Groupes et Géométrie

◦
sgn(σ)σ∈Snsgn(σ) = (−1)ν(σ)
ν(σ)σ(i, j)∈N21≤i < j ≤n σ(i)> σ(j)
+1 −1
σ, τ ∈Snν(σ) + ν(τ)−ν(στ )sgn :Sn→
{±1}
sgn(σ) = sgn(σ−1)sgn(τστ −1) = sgn(σ)σ, τ ∈Sn
φ:G→H G
sgn :Sn→ {±1}AnAn
SnSnn≥2
S3
S3σ∈S3
S3
S3
S4σ∈S4{1,2,3,4}
σ x, y σ σk(x) = y k ≥0
σ σ σ ∈S4
σ
σ σ
S4
σ∈S4
σ σ S4
K4S4
K4(Z/2Z×Z/2Z,+)
S4→S3K4{1,2,3}
K∗
4K4σ∈S4K∗ 4→K∗ 4:τ7→ στσ−1
K∗
4
Gln(R)n×n
σ∈Snφσ∈Gln(R)ieme
σ(i)eme (e1, . . . , en)Rn
φσ(ei) = eσ(i)i= 1, . . . , n
φ:Sn→Gln(R) : σ7→ φσOn(R)
A∈Gln(R)
Rn t A·A=In
det : Gln(R)→(R∗,·)
sgn(σ) = det(φσ)
2
An→SOn(R)
D2nO2(R)
ρ=cos(2π
n)−sin(2π n)
sin(2π
n) cos(2π n), σ =1 0
0−1.
D2(Z/2Z,+) D4(Z/2Z×Z/2Z,+)
n≥3{1,2, . . . , n} {ρk(1
0), k = 1, . . . , n}
D2n→Sn
H N G N G
N∩H H HN =G[G:N] [H:N∩H]
G= O2(R)N= SO2(R)H=D2nN∩H
(Z/nZ,+) n≥3D2n2n