101 - Groupe opérant sur un ensemble. Exemples et - IMJ-PRG

G X
G X
G×X→X; (g, x)7→ g.x
∀g, g0∈G, ∀x∈X, g.(g0.x)=(gg0).x
∀x∈X, 1.x =x
ϕ:G→S(X)
G X
G X ∀x, y ∈X, ∃g∈
G:g.x =y G X ϕ
R X xRy ⇐⇒ ∃g∈G:y=g.x
x∈X x R
x G.x
x∈X Hx
{g∈G|g.x =x}G
G X
x∈X g 7→ g.x
G/H.x G.x O
|X|=X
x∈O
|G|
|Hx|
G

G
G
p|G|
H p G H
G
H G G G/H
G p |G|
H p G H  G G G/H
Sn
Sn
[|1, n|]σ∈Sn[|1, n|]< σ >
σ
Snn−1 (i, i+
1) n−1Sn
Sn
Sn−1
Sn
nSn
Sn−1n6= 6
XG={x∈X|∀g∈G, g.x =x}
X G G p X |X| ≡
|XG|[p]

p
p2p
|G|=pαm p ∧m= 1 α∈N∗
Sp(G)p G
Sp(G)6=∅
H p G p
p
|Sp(G)| | m|Sp(G)| ≡ 1 [p]
pαq p < q
pq2p6=q
G H G H
ϕ:G→Aut(H)HoG
H×G(h, g)(h0, g0)=(h(g.h0), gg0)
G H, K G H
G H ∩K={1}HK =G G
HoK k.h =khk−1
G≈H×K
K  G
ϕ
G pq p q p < q
p6 |q−1G≈Z/pqZ
p|q−1G≈Z/pqZG≈Z/qZoZ/pZ
GlnRn{0}
Rn∗
On(R)Rn

k k∗Sn
E F
k
E
E E
Eu.M =M+u
∀M∈ E,∀u ∈E, ∃!N∈ E :N=M+u
−−→
MN =u
C R2P P ={z∈C|
Im z > 0}Γ = P Sl2(Z) = Sl2(Z)/{±I2}
ΓPa b
c d .z =az+b
cz+d
D={z∈P| |z| ≥ 1,|Re z| ≤ 1/2}
Γ
D D
D
ΓS=0−1
1 0 
T=1 1
0 1 