1 Groupes opérant : rappels

(G, ·) 1 1G
ES(E)E.
G E
G×E→E
(g, x)7→ g·x
∀x∈E, 1·x=x
∀(g, g′, x)∈G2×E, g ·(g′·x) = (gg′)·x
G E.
G×E→E
(g, x)7→ x·g
∀x∈E, x ·1 = x
∀(g, g′, x)∈G2×E, (x·g)·g′=x·(gg′)
g∈G,
φ(g) : E→E
x7→ g·x
E E, φ (g)∈ S (E).1·x=x
x∈E, φ (1) = IdEg·(g−1·x) = (gg−1)·x= 1 ·x=x g−1·(g·x) = x
φ(g)◦φ(g−1) = φ(g−1)◦φ(g) = IdE, φ (g)
φ(g−1).
g·(g′·x) = (gg′)·x, g, g′, x, φ (gg′) = φ(g)◦φ(g′),
φ(G, ·) (S(E),◦).
φ G E.
φ G E
g·x=φ(g) (x)

G
(g, h)∈G×G7→ g·h=gh
G
(g, h)∈G×G7→ g·h=ghg−1
(G, ·) (S(G),◦)
Ad (g) : G→G
h7→ ghg−1
Ad Int (G)G.
Int (G)G/Z (G), Z (G)
G.
Ad : G→ S (G)g∈G
Ad (g) = IdG, g ∈G ghg−1=h h ∈G,
gh =hg h ∈G. Ad Z(G)G.
Im (Ad) = Int (G), G/Z (G) = G/ ker (Ad) Im (Ad) = Int (G).
G H
(g, h)∈G×H7→ g·h=ghg−1∈H
S(E)E
(σ, x)∈ S (E)×E7→ σ·x=σ(x)∈E
G E. x ∈E,
E
G·x={g·x|g∈G}
x G.
x∼y g ∈G y =g·x
E x = 1 ·x y =g·x
x=g−1·y y =g·x, z =h·y z = (hg)·x
x∈E x.
E.
S(E)E x ∈E,
S(E)·x={σ(x)|σ∈ S (E)}=E
y∈E y =τ(x), τ τ = (x, y)y̸=x, τ =Id y =x

G
∀h∈G, G ·h=ghg−1|g∈G
G G ·h={h}h∈G.
H G, G
(h, g)∈H×G7→ h·g=gh−1
1·g=g1 = g h1·(h2·g) = gh−1
2h−1 1=g(h1h2)−1= (h1h2)·g g ∈G
g H
H·g={h·g|h∈H}=gh−1|h∈H
={gk |k∈H}=gH
G/H H.
G
(h, g)∈H×G7→ h·g=hg
H
H·g={hg |h∈H}=Hg
E σ ∈ S (E)
E H =⟨σ⟩E
(σr, x)∈H×E7→ σr·x=σr(x)
x∈E
H·x={γ·x|γ∈H}={σr(x)|r∈Z}
H·x σ. Orbσ(x)
σ∈ S (E)
σ E σ
σ∈ S (E)\{IdE}
σ= (x1, x2,· · · , xr)σ2
r= 2p σ p ≥1. p = 1, σ2=IdE
p≥2,
Orbσ2(x1) = {x1, x3,· · · , x2p−1}Orbσ2(x2) = {x2, x4,· · · , x2p}
σ2

G E
∀(x, y)∈E2,∃g∈G|y=g·x
∀(x, y)∈E2,∃!g∈G|y=g·x
G E
φ:g∈G7→ (φ(g) : x7→ g·x)∈ S (E)
(g∈G∀x∈E, g ·x=x)⇔(g= 1)
GS(E).
G G
S(G).
g∈G, g ·h=gh =h h ∈G
g= 1, φ
n≥1,On(R)Rn
∀(A, x)∈ On(R)×Rn, A ·x=A(x)
0.
x∈Rn,
On(R)·x={A(x)|A∈ On(R)}
y∈ On(R)·x, A ∈ On(R)y=A(x)∥y∥=∥A(x)∥=∥x∥,
On(R)·x⊂S(0,∥x∥).
y∈S(0,∥x∥)x̸= 0, y ̸= 0
B= (ei)1≤i≤nB′= (e′
i)1≤i≤nRne1=1
∥x∥x e′
1=1
∥y∥y.
B B′y=∥y∥e′
1=∥x∥A(e1) = A(x),
y∈ On(R)·x. On(R)·x=S(0,∥x∥)x̸= 0.
x= 0,On(R)·x={0}=S(0,∥x∥).
n, m K
G=GLn(K)×GLm(K)E=Mn,m (K)n
m
∀(P, Q)∈G, ∀A∈E, (P, Q)·A=P AQ−1
Or={A∈E|rg (A) = r}
r0 min (n, m).

A∈ Mn(K)r
Ar=Ir0
0 0 .
r= 0, A = 0 = A0. r ≥1, u ∈ L (Kn)
AKn, H ker (u)Kn,B1= (ei)1≤i≤r
HB2ker (u), u (B1) = (u(e1))1≤i≤rKn
r
k=1
λku(ek) = 0,
r
k=1
λkek∈H∩ker (u) = {0}λk
B={u(e1),· · · , u (er), fr+1,· · · , fn}KnuB1∪ B2
B
Or={A∈E|rg (A) = r}
=A∈E| ∃ (P, Q)∈G|A=P IrQ−1=G·Ir
E=
min(n,m)
r=0
Or=
min(n,m)
r=0
G·Ir
G E. x ∈X,
G
Gx={g∈G|g·x=x}
x G.
GxG
E E
G=S(E)σ·x=σ(x), x ∈E
S(E\ {x}). σ ∈Gx, σ′σ E \ {x},
GxS(E\ {x}).
(G, ·)E. x ∈E
φx:G/Gx→G·x
g=gGx7→ g·x
G
card (G·x) = [G:Gx] = card (G)
card (Gx)
card (G·x) card (G)
g, h G x ∈E, g ·x=h·x
(h−1g)·x=x, h−1g∈Gxg=h G/Gx,
φx
G/GxG·x. G
card (G·x) = card (G/Gx) = card (G)
card (Gx)