Wyklad6 Permutacje

f : A A A = {1, 2, ..., n}
Sn
|Sn| = n!
Sn
Sn
� " Sn
� = (a1, a2, ..., at)
ńł
a1 , i = at
�ł
�(i) = ai+1 , i = aj '" j < t
ół
i , i " {a1, a2, ..., at}
(i, i + 1)
f g
{x | f(x) = x} )" {x | g(x) = x} = "

� " Sn
1 2 3 4 5 6 7 8 9 10
= (1, 3, 5, 7)(2, 8)(6, 4, 10, 9)
3 8 5 10 7 4 1 2 6 9
1 2 3 4 5 6 7 8 9 10
= (1, 8, 2)(3, 7, 9)(4, 6)
8 1 7 6 5 4 9 2 3 10
� " Sn
� " Sn
1 , � jest parzysta
� (-1)� =
-1 , � jest nieparzysta
n
Det : Kn K
n
A = ai,jei,j
i,j=1
Det(A) = (-1)�a1,�(1)a2,� (2)...an,�(n)
�"Sn
� = (a1, a2, ..., at) (-1)� =
(-1)t
f = c1 � c2 � ... � cs ti
1
(-1)� = (-1)t +t2+...+ts
f
n > 0 fn = id f o(f)
� = (a1, a2, ..., at) o(�) = t
f = c1 � c2 � ... � cs
ti o(�) = NW W (t1, t2, ..., ts)

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