2: -

19, , ..

n = 3: ,

.. Rn.

( n = 2 )

. , ,

Banchoff T., Wermer J.: Linear Algebra Through Geometry 20, UTM, 2nd ed.,

Springer-Verlag, 1992.

, -

..

:

..: 21 ( ), , -

, 1979.

..: 22 ( .), , -

, 1985.

.

,

, :

LedermannW.: Introduction to Group Theory, Oliver & Boyd, 1973.

Rose J. S.: A Course on Group Theory, Cambridge University Press, 1978.

19 rotation , ( n = 2) ( n = 3) , , .

20 n = 2 n = 3. . 1-3]. Leader Books.

21. : 2, . 1-3.

22. . , V.

xvii

Scott W. R.: Group Theory, 2nd ed., Dover Pub., 1987.

Humphreys J. F.: A Course in Group Theory, Oxford University Press, 1997.

SmithG., TabachnikovaO.: Topics in Group Theory, SUMS, Springer-Verlag,

2000.

,

J.J. Rotman 6].

,

1], 2], 3], 4], 5], 8], 9] 10], -

, ( -- ) :

Hilbert D., Cohn Vossen S.: Anschauliche Geometrie, Springer, 1932.

Coxeter H.S.M.: Introduction to Geometry, Wiley, 1961.

Fejer oth L.: Regular Figures, Pergamon Press, 1964.

Coxeter H.S.M.: Regular Polytopes, Dover Pub., 1973.

Grove L., BensonC.: Finite Reflection Groups, GTM, Vol. 99, Springer-Verlag,

1971.

BergerM.: Geometry (2 Volumes), Universitext, Springer-Verlag, 1987.

Quaisser E.: Diskrete Geometrie, Spektrum Akad. Verlag, 1994.

Knrrer H.: Geometrie, Vieweg, 1996.

(

-

.)

, -

() -

(, , -

, , ..) -

-

:

. .: ( ), 2 ,

, , 1991.

. , , ( ) - . - .

() x y x, y n,

x = y(mod n) (, , x y(mod n)).

x y ( ) n n. ,

, ,

xviii

(. measure) (. meter),

. , -

( ) -

(. ) (: ,

- - 23 ..). , x

y n.

() commutative // (24 ab = ba),

anticommutative -

// ( ab = ba), , , -

(

, - ) (

- - ,

- ). , ,

, ()

;]25

() -

permutation,

commute permute. ( permutation groups

3 ! ,

transposition.)

() lattice (

) . -

() Gitter Verband. ,

-

V

Zv1 + Zv2 + + Zvn = {1v1 + 2v2 + + nvn | i Z, i, 1 i n} ,

{v1, v2, . . . , vn} V

R, 26. ,

x, y (infimum) xy

(supremum) x y, .

23. (. 79) (. 72).24 ab = ba, a b () -- .

25, -// .26 , , , . lattice n = 2, n 3. (, ). , , ( ) honey comb. , , ( ), . n = 3 , . zome-kits (.http://store.yahoo.com/zome-tool/index.html) () . , , , .

xix

() degree/order/rank,

: /(, )/, .

() , , -

regular/canonical/normal, - - -

, , -

,

. , , -

:(,

)/ /

(, ,

/, ..

),

27. .. regular polyhedron , normal

subgroup . normal -

G ( -

G), . , ,

:

(i) ,

(ii) H G

G/H ( gH = Hg g G),

(iii) G G,

G (. .

23.6),

(iv) 28 G

G 29.

() coset. ()

. ,

complex

. coset

Nebenklasse.

() , ,

, :

unit matrix / unitary matrix /

maximum / maximal element /

minimum / minimal element /

27: orthonormal ( ).

28 G ( ()). , H1

,H2

G, H1

H2

= H1

H2

H1

H2

= G H1

H2

. , G, , .

29 (modular lattice) x, y, z, x z, : x (y z) = (x y) z. ( G .)

xx

() , , G

G , - -

-

.

. , , ,

( , )

,

( -

),

( .. -

-

, -

). , ,

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.

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30. / / -

-

. ( -

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( ) -

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31.

. .

, -

. ,

-

.-. -

30 , . .

31. . ( .), ....,. , 1986, . 24.

xxi

, -

. ,

-

. , ,

Groups and Symmetry, ( -

32)

.

,

, . . -

. .

. ,

..., . ,

.

( ...) -

.

, -

,

33 p6 (. . 26).

,

Leader Books ( . ), -

,

.

. .

, , 2002

32. http://www.math.uoa.gr/~mmaliak/algabibliografia.htm

33 , 12 60. - (La Giralda) (Lion Courtyard). . O. Jones 8]

Field R.: Geometric Patterns from Islamic Art & Architecture. Tarquin Publications, 1998.

, , , -

, ,

.

D Arcy Thompson

(On Growth and Form, Cambridge, 1917)

,

, -

. ,

, ..

, , -

.

Rene Thom

(Paraboles et Catastrophes, Flammarion, 1983)

, . -

,

. ,

.

, -

.

, -

, -

. -

,

, ( ) -

, ,

. 28 ,

. ,

,

-

. Langrange, Cauchy Sylow

, , -

, -

, Nielsen Schreier.

, -

.

xxvi

-

,

-

.

- - -

.

,

.

,

.

, , ,

( -

, -

,

), -

.

,

,

.

, ,

, , -

. ,

H. Wielandt -

Sylow (. 20), J. McKay

Cauchy (. 13) -

J.-P. Serre (. 28).

.

A.M. Macbeath,

. ,

,

.

,

, .

. Andrew Jobbings,

, LyndonWoodward

, S. Nesbitt

-

,

xxvii

: -

Cambridge University Press (

Growth and Form ), Flammarion (

Paraboles et Catastrophes ), Dover Publications ( 2.1

SnowCrystals ), Office du Livre, Fribourg ( 25.3 -

26.2 Ornamental Design ), Plenum Publishing

Corporation ( 26.2

Symmetry in Science and Art ).

M. A. Armstrong

Durham, England, 1987

. . . . . . . . . . . . . . . . . . . . . . . . vii

. . . . . . . . . . . . . . . . . . . . . . . . . xxv

1 . . . . . . . . . . . . . . . . . . . . . 1

2 . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 . . . . . . . . . . . . . . . . . . . . . . . . 28

6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8 Cayley . . . . . . . . . . 52

9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

11 Lagrange . . . . . . . . . . . . . . . . . . . . . . . . 79

12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

13 Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . 96

14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

17 , . . . . . . . . . . . . . . . . . 129

18 . . . . . . . . . . . . . . . . . . . . . . . . . . 138

19 . . . . . . . . . . . . . . . . . . 146

20 Sylow . . . . . . . . . . . . . . . . . . . . . . . . . 158

15 . . . . . . . . . . . . . . . . . . 160 16; . . . . . . . . . . . . . . . . . 164

21 . . . . . . . . . . . . 169

22 . . . . . . . . . . . . . . . . . . . . . . 178

23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

25 . . . . . . . . . . . . . . . . . . 204

26 . . . . . . . . 217

xxx

27 . . . . . . . . . . . . . . . . . . 233

28 Nielsen Schreier . . . . . . . . . . 242

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

1

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coshu sinhu

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cosh (u v) = coshu cosh v sinhu sinh v,sinh (u v) = sinhu cosh v coshu sinh v

, ,

[coshu sinhu

sinhu coshu

] [cosh v sinh v

sinh v cosh v

]=

[cosh (u+ v) sinh (u+ v)

sinh (u+ v) cosh (u+ v)

].

. -

( ) -

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[cosh0 sinh 0

sinh 0 cosh0

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2.1

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b c

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C = {z C : |z| = 1} .

3. 17

z, w C, |zw| = |z| |w| = 1, zw C. 1 C -

. , z C, 1z

= 1|z| = 1, 1z C, C C.

, C .

, , ()

,

C.

,

.. (R,+)

,

.

, , -

, -

R

.

( ).

.

-

, -

. , , Z -

R.

, 5.

( Abel)

xy = yx

x, y. -

,

x+ y = y + x, x y = y x

x, y.

n . {0, 1, 2, . . . , n 1} n. x

y , +n

18

:

x+n y =

{x+ y, 0 x+ y < n,x+ y n, x+ y n.

, 5 +6 3 = 8 6 = 2. ( . ..

2). .

x +n y {0, 1, 2, . . . , n 1}. , (x+n y ) +n z x+n (y +n z)

x+ y + z, 0 x+ y + z < n,x+ y + z n, n x+ y + z < 2n,x+ y + z 2n, x+ y + z 2n,

. -

+n nx () x, x = 0. ,

x+n y = y +n x,

Zn.

n

n. , x

n 0, 1, 2, . . . , n 1, x n.

x (mod n) . ,

x+n y = (x+ y) (mod n) .

, 0, 1, 2, . . . ,

n 1 n,

x n y = xy (mod n) .

, 5 6 3 = 3, - .

; , ,

. . n = 10,

2 10 5 = 0, 10 {1, 2, . . . , 9} 1 9. , . , -

n {1, 2, . . . , n 1}

3. 19

n . ,

{1, 3, 7, 9} ( 0, 2, 4, 5, 6, 8 {0, 1, 2, . . . , 9}) 10.

; ( -

11.)

3.1

.

(i) .

(ii) a+ b2, a, b Z.

(iii) a+ b2, a, b Q.

(iv) a+ bi, a, b Z.3.2 Q(

2) 3.1 (iii).

a+b2 Q(

2) -

1a+b

2 c + d

2, c, d Q. ,

Q(2){0} .

3.3 n G

z, zn = 1.

G

.

3.4 n

n=1

{z C | zn = 1}

-

.

3.5 n .

(x n y) n z = x n (y n z)

x, y, z Z.

20

3.6

{1, 3, 7, 9, 11, 13, 17, 19}{1, 3, 7, 9}{1, 9, 13, 17}

20.

3.7 -

14;

{1, 3, 5}, {1, 3, 5, 7},{1, 7, 13}, {1, 9, 11, 13}.

3.8 , {1, 2, . . . , 20, 21} - 11,

22.

3.9 p x ,

: 1 x p 1. x, 2x, . . . , (p 1)x p. z,

:

1 x p 1 xz(mod p) = 1.

3.10 3.5 3.9

n

{1, 2, . . . , n 1} n . n ;

4

-

( 1). -

, -

. n 3 n . ,

, .

n = 3, . -

. r s 4.1,

:

e, r, r2, s, rs, r2s. ()

4.1

22

, r2 rr,

r . , r3

, r 2,

r . , s2

.

rs s (

) r. 4.1 rs

M. , r2s

N.

() , D3. ( ), -

- - .

rs . sr; 4.2

r2s. , sr2 = rs.

4.2

:

sr2 = s(rr) = (sr)r = (r2s)r = r2(sr)

= r2(r2s) = r4s = r3(rs) = e(rs)

= rs.

.

, ,

, .

r3 = e, s2 = e

sr = r2s ,

4. o 23

(). :

r(r2s) = r3s = es = s,

(r2s)(rs) = r2(s(rs)) = r2((sr)s = r2((r2s)s)

= r2(r2s2) = r4s2 = re = r.

(r2s)(rs) = ((r2s)r)s. , , -

. , r2srs

. ,

x1, x2, . . . , xn , -

( )-

, x1, x2, . . . , xn

. -

4.10.

.

,

,

xyy1x1 = xex1 = xx1 = e,

-- y1x1xy = e. , :

(4.1) . x y ,

(xy)1 = y1x1.

, x1, x2, . . . , xn ,

(x1 x2 xn)1 = x1n x12 x11 .

m x ,

xm =

xx x m

, m > 0

e, m = 0

x1 x1 x1 m

, m < 0

24

xmxn = xm+n, (xm)n = xmn

m n.

, ,

36 xy (x, y)

x, y D3.

e r r2 s rs r2s

e e r r2 s rs r2s

r r r2 e rs r2s s

r2 r2 e r r2s s rs

s s r2s rs e r2 r

rs rs s r2s r e r2

r2s r2s rs s r2 r e

xy x y.

, s(rs).

. -

(. 4.4). , -

,

.

Dn

n .

, D3. r

2n ,

, s ,

. Dn :

e, r, r2, . . . , rn1, s, rs, r2s, . . . , rn1s.

, rn = s2 = e, sr = rn1s. rn1 = r1, sr = r1s. , -

. ,

sr2 = srr = r1sr = r2s = rn2s.

ra ras, a

4. o 25

0 a n 1. ,

rarb = rk

ra(rbs

)= rks

} k = a+n b

(ras) rb = rls

(ras)(rbs

)= rl

} l = a+n (n b)

, r s

Dn. ( 5).

.

. G |G| . x xn = e n,

x , x

m, xm = e. ,

x .

(4.2) . (i) D3 .

3 ( r, r2) 2 ( s, rs, r2s).

(ii) Z6 . 1 5 6, 2

4 3, 3 2.

(iii) H R - -

,

,

.

(iv) , C -

. -

,

. ei

2,

= 2mn, m Z n Z{0}.

4.1 -

D4. 2 D4;

26

4.2 Z5,Z9 Z12.

4.3 1, 2, 4, 7, 8, 11, 13 14 -

15. -

-

.

4.4 g G. g

{gx | x G} , G. K

{xg | x G}.4.5 x x2 = e

x = x1. , ,

2.

4.6 x y G x, y, xy

2, xy = yx.

4.7 G x,

0 x < 1.

x+ y =

{x+ y, 0 x+ y < 1x+ y 1, x+ y 1

G ,

.

4.8 x g G. x gxg1

. xy yx

x, y G.

4.9 2 2 [a b

c d

],

a, b, c, d Z ad bc = 1, .

A =

[0 11 0

], B =

[0 1

1 1],

A,B,AB,BA .

4. o 27

4.10 . G . -

x1x2 xk G 1 k n1. - x1x2 xn n ,

.

-

:

(x1x2 xr) (xr+1 xn) , (1)

(x1x2 xs) (xs+1 xn) , (2)

1 r < s n 1. . (1)

(x1x2 xr) [(xr+1 xs) (xs+1 xn)] ,

(2) -

, ,

.

5

e, r2, r4, s, r2s, r4s

D6 -

. .

, , ,

e1 = e,(r2)1

= r4,(r4)1

= r2, s1 = s,(r2s

)1= r2s,

(r4s

)1= r4s.

5.1 -

, D3

D6, D6 :

G

G, , G,

' .

( , -

).

H G

H G. -

: x y H ,

5. 29

G. H;

G H; H

G. H;

, H

G G. -

.

( (xy)z = x(yz) G,

,

G). H G, H < G.

5.1

(5.1) . (i) Z < Q, Q < R R < C.

(ii) , 2Z,

. ,

nZ n

Z.

(iii) Q{0} < R{0} R{0} < C{0}.(iv) {1} < C C < C{0}.(v) e, r2, r3, r4, r5 D6.

(vi) {e, r, s, rs} D6. r , rr = r2 .

(vii) {0, 2, 4} Z6.To (v) : x G, -

x ( G

xn, n ) G. (

30

xmxn x xm+n,

x, G x0 xn

xn, x). x x . x ,

x = {. . . , x2, x1, e, x, x2, x3, . . .} . x , m,

x = {e, x, x2, . . . , xm1} ., x x - x. x G - G, G .

(5.2) . (i) 1 1 o Z, Z . (

Z , , , 1

1 + 1 + 1 + 1 = 4.)

(ii) 1 Zn, Zn n.

(iii) Z6

0 = {0},

1 = 5 = Z6,

2 = 4 = {0, 2, 4},

3 = {0, 3}. , 4 4, 4+6 4 = 2 4+6 4+6 4 = 0.(iv) D3

e = {e},

r = r2 = {e, r, r2},s = {e, s},

rs = {e, rs},r2s

= {e, r2s}.

5. 31

Dn ,

r s. r s

Dn.

, X G.

xm11 xm2

2 xmkk ()

x1, x2, . . . , xk X ( -

) m1, m2, . . . , mk , (

) X.

G. ( ,

X X.

G x0 x X,

() xmkk xm22 xm11 , X). -

X. G, X

G X ( )

G. X G Y

G. Y X, Y G , X

Y, Y G.

(5.3) . (i) r s Dn.

. ,

rs s Dn, r = (rs)s,

r s -

rs s.

(ii) H C -

. ,

m1z1+m2z2+ +mkzk . {1, i} Gauss, a+ ib, a, b Z.(iii)

5.2. G

,

.

G .

32

5.2

(. 5.9)

G ( ) -

, ,

,

. t

, t(x) = x+1, s ,

s(x) = x. G :. . . t2, t1, e, t, t2, . . .

. . . t2s, t1s, s, ts, t2s, . . .()

e . , t2(x) = x 2, - t2 , ts(x) = t(x) = x+1, ts - 12 . t s

G. , ,

st(x) = s(x+ 1) = x 1,

t1s(x) = t1 (x) = x 1, st = t1s. s2 = e st = t1s - () , .

Dn.

r n t

. G -

D.

.

5. 33

(5.4) . H G

G xy1 H x, y H.

. H x, y H , y1 H, xy1 H. H = xy1 H x, y H, : x H, e = xx1 H x1 = ex1 H. , y H, y1 H, xy = x(y1)1 H. H G.

(5.5) . '

.

. H K G.

, H K = . x y H K, H K. H K , xy1 H K. , (5.4)

H K.

(5.6) . (a) Z .

(b) , , -

.

. (a) H Z. H ,

. H ,

x , H ,

x H. H . d H. d H.

n H, n d n = qd +m, q m 0 m < d, m = n (mod d) . n H d H. H , qd H, qd H,

m = n qd = n+ (qd) H.

d, m

. n = qd,

H d.

(b) G K G.

x G, G,

34

K, x.H = {n Z | xn K}. H Z.

(a) H . d H, xd

K. .

5.1 Z4, Z7, Z12,D4 D5.

5.2 m n m n,

Zn m. -

Zn m 1;

5.3 4 -

rs r2s Dn.

5.4 Dn r2

r2s

n n .

5.5 H ,

G. H G

xy H x y H.5.6 . -

,

, -

.

-

.

5.7 G H

G G, -

. H

G.

5.8 ; -

D;

5.9 f ,

-

.

5. 35

(a) f , f -

.

(b) f ,

. , f

.

(c) , f

.

5.10 Z12 -

Z12. Z5 Z9.

-

;

5.11 Q .

, Q -

.

5.12 a, b Z H = {a+ b | , Z}, H Z. d H . -

d a b. (

a b

a + b

.)

6

-

.

, 1 3,

2 , {1, 2, 3}. X X

. X -

SX .

: : X X : X X , - : X X, (x) = ( (x)) , . ,

, X ,

. , : X X , 1 : X X, X 1 = = 1. X , SX . X = {1, 2, . . . , n}, SX Sn. Sn n

n!.

. S3 :

=

[1 2 3

1 2 3

],

[1 2 3

2 1 3

],

[1 2 3

3 2 1

],

[1 2 3

1 3 2

],

[1 2 3

2 3 1

],

[1 2 3

3 1 2

].

6. 37

-

, () .

,

[1 2 3

3 1 2

]

1 3, 2 1 3 2.

,

[1 2 3

2 1 3

] [1 2 3

1 3 2

]=

[1 2 3

2 3 1

],

[1 2 3

1 3 2

] [1 2 3

2 1 3

]=

[1 2 3

3 1 2

].

()

S3 .

, , Sn n 3. (;) n,

. ,

S6

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

:

[1 2 3 4 5 6

5 4 3 6 1 2

].

= (15)(2 4 6). ,

-

, .

, 1 5 5 1, 2 4, 4

6 6 2. -

. ( ..

3). '

: ,

, ,

.

, ...,

38

-

. , , -

- - ,

,

.

(6.1) . (i)

[1 2 3 4 5 6 7 8 9

1 8 9 3 6 2 7 5 4

]= (2 8 5 6) (3 9 4) .

(ii)

[1 2 3 4 5 6 7 8

8 1 6 7 3 5 4 2

]= (1 8 2) (3 6 5) (4 7) .

(iii) S3

, (1 2) , (1 3), (2 3), (1 2 3), (1 3 2).

() :(1 2) (2 3) = (1 2 3), (2 3) (1 2) = (1 3 2).

(a1 a2 . . . ak), (

) ,

. (a1 a2 . . . ak) a1 -

a2, a2 a3 .., ak1 ak, , , ak a1,- .

k (a1 a2 . . . ak). ,

k k-. , 2- -

.

, Sn

( ,

).

(6.1) (i),

(2 8 5 6) (3 9 4).

2, 5, 6, 8, 3, 4 9. -

, ,

(2 8 5 6) (3 9 4) = (3 9 4) (2 8 5 6) . .

Sn -

, = .

Sn

, ,

.

(6.2) . Sn Sn.

6. 39

. Sn -

-

,

(a1 a2 . . . ak) = (a1 ak) (a1 a3)(a1 a2).

, Sn -

.

-

.

(6.3) . ,

[1 2 3 4 5 6

5 4 3 6 1 2

]= (1 5) (2 4 6) = (1 5) (2 6) (2 4) .

(2 4 6) = (6 2 4) ,

S6

[1 2 3 4 5 6

5 4 3 6 1 2

]= (1 5) (6 2 4) = (1 5) (6 4) (6 2)

= (1 5) (4 6) (2 6) .

(6.4) . (a) (1 2), (1 3), . . . , (1n)

Sn.

(b) , (1 2), (2 3), . . . , (n 1n) Sn.

. (a) (a b) = (1a)(1 b)(1a) -

(6.2).

(b)

(1 k) = (k 1 k) (3 4) (2 3) (1 2) (2 3) (3 4) (k 1 k)

(a).

(6.5) . (1 2), n- (1 2 . . . n),

Sn.

. (6.4) (b) -

(k k + 1)

40

(1 2) n- (1 2 . . . n).

:

(2 3) = (1 2 . . . n)(1 2)(1 2 . . . n)1,

, k, 2 k < n,

(k k + 1) = (1 2 . . . n)k1(1 2)(1 2 . . . n)1k.

Sn -

. , -

.

P = P (x1, x2, . . . , xn)

= (x1 x2)(x1 x3) (x1 xn) (x2 x3) (xn1 xn),

(xi xj), 1 i n , 1 j n i < j. Sn, P (x(i) x(j)), 1 i n , 1 j n i < j. , ()

P,

. , P P P. + ( ),

( ). , , +1, 1, ( ). (..):

sign.]

(6.6) . P ,

. n = 3 = (13 2),

P = (x1 x2) (x1 x3) (x2 x3)

P = (x3 x1) (x3 x2) (x1 x2) = P.

(1 3 2) +1 ( ).

6. 41

, , Sn, - (, ,

(+1)(+1) = +1, (+1)(1) = 1, (1)(+1) = 1 (1)(1) = +1). - (1 2) 1 . , a > 2,

(1a) = (2a) (1 2) (2a) ,

1 (1a). ,

(a b) = (1a) (1 b) (1 a) ,

1 . -, Sn,

, +1 ,

1. Sn -

Sn .

(a1 a2 . . . ak) = (a1 ak) (a1 a3)(a1 a2),

-

.

(6.7) . Sn n!2 ,

An n.

. ,

. -

.

, 1 . , , = (1 2)(12).

Sn. , (1 2)

. Sn

, Sn .

( -

(1 2);)

(6.8) . n 3, 3- An.

42

. 3- , , .

An, (6.4) (a)

(1a).

(1a)(1 b) = (1 b a).

3-.

(6.9) . A4

, (1 2) (3 4) , (1 3) (2 4) , (1 4) (2 3) ,

(1 2 3) , (1 2 4) , (1 3 4) , (2 3 4) ,

(1 3 2) , (1 4 2) , (1 4 3) , (2 4 3) .

S4, ,

(1 2) , (1 3) , (1 4) , (2 3) ,

(2 4) , (3 4) , (1 2 3 4) , (1 2 4 3) ,

(1 3 2 4) , (1 4 3 2) , (1 3 4 2) , (1 4 2 3) .

.. (1 3) (2 4) 3-,

(6.8)

(1 3) (2 4) = (1 3) (1 2) (1 4) (1 2) = (1 2 3) (1 2 4) .

6.1 S3.

6.2 S8

, -

.

(a)

[1 2 3 4 5 6 7 8

7 6 4 1 8 2 3 5

], (b) (4 5 6 8) (1 2 4 5), (c) (6 2 4) (2 5 3) (8 6 7) (4 5).

A8;

6.3 S9 2, 5 7 -

() 2, 5 7 S9.

;

6.4 S4 .

S4;

6. 43

6.5 P (x1, x2, x3, x4), = (1 4 3) = (2 3) (4 1 2).

6.6 H Sn An, -

H .

6.7 , n 4, Sn , 2. (-

.)

6.8 Sn, 11

An, 1 An

. n = 4

= (2 1 4 3), = (4 2 3) .

6.9 n ( , ),

(1 2 3) (1 2 . . . n) ( , (1 2 3) (2 3 . . . n)) -

An.

6.10 , Sn = , , .

, n-,

.

6.11 6.2.

6.12 Sn , -

.

7

(. . 7.1) : -

e, r q1 q2

.

,

. , -

8 {1, 3, 5, 7} , .

e r q1 q2

e e r q1 q2r r e q2 q1q1 q1 q2 e r

q2 q2 q1 r e

1 3 5 7

1 1 3 5 7

3 3 1 7 5

5 5 7 1 3

7 7 5 3 1

, -

.

, -- -

. ,

-

.

G G, : G G , :

e 1, r 3, q1 5, q2 7,

7. 45

, , x G

x. , x x y y, , , xy xy.

7.1

G G. , G

G. , G G , :

G G ( ) ( G G , , G G) : G G, :

(xy) = (x) (y) , x, y G.

( , , .)

: G G , - G

G. , , (xy) = (x) (y) x, y G, G

G G G. G, , G . 1 : G G ,

46

G G. G G , G = G.

(7.1) . (i) : R R>0 (x) = ex.

(x+ y) = ex+y = exey = (x) (y) , x, y R.

R R>0 . ( R -, R>0 .)

(ii) -

, .. , -

G.

: 1, 2, 3, 4 7.2. -

,

{1, 2, 3, 4}. , r (2 3 4), s (1 4)(2 3). -

A4. u, v , -

, uv . ,

( ) ( ( ) )

G A4.

7.2

7. 47

(iii) Z. G x,

: G Z, (xm) = m.

H :

(xmxn) = (xm+n

)= m+ n = (xm) + (xn) .

.

(iv) n Zn. G x,

: G Zn, (xm) = m(mod n)

.

(v) {1,1, i,i} -

i i., (iv),

1 0, i 1, 1 2, i 3,

1 0, i 1, 1 2, i 3

Z4.

(vi) D3 S3. -

(ii), ,

, 1, 2, 3.

(vii) Q Q>0.- : Q Q>0 x, (x) = 2,

(x) = (x2+x

2

)=

(x2

)(x2

)= 2,

(x2

)=2 / Q, .

(7.2) . : G G . |G| = |G| G

G.

48

. x G (x) = x.

x (e) = (x) (e) = (xe) = (x) = x

, , (e)x = x. (e) G. :

(e) (e) = (ee) = (e)

-

(e) G. -, ,

( ) G

G.

(7.3) . : G G . ,

(x)1

= (x1

), x, x G.

. , x G,

(x1

) (x) =

(x1x

)= (e) =

(xx1

)= (x)

(x1

),

(x)1

= (x1

).

(7.4) . : G G G , G ' .

. x, y G (x) = x, (y) = y.

xy = (x) (y)

= (xy)

= (yx)

= (y) (x) = yx,

G .

(7.5) . : G G H G, (H) G.

7. 49

. x, y (H). , |H , x, y H, (x) = x (y) = y. xy1 H, H G. ,

(xy1

)= (x)

(y1

)= (x)(y)1 = xy 1

xy 1 (H). , - (5.4).

(7.6) . : G G - G.

. g G. (7.5)

H G

g. x (H) ,

x = (gm) = (g)m

m., (H) (g)

. H (H) ,

(g) g.

, ,

.

(7.7) . : G G : G G -,

: G G

.

1 : ,

. ,

, ,

: , -

. . -

A4.

D6, (

50

6 ), -

Z12 ( (7.1) (iv)). Z12 , ,

(. (7.4)). , D6,

A4, . D6 A4

(. (7.6)).

7.1 1, 2, 4, 5, 7 8

9

Z6.

7.2 1, 3, 7, 9, 11, 13, 17 19 -

20.

Z8.

7.3 {, (1 2) (3 4) , (1 3) (2 4) , (1 4) (2 3)} A4 - .

7.4 S3 D3.

S3 D3;

7.5 G . x x1 G G G .

7.6 Q>0 Z.

7.7 G g G,

: G G, (x) = gxg1,

.

G = A4 g (1 2 3).

7.8 H G G

{e} H G. , .

7.9 G . x G

: G G , - (x) (x)

7. 51

G.

Z Z, - Z12 Z12.

7.10 R Q R{0} Q{0}. R R{0};

7.11 S6 (1 2 3 4) (5 6)

, 7.2.

7.12 S4 (1 2 3 4) (2 4)

D4.

8

Cayley

() : ( -

), ( ), ( -

), ( )

( ), . 8.1. (...):

. H.

Weyl 1] (p. 74): -

.

.

, ,

, -

,

. ...

1. , -

,

1 (. 415-369 ..). , . (. . - .. . , 1980). . , ,

8. cayley 53

2.

3 , -

, -

4. , , ,

, -

,

: , , , ,

, , , ,

. , , 5.

6

7 (, ) -

8.]

-

A4.

.

. 9 , , .

2 F. Lindemann (12/4/1852-6/3/1939), - ( 1881), . , Zur Geschichte der Polyeder und der Zahlzeichen (Sitzungsb.Bayr. Akad. Wiss., Math.-Phys. Kl., Bd. 26, 1896, 625-768), ( , , ..) .

3 . W.C. Waterhouse: The Discovery of the Regular Solids, Arch. Hist. of Exact Sciences 9, (1972), 212-221.

4 , , : . , --: . , , , 1998. (. . . 249-255.)

5(. , . 55c). , 2400 400 Mysterium Cosmographicum J. Kepler, . , , ( , ), ( ) , ( ) .

6. . : ( , , ). , , -, , , (1957) . 167-187].

7 (, ) (P1

, E1

, F1

)

(P2

, E2

, F2

) P1

, P2

, E1

, E2

() F1

, F2

, Pi

Ei

Fi

i = 1, 2, : R3 R3 : (P1

) = P2

, (E1

) = E2

(F1

) = F2

.

8 , : () , () () . , :() ( ), () ( ) () . .

54

8.1

.

, -

-

.

L,M N 8.2. L,

, M ,

, -

N, 23 43 .

, , (3 3)+(6 1)+(4 2)+1 = 24 ( ).

S8. , ,

-

.

S4.

8. cayley 55

. 8.2 Nk k k, 1 k 4. N1, N2, N3, N4,

1, 2, 3, 4. , . 8.2, -

r N1 N2, N2 N3, N3 N4 N4 N1, (1 2 3 4), s

(1 4 3). t N1 N2, N3

N4 . ( N3 N4

, ). , t

(1 2).G

: G S4 .

( ),

.

8.2

, , , -

4 , .

, -

, ,

56

. -

. (1 2 3 4) (1 2) -

(G) , S4,

G S4. ,

(1 2 3 4) (1 2) (G).

(1 2 3 4) (1 2) S4,

(G) = S4, .

-

,

(. . 8.3). ,

, .

. ,

( ) ,

. ,

,

.

8.3

, -

.

. , , -

, . 8.4

.

-

.

.

,

,

.

-

.

8. cayley 57

8.4

,

, -

A5.

, .

(i)

.

(ii) A5 60.

(iii)

1 5,

S5.

(iv) 3- S5

, - -

.

(v) 3- S5 A5.

:

(8.1) .

A4.

S4.,

-

A5.

58

.

-

.

10.

-

. , ,

.

(8.2) Cayley. G

SG.

. g G Lg :

Lg : G G, Lg(x) = gx.

( Lg ,

Lg (x) = Lg (y) = gx = gy = g1gx = g1gy = ex = ey = x = y,

, z G, Lg(g1z

)= gg1z = ez = z). H Lg

g. ( , G = R,

Lg g). G -

{Lg | g G} SG. SG . ,

Lg (Lh (x)) = Lg (hx) = ghx = Lgh(x), x, x G.

, G G.

SG G Le, -

Lg SG Lg1 G.

G SG.

G G, g Lg,

G -

G (gh Lgh = LgLh). , , Lg = Lh g = Lg(e) = Lh(e) = h.

, G G

SG.

(8.3) . G n, G

Sn.

8. cayley 59

. , , G 1, 2, . . . , n,

G 1, 2, . . . , n. , -

SG Sn, G

SG G Sn. G

G , G

G.

(8.4) . G G

, -

.

Lr (e) = r, Lr (r) = r2 = e,

Lr (q1) = rq1 = q2, Lr (q2) = rq2 = q1.

, Lr e r, q1 q2.

Lq1

Lq2

e, r, q1, q2 - -

1, 2, 3 4, G

{, (1 2), (1 2)(3 4), (1 3)(24), (1 4)(2 3)} S4.

8.1

1, 2, . . . , 6,

S6.

S6, , -

S6.

8.2 -

-

, , -

.

8.3 1, 2, . . . , 6.

S6, r, s t -

8.2.

8.4

, 18.1.

-

; S4 ();

60

8.5 ,

.

-

.

(a) 25 ,

.

(b) , -

.

(c) 23 ,

.

8.6 Cayley -

S6 D3.

8.7 Cayley R - SR, -

.

8.8 Sn Sn+2 :

1, 2, . . . , n. , n+1

n+ 2 , , ,

n + 1 n + 2. -

Sn An+2.,

n = 3.

8.9 G n, G -

An+2.

8.10 Sn # S2n

#(k) =

{(k), 1 k n,(k n) + n, n+ 1 k 2n.

# -

# Sn - A2n. n = 3.

8.11 G -

T -

7.2. q T ,

8. cayley 61

(1 2) , qr 4-

(1 2 3 4). , qr

, .

, T G -

S4.

8.12 -

.

9

nn , - , -

. , A = [ai j ] B = [bi j ]

, i- j-

ai 1b1 j + ai 2b2 j + + ai nbnj .

, nn In , AB

B1A1.

A -

fA : Rn Rn, fA(x) = xAt

x = (x1, . . . , xn) Rn, At - A.

fAB(x) = x (AB)t= xBtAt = fA (fB (x)) ,

9. 63

AB fAfB

fA fB . f : Rn Rn - A

Rn, A f = fA. , -

, GLn.

-

, GLn(R). R C GLn(C) n n .

n 2, GL2, GL3, . . . . n = 1,

,

( ), -

. -

GL1 = R{0}. A GLn, (n+ 1) (n + 1)

A =

[A 0

0 1

]

GLn+1. A

A (

n (n+ 1) ), , , ,

1. o

GLn+1,

GLn GLn+1, A A, GLn . Rn Rn+1 , ,

fA fA Rn, -

. ,

Rn+1 = Rn R, fA: Rn+1 Rn+1, f

A(x, z) = (fA(x), z).

n n A = (ai j) AtA = In,

a1 ia1 j + a2 ia2 j + + an ian j ={

1, i = j

0, i = j

64

, A

( ).

Rn. , A Rn ,

det(AtA

)= (det(A))

2,

A {1, 1}. A B , (

AB1)tAB1 =

(B1

)tAtAB1 =

(Bt)tAtAB1 = BAtAB1 = In.

, AB1 , ( (5.4))

n n GLn. On.

On 1 On,

SOn.

(9.1) . A On, fA .

. x y Rn fA (x) fA (y) fA (x) fA (y) .

fA (x) fA (y) =(xAt

) (yAt

)t= xAtAyt

= xyt = x y.

x = x x, x y fA (x) = x , fA . ,

fA (x) f (y) = fA (x y) = x y ,

fA x y. ,

fA (x) fA (y) = 0 x y = 0,

, x y, fA (x)

fA (y).

.

(9.2) . f : Rn Rn - f = fA A On.

9. 65

. -

,

f (x) f (y) = f (x y) = x y ,

,

f (x) f (y) = 12[f (x)2 f (x) f (y)2 + f (y)2

]= 12

[x2 x y2 + y2

]= x y.

, f Rn

. A, f ,

, A .

f = fA A On.

n {2, 3}, - ( -

) . ,

, n = 2.

(9.3) . 22 - ,

. 22 1 ().

. A O2, A .

A =

[a c

b d

],

(a, b) , a = cos

b = sin , 0 < 2. (c, d) ,

(a, b), c = cos

d = sin, { + 2 , 2 }.

A =

[cos sin sin cos

],

66

SO2

( ),

A =

[cos sin

sin cos ],

1, ,

2 x.

(9.4) . SO2, -

, C .

C ei, 0 < 2,

C SO2, ei [

cos sin sin cos

],

, C = SO2. ,

n = 3.

(9.5) . SO3

R3 - . ( () R3, , SO3.)

. A SO3. det (A I) , -

. A .

,

1 . v ,

1, ,

v, -

fA. , fA

, , v

, . , -

R3, vv , fA

SO3 1 0 00

0 B

9. 67

, B SO2, fA v.

9.11

() R3, , SO3.

(9.6) . A O3SO3 , .

. A O3SO3

AU SO3,

U =

1 0 00 1 0

0 0 1

.

U (x, y)-

. A = (AU)U,

fA = fAUfU ,

o fAU (

(9.5)).

(9.5), SO3

. -

R3, , O3. ,

SO3,

O3.

(9.7) . P = (1, 1, 1), Q = (1,1, 1), R = (1,1,1) S = (1, 1,1) , (. 9.1).

, P ,

0 0 11 0 0

0 1 0

,

0 1 00 0 1

1 0 0

.

68

9.1

PQ RS

z,

1 0 00 1 0

0 0 1

.

,

P , Q (0, 0, 0) P Q , R S.

0 1 01 0 0

0 0 1

.

, P,Q,R, S

() , -

. ;

SO3 19. -

.

GLn(C) n- Cn. z Cn,

9. 69

z z z, . U

o U tU . ,

Cn. nn GLn (C) , Un. Un, 1,

SUn.

9.1 nn ;

(a) .

(b) .

(c) ,

.

(d) ,

.

9.2

[a b

0 c

], a, b, c R ac = 0,

GL2(R).

9.3 GLn(R), +1 1, GLn(R). GLn(Z).

9.4 (1,3), (1,3) (2, 0) -. O2,

. -

,

(2, 0), (1,3), (1,3), (2, 0),

(1,3) (1,3).

70

9.5

A =

[cos sin sin cos

] B =

[cos sin

sin cos].

AA = A+, AB = B+, BA = B BB = A,

mod2. -

.

9.6

ABA1 , BAB ABA

1 B.

= 3 =4 .

9.7

12

0 .

0 1 .

12

0 .

SO3, ,

O3SO3. , .

9.8 23 13 2323 23 1313 23 23

,

1

213

16

12

13

16

0 13

26

.

9.9 v1,v2,v3

R3 A , v1 , v2 v3 .

B =

1 0 00 1 0

0 0 1

ABA1 -, v1 v2, ABA1 v3. -

x+3y = z.

9. 71

9.10

A

[A 0

0 1

], A SO2,

[A 0

0 1], A O2SO2,

O2 SO3.-

9.4 -

SO3,

.

9.11 R3, , SO3.

z- -

.

9.12 1 0 00 1 0

0 0 1

,

1 0 00 1 0

0 0 1

1 0 00 1 0

0 0 1

,

1 0 00 1 0

0 0 1

SO3 -

.

{(x, y, z) R3 x2 + (y 3)2 25, x2 + (y + 3)2 25, 1 z 1} , -

-

.

D2.

9.13 n n

1.

72

9.14 U2 [z w

eiw eiz]

z, w C, R zz + ww = 1. SU2;

10

() G H G H : GH (g, h), g G h H,

(g, h)(g, h) = (gg, hh).

g, g G , G, . -

h h H., (gg, hh) GH . - -

G H. , (e, e) , (g1, h1) (g, h). (

G H.)

GH H G, (g, h) (h, g),

G H = H G. G H , G H , GH G H . G H , GH . G {(g, e)| g G} G H ( g (g, e)), H {(e, h)|h H} GH

74

( h (e, h)), , G H , G H . G1G2 Gn , GH, n- (x1, x2, . . . , xn), xi Gi, i, 1 i n,

(x1, x2, . . . , xn)(x1, x

2, . . . , x

n) = (x1x

1, x2x

2, . . . , xnx

n).

, -

.

(10.1) . (i) Z2 Z3 :(0, 0), (1, 0), (0, 1), (1, 1), (0, 2), (1, 2).

(x, y) + (x, y) = (x+2 x, y +3 y

) .

,

() . , -

(1, 1)

Z2Z3. , Z2Z3 , , Z6. Z2 Z3 Z6 :

(0, 0) 0, (1, 1) 1, (0, 2) 2,(1, 0) 3, (0, 1) 4, (1, 2) 5.

(ii) ,

Z2 Z2 (0, 0), (1, 0), (0, 1), (1, 1), , , 2 . (0, 0)

2, Z2 Z2 . Z2 Z2 (. 7),

:

(0, 0) e, (1, 0) q1,(0, 1) q2, (1, 1) r,

Klein.

(iii) Rn n R -, , Rn x = (x1, . . . , xn)

x+ y = (x1 + y1, . . . , xn + yn)

10. 75

x = (x1, . . . , xn), y = (y1, . . . , yn) Rn.

(10.2) . Zm Zn (m,n) = 1.

. k (1, 1) ZmZn. (1, 1) k (0, 0),

(k(mod m), k(mod n)) = (0, 0).

m n k.

(m,n) = 1, mn k, k = mn.

, , Zm Zn (1, 1) .

d m n

d > 1. Zm Zn, , . m = m

d n = n

d. (x, y) ZmZn

mdn(x, y) = (mdnx(mod m),mdny(mod n))

= (mnx(mod m),mny(mod n))

= (0, 0),

(x, y) mdn. Zm Zn -, mn.

(10.3) . I 3 3 J = I. I J O3, -

O3 2. O3

SO3 .

: SO3 {I, J} O3, (A,U) AU.

,

((A,U) (B,V )) = (AB,UV ) = ABUV = (A,U) (B,V )

A,B SO3 U, V {I, J}. (A,U) = (B,V ) , AU = BV, det(AU) = det(BV ).

det(AU) = det(A) det(U) = det(U)

A SO3 -- det(BV ) = det(V ). , U = V A = B, .

76

. A O3, A SO3, A = (A, I) , AJ SO3, A = (AJ, J) . - .

( I 0, J 1) {I, J} Z2. SO3 Z2 = O3.

SOn Z2 = On n. , n , -

SOn Z2 On (. 10.9). : H K -

G, HK

xy, x H y K.

(10.4) . H K G,

HK = G H K = {e}, -- H K,

G = H K.. , (10.3). -

: H K G, (x, y) xy.

((x, y)(x, y)) = (xx, yy)

= xxyy

= xyxy(

H

K

)

= (x, y) (x, y) .

, HK G.-, (x, y) = (x, y) ,

xy = xy = (x)1 x = yy1. H

K, H K = {e}. (x)

1x = e = yy1 = x = x, y = y,

10. 77

. ,

HK = G G xy,

x H y K. .

(10.5) .

fJ : R3 R3, x x,

. R3 , , ,

, -

. G

H -

( -

SO3), ,

(10.4),

G = H fJ = H Z2.

:

(10.6) . -

S4 Z2, A4 Z2.

10.1 G H . G H , G H .

10.2 Z Z Z.10.3 C RR C{0}

R>0 C.10.4 , Klein, -

(Vierergruppe)

V . Z3 V Z2 Z6.10.5 {(x, x) | x G} G

GG, G.

78

10.6 G H . A

G B H, A B G H. Z Z, (

Z).

10.7 ;

Z24, D4 Z3, D12, A4 Z2,

Z2 D6, S4, Z12 Z2.

10.8 (, 1) An Z2 An Z2. An Z2 Sn n 3.

10.9 (10.3) -

SOn Z2 On n ; On On;

( n ) SOn Z2 On - .

10.10 G ,

(a1, a2, . . . ) . ( -

)

(a1, a2, . . . )(b1, b2, . . . ) = (a1 + b1, a2 + b2, . . . ).

G Z = GG = G.10.11 , n , D2n Dn Z2.10.12 G 4, G

Klein.

10.13 G , -

, 2. G

Z2.

11

Lagrange

G H. -

G H ; o

H G, g1 GH H g1

g1H = {g1h | h H}. () g1H H

() H g1H = . () H g1H, h g1h,

. ( -

g1H - - g11 ).

() x H g1H. h1 H , x = g1h1. g1 = xh

11 ,

g1 (g1 GH). H g1H = . H g1H G (, -

, ), |G| = 2 |H| . , - g2 G(H g1H) () g2H.

H, H . -

g1H g2H = . x g1H g2H, h1 h2 H ,

x = g1h1 = g2h2 = g2 = g1(h1h

12

).

80

, g2 / g1H. g1H , g2H H G, |G| = 3 |H| . , g3 G(H g1H g2H) . G ,

. k

. G

k+1 ( k+1 - - )

H, g1H, g2H, . . . , gkH

H. ,

|G| = (k + 1) |H| . :

(11.1) Lagrange. -

.

: G m

G, G m. -

, , A4

6. , -

. 13 , p

G, G

p. 20,

Sylow.

-

. G

S3 H {, (1 3)}. g1 H, g1 = (12 3).

g1H = {(1 2 3), (1 2 3)(13)} = {(1 2 3), (1 3)}.

g2 G(H g1H), g2 = (1 2).

g2H = {(1 2), (1 2)(13)} = {(1 2), (1 3 2)}.

-

H, g1H g2H , .

Lagrange.

(11.2) . G

G.

11. lagrange 81

.

, ,

Lagrange.

(11.3) . G , G

.

. x G{e}, x G( (11.2)). , x = G.

(11.4) . x G

x|G| = e.

. m x. (11.2) |G| = km k. , x|G| = xkm = (xm)k = e.

n Rn

Rn = {m Z | 1 m n 1 (m,n) = 1} . n Rn .

( ,

3). m1 m2

Rn. (m1m2, n) = 1, (m1m2(mod n), n) = 1, ,

Rn -

n.

(. 3.5), 1 -

. , m Rn, x y, xm+yn = 1. n,

m, x(modm).,

Rn (n),

- Euler . (...): (n)

Rn.]

(11.5) . (i) R9 1, 2, 4, 5, 7, 8, (9) = 6.

22 = 4, 23 = 8, 24 = 16(mod 9) = 7 25 = 32(mod 9) = 5,

R9 2.

(ii) R16 1, 3, 5, 7, 9, 11, 13, 15, (16) = 8. -

H = 3 K = 15, H = {1, 3, 9, 11} K = {1, 15}, HK = R16 H K = {1}. , (10.4),

R16 = H K = Z4 Z2.

82

(11.6) Euler. (x, n) = 1, x(n) 1

n.

. x n m, m Rn. (11.4) m(n) 1 n.

x(n) m(n) n, .

(11.7) Fermat. p

x p, xp1 1 p.

. Euler

(p) = p 1.

Lagrange, H

A4 12 = |A4| . , |H| = 1 |H| = 12, H = {} H = A4, . |H| = 2, H , 2, :

{, (1 2)(3 4)} , {, (1 3)(24)} , {, (1 4)(2 3)} .

H A4 3,

{, (1 2 3), (1 3 2)} , {, (1 2 4), (1 4 2)} ,{, (1 3 4), (1 4 3)} , {, (2 3 4), (2 4 3)} .

, A4 4,

H = {, (1 2)(34), (1 3)(2 4), (1 4)(23)} .

A4 3-

, (11.2), 3-

4.

, A4 6.

|H| = 6, . 3- H, , 3-

H .

6,

4 3- ,1, 1,

, , 1, , 1, , 1

11. lagrange 83

H, |H| = 6.,

H 3-. , H

{, (1 2)(3 4), (1 3)(24), (1 4)(2 3)} .

4 6, Lagrange

. ,

H A4 6.

11.1 , o -

Lagrange, G = D6, H = r , G = D6,H =

r3, , G = A4,H = (2 3 4) .

11.2 H G.

g1H = g2H g11 g2 H.

11.3 H K G

, H K ( ).

11.4 G -

. G

.

11.5 X Y G, XY

xy, x X y Y. X Y , Y G , ,

XY X, X

Y.

11.6 m n 1 , Rmn -

Rm Rn. R20 Z2Z4 .

84

11.7 n m

2n. Dn m.

11.8 A5 m m 60;

11.9 G m

.

G m.

11.10

.

11.11 H G

|G| = m |H| , o Lagrange, gm H g G.

11.12 Rp p -

.

12

( ) X

X , ,

X. Lagrange

,

. -

.

X x

y X. x y ( x

y ( ) ) x ,

y. :

(a) x X .

(b) x y, y x,

x, y X.

(c) x y y z, x

z, x, y, z X.

(a), (b), (c) ,

12.1 . -

86

.

12.1

. X R X X. , R (x, y), -

X. x y X, x

y R.-- (a), (b), (c) ,

R X. x X

R (x) = {y X | y x}

X x -

x.

(12.1) . R (x) = R (y) (x, y) R.

. (x, y) R z R (x) . z x x y, , (c), z -

y. , z R (y) , R (x) R (y) . , (b), (y, x) R, x y R (y) R (x) .

(12.2) . X = Z R - (x, y) Z Z x y 3., xx = 0 3, xy 3,

12. 87

yx 3, , xy, yz 3, xz = (x y)+(y z) 3. - 0 1 2. ,

. 0

3, 1

1 3, , 2

2 3. R (0), R (1) R (2) Z (. . 12.2). .

12.2

(12.3) . -

X X.

. , R (x) x ( (a)). R (x)R (y) = , z R (x)R (y) , z x y. (b), x z, y

( (c)). R (x) = R (y) , , ,

. , x R (x) , X.

(12.4) . 3

n Z, Z n

R (0) ,R (1) , . . . , R (n 1) .

.

x R (m) x m n.

88

(12.5) . H G R (x, y) GG y1x H. - R G. (- x G x1x = e H, y1x H,

x1y = (y1x)1 H,

, y1x z1y H, z1x =(z1y

) (y1x

) H). g G x G, : g1x H . g1x H x = gh h H. ,

R (g) = gH = {gh | h H}.

gH H

g. (12.3),

H G

G, ,

Lagrange. R (x, y) G G xy1 H, G.

g G Hg = {h g | h H}.

. -

Lagrange (

(12.3)) .

(12.3) .

.

(12.3) ,

.

(12.6) . x y G.

x y

gxg1 = y

g G. , , . -

G. R - GG (x, y), x y. x G , exe1 = x. x y, gxg1 = y, y

12. 89

x, g1yg = x. , x y y z,

g1xg1

1= y g2yg

1

2= z, x z,

(g2g1)x (g2g1)1 = g2

(g1xg

1

1

)g12

= g2yg1

2= z.

, R G - , , -

G. -

14.

(12.7) . X G SX ,

G X.R X X :

(x, y) R [g, g G : g (x) = y] .

, -

G. ( , . x X (x) = x, x . x

y, g (x) = y, y x, g1(y) = x.

, x y y z, g(x) = y

g(y) = z, x z, gg(x) = g(y) = z).

X,

-

(12.3), .

, .. X R3 G fA A SO3. ,

0 {0}. - ,

x ,

x . - , ,

R3.

,

.

(12.8) . ,

,

-- () .

90

, ,

(. . 12.3).

, ,

()

.

12.3

. -

b1 b2,

b1b2, b2 b1,

. 12.4.

12.4

, e , -

, .

() ( ) () ( ),

12. 91

( ) -

(. . 12.5).

12.5

b , -

b1. b1b

(. . 12.6).

12.6

-

.

b1 b2 , b1 b2, -

b1 1,

b2.

-

, ,

. R 1(...): - .

92

-

:

R (b1)R (b2) = R (b1b2) ,

. -

R(e) R (b1) R (b) .

B3 .

, ,

Bn n , n .

(...): (. braid groups, . groupes de tresses, .

Zopfgruppen) 1925 E.Artin2(1898-1962)

. , W.Magnus3, A. Markoff4 F.

Bohnenblust5 6

Bn Bn, -

Fn n .

F. Klein7, W. Magnus, A. Karrass & D. Solitar8, J.S. Birman9 S. Moran10.]

12.1 R R - R;

(a) {(x, y) | x y } ,2E. Artin: Theorie der Zpfe, Abh. Math. Sem. Univ. Hamburg 4, (1925), 47-72, Theory of braids, Ann. of Math.48, (1947), 101-126.

3W. Magnus: Uber Automorphismen von Fundamentalgruppen berandeter Flachen, Math. Ann. 109,(1934), 617-646.

4A. Markoff: Foundations of the Algebraic Theory of Tresses, (Russian), Trav. Inst. Math. Stekloff, Vol. 16, (1945).

5F. Bohnenblust: The algebraic braid groups, Ann. of Math. 48, (1947), 127-136.

6 , . 27.

7F. Klein: Vorlesungen uber hohere Geometrie, 3. Auflage, Springer, (1926). (. . 89.)

8W. Magnus, A. Karrass and D. Solitar: Combinatorial Group Theory. (Representations of Groups in Terms ofGenerators and Relations ), 2nd rev. ed., Dover Pub., (1976). (. . 3.7)

9J.S. Birman: Braids, Links and Mapping Class Groups, Ann. Math. Studies, Vol. 82, Princeton University Press,(1974).

10S. Moran: The Mathematical Theory of Knots and Braids: An Introduction. North-Holland, (1983).

12. 93

(b) {(x, y) | x y } ,(c) {(x, y) | x+ y } ,(d) {(x, y) |x y 0} .

12.2 R ,

, ;

(a) (z, w) R zw R,(b) (z, w) R z/w R,(c) (z, w) R z/w Z.

12.3 G H ,

{(x, y) |xy H } G.12.4 G H, {

(x, y)xyx1y1 H } G.

12.5 R X. x X, y X, (x, y) R. (b) (y, x) R. , (c) (x, x) R. , (a) .

;

12.6 n

Z - n. [x] x

:

[x] + [y] = [x+ y] .

, ,

-

. , .

,

[x] = [x] [y] = [y] ,

[x+ y] = [x + y] .

, n

Zn. ( , Zn .)

94

12.7

H G

G = A4, H = {, (1 2)(34), (1 3)(2 4), (1 4)(23)}

G = A4, H = {, (1 2 3), (1 3 2)} .

12.8 G H G

|H| = 12 |G| . gH = Hg g G.

12.9

. -

(m,n), m n

. (m,n) -

m/n. , -

. ,

(2, 3), (4, 6), (6,9) 23 . , - - .

-

. (m,n)

(m, n) mn = mn.

X. [(m,n)] -

(m,n) -

-

[(m1, n1)] + [(m2, n2)] = [(m1n2 +m2n1, n1n2)] ,

[(m1, n1)] [(m2, n2)] = [(m1m2, n1n2)] .

,

, ,

[(0, n)], ,

.

, -

.

12.10 () -

(12.6) ( ) -

G = D4.

D.

12. 95

12.11 B3 ,

b1, b2 12.4.

12.12 B3 : ,

-

, -- ()

. , -

, 1, 2, 3, , -

.

S3.

B3 S3,

, B3

S3. , ,

(1 2 3).

13

Cauchy

11, -

Lagrange, :

(13.1) Cauchy. G ,

p , G -

p.

. x G{e} xp = e.

X = {x = (x1, x2, . . . , xp) GG G | x1x2 xp = e} .

p- , -

(e, e, . . . , e), .

X.

X; p- (x1, x2, . . . , xp) X, -

x1, x2, . . . , xp1 G, xp

xp = (x1x2 xp1)1 .

p- X |G|p1 , - p.

13. cauchy 97

R XX : - (x,y) R y x, y p-:

(x1, x2, . . . , xp)

(xp, x1, . . . , xp1)...

(x2, . . . , xp, x1)

()

p- X. ,

xpx1 xp1 = xp (x1x2 xp1xp)x1p= xpex

1p

= e,

(xp, x1, . . . , xp1) X, - . R X, -

R (x) p- x = (x1, x2, . . . , xp) (). p- p -

p-; p-

e = (e, e, . . . , e),

R (e) . - R X,

X. - R (e)- p , X 1 p,

.

p- x = (x1, x2, . . . , xp), e,

R (x) < p. , , (),

(xr+1, . . . , xp, x1, . . . , xr) = (xs+1, . . . , xp, x1, . . . , xs) .

r > s, () pr

(x1, x2, . . . , xp) = (xk+1, . . . , xp, x1, . . . , xk) ,

k = p r+ s.

xi = xk+i(mod p), i, 1 i p,

98

x1 = xk+1 = x2k+1 = = x(p1)k+1,

() p.

bk + 1 = ak + 1 (mod p)

0 a < b p1. p (ba)k, , p b a < k, k < p. ,

1, k + 1, 2k + 1, . . . , (p 1) k + 1

p. p

p, 1, 2, . . . , p, -

, . -

x1 = x2 = = xp1 = xp,

xp1 = e, . ( -

,

, . 17.)

Cauchy -

6 . , -

:

(13.2) . 6 Z6 D3.

. G .

Cauchy x 3 y

2. x x y e, x, x2, y, xy, x2y, , , G. yx

. , yx / x yx = y. yx = xy yx = x2y. ,G = xy = Z3Z2, G = Z6 (10.2). , 4

x r y s, G

D3.

, p -

, 2p (. -

15).

13. cauchy 99

-

. 2, 3, 5 7 (11.3).

4 Z4 Klein(. 10.2), 6 Z6 D3 (13.2). - -

8 . 8,

Z8, Z4 Z2, Z2 Z2 Z2 D4. . ( )

a + bi + cj + dk, a, b, c, d i, j, k

i2 = j2 = k2 = 1, ij = ji = k. ()

H. -1,i,j,k, (), Q, .

:

1 1 i i j j k k1 1 1 i i j j k k

1 1 1 i i j j k ki i i 1 1 k k j j

i i i 1 1 k k j jj j j k k 1 1 i i

j j j k k 1 1 i ik k k j j i i 1 1

k k k j j i i 1 1

Q ( -

Z8, Z4 Z2, Z2 Z2 Z2). , 1 2, D4, D4 2.

(13.3) . 8

: Z8, Z4 Z2, Z2 Z2 Z2,D4, Q.

. G .

G 8, G = Z8. , , G 4. , x, G 4, y G{x}. x x y G

100

e, x, x2, x3, y, xy, x2y, x3y.

yx / x (), yx = y ( yx = y x = e) yx = x2y ( yx = x2y x = y1x2y, , , x2 = y1x2yy1x2y = e). yx {xy, x3y} ., y 2 4. y2 / x y (y / x) y2 / {x, x3} ( y 8). , y 4, y2 = x2. :

(i) yx = xy y2 = e, G

x (1, 0), y (0, 1)

G = Z4 Z2.(ii) yx = x3y y2 = e, ( )

x r, y s G D4.

(iii) yx = xy y2 = x2, G , xy1 2

x (1, 0), xy1 (0, 1) G Z4 Z2.

(iv) , yx = x3y y2 = x2, x i, y j G Q.

G{e} 2; G .

x, y, z G{e}, xy = z. H = {e, x, y, xy} Z2 Z2 , K = z , HK = G H K = {e}. ,

G = H K = Z2 Z2 Z2 (10.4).

13.1 R, Cauchy, .

13. cauchy 101

13.2 p1, p2, . . . , ps ,

p1p2 ps .13.3 (13.2) 6

. -

e, x, x2, y, xy, x2y.

xy. Z6, D3 .

13.4 10 Z10 D5.

13.5 G 4n+ 2. Cauchy,

Cayley 6.6 G

2n+ 1.

13.6 Q ( )

.

13.7

q = a+ bi+ cj + dk, q = a + bi+ cj + dk,

:

q + q = (a+ a) + (b+ b) i+ (c+ c) j + (d+ d) k,

q q = (aa bb cc dd) + (ab + ba + cd dc) i

+(ac bd + ca + db) j + (ad + bc cb + da) k. H - H{0} ( ) . ,

a+ bi+ cj + dk (a, b, c, d)

H R4.

13.8 q = a+ bi+ cj + dk

q = a bi cj dk.

, q

q q =

a2 + b2 + c2 + d2.

102

1 H{0}. - () S3, H R4.

13.9

a+ bi+ cj + dk [

a+ bi c+ di

c+ di a bi]

S3 SU2.

13.10 SU2

Q S3. S3 - C.

13.11 H, bi+cj+dk, .

q (bi+ cj + dk) q1

q H.13.12 x = (x1, x2, x3) R3, q(x)

x1i+ x2j + x3k. x,y R3,

q(x y) = x y+q (x) q (y) .

14

12 ,

. . -

x y G, x y

gxg1 = y g G. -.

.

g G,

G G, x gxg1,

, g.

( ,

g1. ,

G,

g(xy)g1 = (gxg1)(gyg1)

x, y G). ,

.

(14.1) . G x

G,

gxg1 = x, g, g G.

104

, x G

{x}, x G.

(14.2) . G D6,

4. D6

e, r, r2, r3, r4, r5,

s, rs, r2s, r3s, r4s, r5s,

r6 = e, s2 = e, sr = r5s.

r, ra,

1 a 5, grag1 g D6. g = e g r, ra. g = s ( s = s1),

sras = r6as2 = r6a.

, g = rbs, 1 b 5, (rbs)ra(rbs)1

= rb (sras) r6b

= rb(r6a

)r6b

= r6a.

, ra {ra, r6a}.

rbsrb = rbrbs = r2b1s

rb (rs) rb = rb+1rbs = r2b+1s.

, rbs s

r2bs rs r2b1s. , s, r2s, r4s

. rs, r3s r5s. ,

D6

{e}, {r, r5}, {r2, r4}, {r3},

{s, r2s, r4s}, {rs, r3s, r5s}.

14. 105

14.1

14.2.

(14.3) . Sn

, ,

2-, 3- ... -

, Sn , - () -

, -

.

1. g Sn, -

-- . gg1 = ,

, ,

, , -

, , ()

. , , ,

Sn.

.

= (6 7) (2 5 3 9) (1 4) , = (1 2) (3 8) (5 4 6 7)

S9 -

4-.

(2 5 3 9) (6 7) (1 4) (8)

g(5 4 6 7) (1 2) (3 8) (9)

g = (1 3 6) (2 5 4 8 9 7) . ,

gg1 (1) = g (6)

= g (7)

= 2 = (1) ..

g . ..

(2 5 3 9) (1 4) (6 7) (8) , g = (2 5 4) (3 6) (7 8 9) .

.

,

= 12 t

106

Sn

. g Sn

gg1 = g (12 t) g1=

(g1g

1) (g2g

1) (gtg1) .

i k, i = (a1 a2 . . . ak) ,

gig1(g (a1)) = gi (a1) = g (a2)

gig1(g (a2)) = gi (a2) = g (a3)

...

gig1(g (ak)) = gi (ak) = g (a1) .

, m / {g (a1) , g (a2) , . . . , g (ak)}, i g1(m)

gig1(m) = gg1(m) = m.

, gig1 = (g (a1) g (a2) . . . g (ak)),

i. g1g1, g1g

1,. . . , gkg1

, gg1

.

(14.4) .

S4

{},

{(1 2) , (1 3) , (1 4) , (2 3) , (2 4) , (3 4)},

{(1 2 3) , (1 3 2) , (1 4 2) , (1 2 4) , (1 3 4) , (1 4 3) , (2 4 3) , (2 3 4)},

{(1 2 3 4) , (1 4 3 2) , (1 2 4 3) , (1 3 4 2) , (1 3 2 4) , (1 4 2 3)},

{(1 2) (3 4) , (1 3) (2 4) , (1 4) (2 3)}.

, , A4;

. , A4 , g S4, gg1 = , g

14. 107

. , g (1 2 3) g1 = (1 3 2) , (g (1) g (2) g (3)) =

(1 3 2) , g (2 3) , (1 3) (1 2) .

g A4.

A4

{},

{(1 2 3) , (1 4 2) , (1 3 4) , (2 4 3)},

{(1 3 2) , (1 2 4) , (1 4 3) , (2 3 4)},

{(1 2) (3 4) , (1 3) (2 4) , (1 4) (2 3)}.

. -

A4 -

. ,

23 , ,

,

( ).

() ,

,

( ).

3-. -

, ( )

, -

( ) .

(14.5) . G O2

A =

[cos sin sin cos

], B =

[cos sin

sin cos].

A

( ), B -

, 2

x. ,

.

ABA1 = ABA = B+2,

108

B... . ,

AAA1 = A

BAB1 = BAB = A,

-

{A, A}. , O2 :

{I},{A, A}, 0 < < ,{A},{B | 0 < 2}.

.

.

Z(G) = {x G | xg = gx, g, g G}

G ,

G.

(14.6) . G G -

, .

. x, y Z (G) g G,

gxy1 = xgy1 ( x Z (G) )= x(yg1)1

= x(g1y)1 ( y Z (G) )= xy1g.

xy1 Z (G). e Z (G) , Z (G) G G (5.4). ,

xg = gx gxg1 = x,

x Z(G) x

{x}.

14. 109

(14.7) .

.

(14.8) . n 3, Sn {}. (14.3).

(14.9) . (14.2),

D6 {e, r3}. 14.10 Dn,

n n

.

(14.10) . GLn ( )

(. 14.11).

14.1 D5.

14.2 Dn -

n

n .

14.3 : G G . G G.

14.4 S6 -

.

, g S6,

g (1 2 3) (4 5 6) g1 = (5 3 1) (2 6 4) .

, (1 2 3) (4 5 6) (5 3 1) (2 6 4) A6,

(1 2 3 4 5)(67 8) (4 3 7 8 6)(21 5) A8.

14.5 3-

A5. , 5- A5,

A5.

14.6 S8

(1 2)(34 5)(6 7 8);

110

14.7 Q

. S3;

14.8 6.10 Sn

{} n 3.14.9 A3 , Z(A3) = A3.

Z(An) = {} n > 3.14.10 Dn

14.2. :

Z(Dn) =

{ {e}, n ,{e, r n2 }, n .

14.11

nn 1 . -

GLn(R).

14.12 On SOn. Un eiIn, R In n n .

15

-

.

H G G H

G.

, -

. X Y

G,

XY = {xy | x X, y Y }.

(15.1) . H G,

H G

.

. H G

,

(xH) (yH) = xyH (*)

x, y G. , G, -

eH = H ,

x1H