Введение в теорию Галуа. Ермолаев Ю.Б. - 14 стр.

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A
n
n > 4
A
n
n n ≥ 5
S
n
c = (1, 2, ..., k−1, k) =
(1, k)(2, k) ···(k − 1, k) ⇒ (1, ..., k) (−1)
k−1
σ =
c
1
c
2
···c
s
(−1)
k
1
+···+k
s
−s
A
n
A
n
(1, i)(2, j) = (1, 2, j)(1, 2, i); (1, i)(2, i) =
(1, 2, i)
N C A
n
, (i, j, k) ∈ N ⇒ N = A
n
(1, 2, 3) ∈ N ⇒ (1, 2)(3, k)(1, 2, 3)(1, 2)(3, k) = (1, 2, k)
NCA
n
, (i, j)(k, l) ∈ N (n > 4) ⇒ N = A
n
τ = (1, 2)(3, 4), τ
1
=
(3, 4, 5)τ(3, 5, 4) = (1, 2)(3, 5) ∈ N; ⇒ τ
1
τ
−1
= τ
1
τ = (3, 5, 4) ∈ N
τ = c
1
···c
s
(`(c
i
) ≥ `(c
i+1
)); τ = c
0
τ
0
c
0
= c
1
· c
q
, τ
0
=
c
q+1
·c
s
τ ∈ N C A
n
σ τ
0
τ
0
στ
0
= τ
0
σ τ
1
= τ
−1
(στσ
−1
) = c
−1
0
σc
0
σ
−1
∈ N
N C A
n
τ = c
1
···c
s
∈ N (`(c
i
) ≥
`(c
i+1
))
o
`(c
1
) ≥ 4 ⇒ N = A
n
c
0
= c
1
= (1, 2, ..., k)
k > 4 σ = (2, 3, 4) τ
1
= c
−1
0
σc
0
σ
−1
= (1, k, k −
1, ..., 2)(2, 3, 4) ·(1, 2, ..., k)(2, 4, 3) = (2, 4, 5) ∈ N
k = 4 τ
1
= c
−1
0
σc
0
σ
−1
= (1, 4, 3, 2)(2, 3, 4)(1, 2, 3, 4)(2, 4, 3) = (1, 2, 4) ∈
N
o
`(c
1
) = 3 ⇒ N = A
n
c
0
= c
1
c
2
= (1, 2, 3)(4, 5, 6), σ = (2, 3, 4) τ
1
= c
−1
0
σc
0
σ
−1
=
(1, 3, 2)(4, 6, 5) ·(2, 3, 4)(1, 2, 3)(4, 5, 6)(2, 4, 3) = (1, 5, 2, 4, 3) ∈ N
o
c
0
= c
1
c
2
= (1, 2, 3)(4, 5), σ = (2, 3, 4) τ
1
= c
−1
0
σc
0
σ
−1
=
(1, 3, 2)(4, 5) · (2, 3, 4)(1, 2, 3)(4, 5)(2, 4, 3) = (1, 5, 2, 4, 3) ∈ N
o
o
`(c
1
) = 2 ⇒ N = A
n
s > 2 c
0
= (1, 2)3, 4), σ = (2, 3, 4); τ
1
= (1, 2)(3, 4)(2, 3, 4)(1, 2)(3, 4)(2, 4, 3) =
(1, 3)(2, 4) n > 4
s = 2 τ = (1, 2)(3, 4) ∈ N
`(c
1
) = 1 N = 1 
                         7. Ïðîñòîòà ãðóïïû An (n > 4)

     Òåîðåìà 5. Ãðóïïà An ÷åòíûõ ïîäñòàíîâîê ñòåïåíè n ïðè n ≥ 5 ïðîñòàÿ
(íåêîììóòàòèâíàÿ).
     Ïðåäâàðèòåëüíûå çàìå÷àíèÿ.
 Ãðóïïà ïîäñòàíîâîê Sn . Âûðàæåíèå ïîäñòàíîâîê íåçàâèñèìûìè öèêëàìè.
 Ïðåäñòàâëåíèå öèêëà ïðîèçâåäåíèåì òðàíñïîçèöèé: c = (1, 2, ..., k−1, k) =
(1, k)(2, k) · · · (k − 1, k) ⇒ ÷åòíîñòü (1, ..., k) ðàâíà (−1)k−1 , ÷åòíîñòü σ =
c1 c2 · · · cs ðàâíà (−1)k +···+k −s .
                          1      s

 Îïðåäåëåíèå An  ìíîæåñòâî âñåõ ÷åòíûõ ïîäñòàíîâîê.
     Ëåììà 1. An ïîðîæäàåòñÿ 3-öèêëàìè: (1, i)(2, j) = (1, 2, j)(1, 2, i); (1, i)(2, i) =
(1, 2, i).
     Ëåììà 2. N C An , (i, j, k) ∈ N ⇒ N = An :
(1, 2, 3) ∈ N ⇒ (1, 2)(3, k)(1, 2, 3)(1, 2)(3, k) = (1, 2, k).
     Ëåììà 3. N CAn , (i, j)(k, l) ∈ N (n > 4) ⇒ N = An : τ = (1, 2)(3, 4), τ1 =
(3, 4, 5)τ (3, 5, 4) = (1, 2)(3, 5) ∈ N ; ⇒ τ1 τ −1 = τ1 τ = (3, 5, 4) ∈ N .
     Ëåììà 4. τ = c1 · · · cs (`(ci ) ≥ `(ci+1 )); τ = c0 τ0 , ãäå c0 = c1 · cq , τ0 =
cq+1 · cs . Åñëè τ ∈ N C An , òî äëÿ âñÿêîãî σ íåçàâèñÿùåãî îò τ0 (ò.å. ñòàáèëü-
íîãî íà ÷èñëàõ èç τ0 : στ0 = τ0 σ) èìååì τ1 = τ −1 (στ σ−1 ) = c−1               0 σc0 σ
                                                                                         −1
                                                                                            ∈ N.
     Äîêàçàòåëüñòâî òåîðåìû. Ïóñòü N C An è τ = c1 · · · cs ∈ N (`(ci ) ≥
`(ci+1 )).
     1o `(c1 ) ≥ 4 ⇒ N = An c0 = c1 = (1, 2, ..., k)
     a) k > 4. Ëåììà 4 äëÿ σ = (2, 3, 4) äàåò: τ1 = c−1                  0 σc0 σ
                                                                                  −1
                                                                                      = (1, k, k −
1, ..., 2)(2, 3, 4) · (1, 2, ..., k)(2, 4, 3) = (2, 4, 5) ∈ N .
     b) k = 4. τ1 = c−1    0 σc0 σ
                                     −1
                                         = (1, 4, 3, 2)(2, 3, 4)(1, 2, 3, 4)(2, 4, 3) = (1, 2, 4) ∈
N.
     2o `(c1 ) = 3 ⇒ N = An
     a) c0 = c1 c2 = (1, 2, 3)(4, 5, 6), σ = (2, 3, 4). Èìååì τ1 = c−1               0 σc0 σ
                                                                                             −1
                                                                                                  =
(1, 3, 2)(4, 6, 5) · (2, 3, 4)(1, 2, 3)(4, 5, 6)(2, 4, 3) = (1, 5, 2, 4, 3) ∈ N . Ñâåëè ê ñëó-
÷àþ 1o .
     b) c0 = c1 c2 = (1, 2, 3)(4, 5), σ = (2, 3, 4). Èìååì τ1 = c−1                  0 σc0 σ
                                                                                             −1
                                                                                                  =
(1, 3, 2)(4, 5) · (2, 3, 4)(1, 2, 3)(4, 5)(2, 4, 3) = (1, 5, 2, 4, 3) ∈ N . Îïÿòü ñâåëè ê
ñëó÷àþ 1o .
     3o `(c1 ) = 2 ⇒ N = An
     a) s > 2. c0 = (1, 2)3, 4), σ = (2, 3, 4); τ1 = (1, 2)(3, 4)(2, 3, 4)(1, 2)(3, 4)(2, 4, 3) =
(1, 3)(2, 4) (n > 4, ëåììà 3).
     b) s = 2. τ = (1, 2)(3, 4) ∈ N . íåïîñðåäñòâåííîå ïðèìåíåíèå ëåììû 3.
      ñëó÷àå `(c1 ) = 1 N = 1. 




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