101: groupe opérant sur un ensemble. Applications

(G, .) 1 X
G X ψ :G×X→X
∀x∈X, ψ(1, x) = x,
∀(g, g0)∈G2,∀x∈X, ψ(gg0, x) = ψ(g, ψ(g0, x)).
X G G X
ϕ:G→Bij(X)
ϕ(g) := x7→ ψ(g, x)ϕ
G/Kerϕ
X
g.a := ga
n
Sn
GLn(k)Mn,m(k)G, M 7→ GM
G/H g.(aH) :=
gaH H
G H G G
G G/H G
g.a := gag−1
g.H := gHg−1
GLn(k)Mn(k)On(k)
Sn(R)

G E E G
F f.x =f(x)
GL(E)GLn(k)E∗k∗
Sn{1, . . . , n}
Dn
n PnDnPn
∆
D2∆σ.d =σ(d)
GL(E)E
FGL(E)FP GL(E)F
G
3X g
G X g, x 7→ g(x)
G X R
xRy ⇔ ∃g∈G|y=g.x R
X G x
Orb(x)Orb(x) = {g.x|g∈G}
x y X g
g y =g.x
a1, . . . , anX b1, . . . , bng∈G∀i∈ {1, . . . , n}
g.ai=bi
D2∆
Orb(H)
H
GLn(k)Mn(k)
A
On(R)
Ann−2{1, . . . , n}
GL(E)
SO(E)
GL(E)n+ 1
n

DnPn
σ∈ Sn< σ >
{1, . . . , n}
σ
E
EEEE
E n E
E
G X x
Stab(x) := {g∈G|g.x =x}G
Stab(g.x) = gStab(x)g−1
H
x x
CG(x)CG(x) := {g∈X|gx =xg}
H
H H NG(H)NG(H)
x
x GL(E E Ker(f−Id)
F GL(E)
GL(E)F F
E
∆Delta
u
ϕ Ker ϕ =T
x∈X
Stab(x)
X X0G G
f:X→X0∀g∈G, ∀x∈X, f (g.x) = g.(f(x)) G
f:X→X0f f−1G
f
x∈X G G/Stab(x)
G Orb(x)
G X ω
X|X|=P
x∈ω
|G|
Stab(x)
G pα∀i6α
pipZ/pZ
G n p
nZ/pZGpp
G

G pαm p ∧m= 1
pα
m1p
k
k∗
GL(2,F2) = SL(2,F2) = P SL(2,F2)' S3.
P GL(2,F3)' S4;P SL(2,F3)' A4.
P GL(2,F3) = P GL(2,F3)' A5.
P GL(2,F5)' S5;P SL(2,F5)' A5.
G X g ∈G
fix(g)g G X
1|GX
g∈G
|fix(G)|
4 3 2
76 4 6
4 7575
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)
2
Dn/2A4A5S5
npn p
N H ϕ
N×H(n, h)(n0, h0)=(nϕ(h)n0, hh0)
N×H N H N oϕH
N N oϕH H
NoϕH⇔ϕ⇔
N G H G NH
G H N N G NH =G N ∩H= 1
G H oK
Z/ZD4
H8

Aut(S6)
SnAnoZ2
DnZ/nZ o Z2
8
Aut(S6)S6oZ2
p p2
p Zp
NA(1) = N A(3) = . . . =NA(59)
p2Z/p2Z Z/pZ2
Z/pZ×Z/qZ
Z/pZ×Z/qZ
Z/pZoZ/qZ
S3Z/3Z