Groupe symétrique

ES(E)E E ={1, . . . , n}
SnA(E)AnS(E)
Sn1
n>1σSnH:= hσi ⊂ Sn
σ H {1, . . . , n}
12345678
36528471.
σ
σ2σ
n
n>3Sn
k6n k Sn
Z/k ×Sn−k
(12)(34) S4
{(i, j)}16i6=j6nSn
{(1,2),(1,3),...,(1, n)}
{(1,2),(2,3),...,(n−1, n)}
n(1,2, . . . , n) (1,2) Sn
n>3 3 AnSn
n>5An
AnSn
AnSn
γ= (γ1, . . . , γp)σSn
σγσ−1Sn
S4S5S6
An{1, . . . , n}(n−2)
n>5 3 Ann64
σ∈AnConjSnSnConjAnAn
|ConjSn|
|ConjAn|∈ {1,2}
H, K G H [K:K∩H]6[G:H]
|ConjSn|
|ConjAn|= 2 σ
1
σ1σ2σ∈Sn
σ1σ2Anσ∈An

n>1A:= k[X1, . . . , Xn]n
Sn
Xi1
1. . . Xin
ni1,...in>0
Sn→Z/2
{1,2,3,4}
S4→S3
S4∼
=VoS3,
V= (Z/2Z)×(Z/2Z)
nSn→Sn−1
n>5HSnAn
[Sn:H]n[Sn:H] = n
H∼
=Sn−1
SnSn/H
TI
T
ITI
S4
I∼
=S4.
S4∼
=Id´et
−−→ Z/2
T
C⊂R30ICC
ICO(3) I+
C:= IC∩SO(3)
IC4C
ϕ:IC→S4
ker ϕ={± id}
ϕ
ϕ×d´et : IC→S4×Z/2
ϕI+
CS4
T1T2C
CIT1IT2
IT1=IT2ICS4
I+
C
G G
ϕ:G →S(G)
ϕA(G)
S3S4