Upload
phillip-short
View
32
Download
2
Embed Size (px)
Citation preview
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: [email protected]
: , e-mail: [email protected]
:
, e-mail: [email protected]
:
: e-mail: [email protected]
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: [email protected]
: 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 ..)
, - - -
,
- - -
, ,
,
--
. (, ,
. -
,
-
.)
. -
.
,
( , ) -
. , 28 , -
,
, :
()
,
() 300 ( ) -
,
()
,
() .
:
(i)
(. 8), (. 12), -
(. 15), (. 19),
xi
(. 20) , Nielsen Schreier (. 28),
(ii) ,
(iii) , -
.
. :
() . -
-
( ). -
. (). (A B), (A B) (AB) A B, A B, . -
.
. , ,
9. .
() .
, .. , ,
, =, , , , , , , , () -
, ,
( ).
() 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. ,
, , , , , -
, . . . , , -
, . ( ,
, , ..,
. . . 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 , - -
-
.
. , , ,
( , )
,
( -
),
( .. -
-
, -
). , ,
, -
.
,
,
(
) .
30. / / -
-
. ( -
, , ).
( ) -
( ) 2 ()
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
; T
, , .
1.1 . , L,
,
M ,
.
1.1
2
L
, 23 4
3, , -
.
-
,
( ). , ,
23( , 4
3) (-
), T
43( , 2
3). , , M,
. (
).
(4 2) + 3 = 11 . , T
, .
.
, ,
.
(. 1.2).
1.2
1. 3
(. 1.3).
1.3
, (
3, 23, , 4
3 5
3)
. , -
, -
,
.
, -
. ,
,
. -
( k6, 1 k 12, -
) .
,
, -
.
. 6
(, , )
. -
4
. ,
, , ,
, .
, ,
, . -
, .
, ()
, -
. ( ,
, , , 3
56, 7
6, -
56
3).
. . -
T , 1.4,
.
1.4
r 23 L (
) s M.
r s, N , -
2 . , s
1. 5
r, 2
4, .
-
s r. r s
,
, r s
() T .
.
, ,
. , -
. ( )
, -
6. ( ; -
.)
,
-
,
, -
. ,
, -
, T .
T -
.
u v, v u, -
, , T
, uv. ( uv - vu-
, fg f g g f). , -
e, .
e u, u
e, ,
u. ,
ue = u = eu,
u T. , u ,
u1,
u1u = e = uu1.
6
u1
u u, -
. ( , r rr, ,
r , ). , -
u, v w,
w uv vw u.
, :
(uv)w = u(vw)
( u, v w - -
T ).
T , -
, .
1.1 ,
, -
.
1.2 .
1.3 1.4 -
srs 4,
rsrr
1 2 3 4.
1.4 -
r s.
1.5 1.4,
r1 = rr, s1 = s, (rs)1
= srr
(sr)1
= rrs.
1.6
, -
. -
, . , -
.
1. 7
1.7 q
( )
. ,
uq, u ,
.
1.8 ( -
) .
1.9 1.2
. .
1.10 () , -
(. 8.1).
.
2
-
.
G ,
(x, y) G G xy, . (
)
G.
G, G,
:
(a) ,
(xy)z = x(yz)
( ' ) x, y, z G.
(b) e G, ( ) ,
xe = x = ex, x, x G.
2. 9
(c) x G x1,
G
x1x = e = xx1.
; ,
,
. .
.
, , -
. , -
(x, y) -
x + y. ,
x, y, z
(x+ y) + z = x+ (y + z),
0 ( )
x (, , -) x.
.
, ,
, -
.
-
. , -
(CH4) ,
( ) -
. , ()
(C6H6) ,
, ( ) -
. (
,
, ,
, . 2.1). -
.
10
-
.
x, y -
y, x. , xy,
yx, .
2.1
2. 11
, , -
Lorentz,
[coshu sinhu
sinhu coshu
](*)
. -
coshu sinhu
, x = coshu, y = sinhu
x2 y2 = 1.
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)
].
. -
( ) -
,
[cosh0 sinh 0
sinh 0 cosh0
].
(*)
[cosh (u) sinh (u)sinh (u) cosh (u)
]
. , ,
, .
, , ,
()
,
, .
, . ,
, ,
. ,
,
. ,
12
(.
2.4).
-
. ( )
,
, -
, .
,
.
(2.1) . .
. e e .
ee = e, e , ee = e, e .
e = e.
(2.2) . -
.
. y z -
x.
y = ey ( e )
= (zx)y ( z x)
= z(xy) ( )
= ze ( y x)
= z ( e ).
, x .
, .
, -
.
. : ,
- - -
.
2. 13
2.1
1.2.
2.2
.
2.3 22 ( ) -
;
(i)
[a b
b c
] ac = b2.
(ii)
[a b
c a
] a2 = bc.
(iii)
[a b
0 c
] ac = 0.
(iv) , ,
.
2.4 f . f
f1 .
-
.
2.5 , -
, .
-
.
2.6 -
P
.
P ;
;
2.7 x y G. G
w z, wx = y xz = y,
w z (
).
14
2.8 x y ,
(xy)1 = y1x1.
3
. -
-
.
( )
.
Z
Q
R
C
, ( ) , x ( ) x.
, ( -
) .
16
Q{0}
R{0}
Q>0 R>0
{1,1} {1}
C{0}
1 C
{1,1, i,i}, i {1,i}
, 1 ( ) , 1x -
x.
,
.
.
. , , - -
( -
),
,
.
,
. , -
x x 0 = 1. . -
- ,
. -
- ; ,
.. x 2 x = 1 12 / Z. , 2 Z.
C
,
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