Groupe des isométries du cube

Γ = ABCDA0B0C0D0E3ABCD
−−→
AA0=−−→ BB0=−−→ CC0=−−→ DD0
G={f∈ A(E3)/ f(Γ) = Γ}
G Iso E3
A GAG
H3G H
3
G
f∈G σf
[AC0],[BD0],[CA0],[DB0]ϕ:f7→ σf
GS4
(ϕ)G
G+=G∩Iso+(E3)G
S4
f∈GE3
Γsff7→ sfG
S8(A, −−→ AB, −−→ AD, −−→ AA0)
fPABCD
ΓP
f(Γ) = Γ P
Γ Γ f(A, −−→ AB, −−→ AD, −−→ AA0)
f(A)f
E3
f∈GAR= (A, −−→ AB, −−→ AD, −−→ AA0)
A−−→ AB, −−→ AD, −−→ AA0
R0fE3f(R) = R0f
ΓGA
3! = 6 6 GAGAId
3 (−−→ AA0,−→ AC)
r, r2(AC0)2π
3
4π
3f∈G
O
O
G E
[AB]A B
A
A E
8 = Card(E) = |G|
|GA|=|G| 6|G|= 48

H3E1 3
E H H {Id}
E8 3 5 2
3 2 H
A C0H⊂GA'S3
H3GA{Id, r, r3}3G
8±2π
34 4
3|G|= 4824×3 3
241 3 4 16
f∈G f ◦r(AC0),2π
3◦f−13G
Γ2π
3f(AC0)f
Γ
f−1f σf
[AC0],[BD0]O
([CA0],[DB0]
S4ϕ:f7→ σf
S4ϕ(G) = S4ϕ
ϕ G/ (ϕ)S4[G: (ϕ)] = |S4|= 24 |(ϕ)|= 2 (ϕ)
Id sO(ϕ) = {Id, sO}
f Z(G)G f ◦r(AC0),2π
3◦f−1=r(AC0),2π 3f
(AC0)
Z(G)⊂(ϕ)SO∈Z(G) (O, −→ i , −→ j , −→ k)
O sO−IdE3GL3(R)
Z(G) = {Id, sO}= (ϕ)
Iso+Iso G+=G∩Iso+
G G∩Iso−=G−=sOG+sO∈G−s2
O=Id
G+sG
O+|G|
2= 24 (ϕ)∩G+={Id}sO∈G−
ϕ G+G+S4|ϕ(G+)|= 24 = |S4|
ϕ(G+) = S4G+S4