105: Groupe symétrique. Applications

E
E E Bij(E)E
{1,...,|E|} S|E|Sn
n>2
|Sn|=n!
σ∈ Snσ=1. . . n
σ(1) . . . σ(n)
S3123
123 123 133 123 213 123 231 123 321
123
312
Snn>3
σ∈ Snp
∃a1, . . . , ap∈ {1, . . . , n}p|σ(ai) = ai+1(modp)
∀x6=a1, . . . , ap, σ(x) = x
p(a1. . . ap){a1, . . . , ap}
2
τ∈ Snσ:= (a1. . . ap)p τστ−1= (σ(1) . . . σ(p))
pSn
SnGLn(k)σ
δi, σ(i)
Gln(Fp)
Fpε(u) = det(u)/p

Sn
Sn
Sn
n−1
(1 2),...,(1 n)Sn
(1 2),...,(n−1n) (1 2),...,((n−1) n)Sn
xk=xk+ 1 ⇒
f(x1, . . . , xp)=0
Sn
Snn
(1 2) (1 2 . . . n)Sn
12345678
25431687= (1 2 5)(3 4)(7 8)
1σ
σ∈ Sni(σ)
σ(i, j)i < j σ(i)> sigma(j)
ε:Sn→({−1,1})
σ7→ (−1)i(σ)
ε(σ) = Q
i<j
σ(j)−σ(i)
j−i
Sn{−1,1}ε−1
a ε(a)=(−1)p+1
Kerε
SnAnn!/2
2Sn
n= 2 An={Id}n>3An
(1 2 3),...,((n−2) (n−1) n)Sn
Sn
σ∈ Snσ
Sn
n1k1+ 2k2+
. . . +nkn

σ
σ σ Q
16i6n
ki!iki
kii σ
An
n6= 2,4An
S4A4
n>5D(An) = Ann>2D(Sn) = An
SnSnAn{Id}
HSnn H Sn−1
n>5Sn
n>3Z(Sn)
Sn
SnSnSn
SnAut(Sn)Sn
Snx7→ σxσ−1σ∈ Sn
SnInt(Sn)
n6= 6 Aut(Sn)'Int(Sn)(' Sn)
Int(S6) 2 Aut(S6)
σ∈SnP(Xσ(1), . . . , Xsigma(n))
Sp=PXp
iΣp=PXi1Xip
p p
w(P)
PMσMsigma MσM
σPσ
G K[X1, . . . , Xn]
1 Φ G(X1,...Xn) = Φ(Σ1(X),...,Σn(X))
ΦGΦPnαn=p
F
G
K F
X3
1+X3 2+X3 3=σ3 1−3σ1σ2+ 3σ3
CP=PaiXixi
Σp= (−1)pap/a0

Mn(C)
T r(Mk)A k tr(Mk)=0
GLn(C)
2
0
E1,0,−1
P
E n p f
Epk
f(x1, . . . , xn)=0 i6=j xi=xj
E n BE
1B
BdetBv1, . . . , vn(aij )
detB(v1, . . . , vn)n×n
Rn
E n f ∈ LEdetB(f(B))
Bf det(f)
B
n E
E E
A
A
r
p f
∀σSn, f(xσ(1), . . . , xσ(n)) = ε(σ)f(x1, . . . , xn)
f E g(x1, . . . , xn) := P
σ∈Sn
ε(σ)f(x1, . . . , xn)
g f
f1, . . . , fp(x1, . . . , xn)7→ f1(x1),...fn(xn)
f1, . . . , fpf1∧. . . ∧fp
p
p>n
p6n(e1, . . . , en)E e∗
i1∧. . .∧e∗ ipi1< i2. . . < ip
pn//p
n∈N∗URnp U
URn

n
Sn
SnGLn(Z/pZ
Z/6Z,
S3.
P SL2(F2)' S3
P SL2(F3)' A4
P SL2(F4)' A5
P SL2(F5)' A5
Is+(T)'A4Is+(C) = Is+(O)'S4Is+(D) = Is+(I) = A5
Is(T)'S4Is(C) = Is(O)'S4×Z/2Z Is(D) = Is(I) = A5×Z/2Z