107:sous-groupes finis de O(2,R)

O(2, R)SO(3, R)
E(.|.)
O(2, R)O(E)
O(E)cos(θ)−sin(θ)//sin(θ)cos(θ)θ
1 0//0−1
O(E)SO(E)
θ θ0θ+θ0
SO(E) 2kπ/n τ r
τ0u=ττ0τuτ−1=u−1
O−(E) 2
n
2π/n
n O n
P O 2π/n
n
n0
Dnn
2kπ/n Dnn
G O(E)
G⊂SO(E)G2kπ/n G
n Z/nZ

G6⊂ So(E)G Z/2Z
Id
GL(E)O(E)
E
GL(E)
DnZ/nZ oϕZ/2Z
ϕ(0) = (k7→ 0) ϕ(1) = (k7→ −k)
n= 3 n= 4 2p p
6S3'D3
8D4H82p p
Dp
r s < r, s >=Dn
r rkk/kn
K K =< a, b > n b
2 (ab)2= 1 b2= 1 K'Dn
Z(Dn)Id −Id n D(Dn) = r2n
n/2
rks rk
n n
Id, s Id, s, −s, −Id
rk
rk, r−k
G X k
fix(g) := {x∈X|g.x =x}k= 1/|G|P|fix(g)|
1/n Pd|nϕ(d)mn/d

SO(3, R)
SO(3, R)SO(3)
u
Z(SO(E)) = Id D(SO(E)) = SO(E)
D x
D B = (x, b B)b B D B
D
D100//0cos(θ)−sin(θ)//0sin(θ)cos(θ)
θ
P P
S A F
pF = 2A=qS
S−A+F= 2
S= 4, A = 6, F = 4,(p, q) = (3,3)
S= 8, A = 12, F = 6,(p, q) = (4,3)
S= 6, A = 12, F = 8,(p, q) = (3,4)
S= 20, A = 30, F = 12,(p, q) = (5,3)
S= 12, A = 30, F = 20,(p, q) = (3,5)
0
P Is+(P)SO(E)P

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
G X G
X ω X |X|=P
x∈ω
|G|
Stab(x)
u U
G SO(3, R)
n X G
X2(n−1)
G X g.x =g(x)
SO(3, R)
GL(3, R)
2
1/24 ∗(|C|6+ 6 ∗ |C|3+ 8 ∗ |C|2+ 6 ∗ |C|3+ 3 ∗ |C|4)C
1/g P|C|γ(g)γ(g)