1. M f: M M → M.
, (m1, m2) m1, m2 M f((m1,m2)) M. , f((m1,m2)) f.
X , , .
, : (X, *), ,
* X
(X,*) ( ).
2. * X , (x * y) * z = x * (y * z) x, y, z M; , x * y = y * x.
3. X ( ) *, * x = x * = x
X. ' , , , ' = ' * = . , (X,*) .
4. x' X *, x' * x = x * x' = e X. x , , ,
= x'*x x = *= (x'*x)*x=x'*(x*x)=x'*e=x'. , (X,*) .
5.. , ( ), x, y , , :
1. , .. x, y, z
(x * y) * z = x * (y * z)
2. e
3. -1 .
( ), .
, , | |< ∞. . | |.
.
, .
:
I. ,
.
II. .
III. .
IV.
1. (M, ∙) ∈ M. * :
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x * y = xay (x, y ∈ M)
(M, *) . ,
1) : (x * y) * z = (xay) * z = xayaz
x * (y * z) = x * (yaz) = xayaz
2) : a-1
x * a-1 =xaa-1 = x
3) : x-1
x * x-1 =xax-1 = a
4) :
x * y = y * x = xay
2. n n Z/nZ ( n), n
n n. , n, 0, 1, 2, , n-1.
Z n :
0) , n 0, .. n;
1) n 1;
n - 1) (n - 1) , n n-1
○ Zn . (k) (m) . : = k (mod n),
b = m (mod n). a + b = k + m (mod n). ,
(k) ○ (m) =
Zn ○ , n. : , , ○ , .. , (0), , (k), (n - k). , .
3. n - .
n - .
n - n - . , ε n = 1 η n = 1.
(εη) n= ε n η n = 1. , . .
1. ,
2. 1, , , .
3. , n - , .
, ε n = 1, ε ε -1= 1, , ε n (ε -1) n = 1,
(ε -1) n = 1.
4. .
, n - , n.
4. n- 2n , 2. , = (101110), b = (111011), = + b = (010101). (000000), , 110010 + 110010 = 000000
5 .
:
, (.. 360). , , : 120, 240 , , , . , , ( , , , , ). , , (. . , , ).
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, .
, 120, , 240, 240 360, . . . 240 480 = 360 + 120, . . 120.
0, 120 1 240 2, :
0 0 = 0, 0 1 = 1 0 = 1,
0 2 = 2 0 = 2, 1 1 = 2,
1 2 = 2 1 = 0, 2 2 = 1.
, . . ,
(0 1) 2 = 1 2= 0
0 (1 2) = 0 0 = 0
0,
0 = 0 =
.
, : , , , 0-1= 0, 0 0 = 0, 1-1 = a2 2-1 = a1 ( a1 a2 = 0).
, . ,
0 1 = 1 0 = 1,
0 2 = 2 0 = 2,
1 2 = 2 1 = 0
, , .
6 n -.
:
F , F , . . F .
.
, . , - , , .
, , . . , .
, , , . , , .
, . , , . . ,
: , , .
, , , (, , ) , . .., , . , .
n- :
1. .
2. .
3. , .
, . , n- .
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6 F .
:
F. , ( ), .
, , . , F , , .
F . , ; , , , ( , ). , g ( , , , g).
7 n- ( ).
:
, n- n-, ; , , n. ,
( ) , .
V. , n- n- .
n- .
, :
1) ( );
2) , . . , . . , , . , , n, .
, , 2 . , ( = 4) , n- , , S S' ( ), ( ). , . , , (. . ) , , .
, , , : , b , .
. , ; , , :
.
, .
, ( , . . ) . ,
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, . . , S S' .
, . . , -, , . ( , ), . . .
, . : , ( ) . , , .. . .
, , . , , , .
, : - 2 n, SS' , . . . SS'.
, , : - .
= 4 , , 8, ( ), 24 . , , , S S', . ( ), , , , . , , , .
, .
:
I. () .
II. () (. . ).
III. , , : . - .
IV. , , : . - .
V.
1. .
φ.
φ1 φ2 φ3 φ1+ φ2= φ3. , . .
1. : (φ1+ φ2)+ φ3= φ1+ (φ2+ φ3)
2. 0 (+ 2πn)
3. ( φ) (+2πn)
4. : φ1+ φ2 = φ2+ φ1
, , , .
2.
, , b, , . :
1. : (a + b) + c = a + (b + c)
2. : 0 .,. a + 0 = a
3. : a + (- a) = 0
4. : a + b = b + a
, .
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4. ( ), .
, . .
1. n . , , , , , , , , .
2. n, , 1, .
, , :
:
1. :
2. : ( )
3. : , 1, 0, 1.
4. :
3. n- , ½ n!
, . : , , .
, . . , -1 , , .. .