Groupes et Géométrie

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12 D12,A4
S3oS2∼
=Z/3ZoZ/4Z12 2 2
2 3
15
ρi:S2→Aut(Z/15Z,+)
Aut(Z/15Z,+) ∼
=(Z/15Z,+,·)×∼ =F×
3×F× 5A×A
G30 n2(G)n3(G)n5(G) 2 3 5 G
G2 3 5 n3(G)6= 1 G20
3n5(G)6= 1 G24 5 n3(G)=1 n5(G)=1
n3(G) = n5(G) = 1
n2(G)n2(Gi)
Gi=Z/15Z oρiS230 30
pq p2q p q q 6≡ 1 mod p
pq
p2q
G n np(G)6= 1 p n
56
A5n2(A5), n3(A5)n5(A5)
A5A515 30
A56 10 12 A5
O(2) O(3)
O(2) ∼
=SO(2) oS2O(2)
S4R4
S4→O(4) (1,1,1,1)⊥∼
=R3
S4→O(3) A4→SO(3)
HRnsHH
sH(x) = x−2< x, xH>
< xH, xH>xH
xHH⊥
sH2sH1H1∩H2
(H1∩H2)⊥∼
=RxH1⊕RxH2RxH1RxH2
GO(n)
AG={H⊂Rn|sH∈G}.
H1, H2∈ AGsH1(H2)∈ AGx∈T
H∈AG
H g(x) = x∀g∈G
AGG n T
H∈AG
H= 0
sH2sH1sH1, sH2∈G
Snn−1
ASnRn−1n= 3 Sn⊂O(n)
(i j)Sn(ei−ej)⊥Rn
(1,...,1)⊥∼
=Rn−1Rn−1
H1, . . . , Hk+1 ∈ AG
Hk+1 ⊃H1∩ · · · ∩ HksHk+1 ∈< sH1, . . . , sHk>⊂G.
Snn−1 (1 2),...,(n−1n)