Fiche 1 - Groupe opérant sur un ensemble

(G, ∗)G∗
G
∗:G×G−→ G
(a, b)7−→ a∗b
∀a, b, c ∈G(a∗b)∗c=a∗(b∗c)
∃e∈G / ∀x∈G, x ∗e=x=e∗x
∀x∈G∃x−1∈G/x∗x−1=x−1∗x=e
∀a, b ∈G, a ∗b=b∗a
[n] = {1,2, . . . , n} Snσ: [n]→[n]
◦ Sn
σ∈ Sn
µ1 2 · · · n−1n
σ(1) σ(2) · · · σ(n−1) σ(n)¶
i6=j tij
tij (i) = j
tij (j) = i
∀k∈[n]\{i, j}, tij (k) = k
(Sn) = n!
S3
◦1c c0t12 t13 t23
1 1 c c0t12 t13 t23
c c c01t13 t23 t12
c0c01c t23 t12 t13
t12 t12 t23 t13 1c0c
t13 t13 t12 t23 c1c0
t23 t23 t13 t12 c0c1
1 = µ1 2 3
1 2 3 ¶
c=µ123
231¶
c0=µ1 2 3
3 1 2 ¶
S3
A3={1, c, c0} S3
nSn
i.e. [n]
tij i6=j∈[n]

1≤i1, i2, . . . , ik≤n k Sn
(i1i2. . . ik) = µi1i2· · · ik−1ik
i2i3· · · iki1¶
2kSn
k
GZnZn∈N
G
Unn n ∈N∗
C∗
G e X
ϕ:G×X−→ X
(g, x)7−→ ϕg(x)
G X
∀x∈X, ϕe(x) = x
∀g, g0∈G, ∀x∈X, ϕg∗g0(x) = ϕg¡ϕ0
g(x)¢
S(X)X
ϕ¯
¯
¯
¯
G→ S(X)
g7→ ϕg
∼X
x∼x0⇐⇒ ∃g∈G / x0=ϕg(x)
x∈X
G·x={g(x), g ∈G}
Gx={g∈G / ϕg(x) = x}
x
XG={x∈X / ∀g∈G, ϕg(x) = x}
X G
∀x, x0∈G, ∃g∈G/x0=ϕg(x)
i.e.
∀x∈X, Gx={e}

GL2(R) 2 ×2R2
ϕ:
GL2(R)×R2−→ R2
µA, µx
y¶¶ 7−→ Aµx y¶
{(0,0)}
R2\{(0,0)}
(1,0)
n
G GL2(R)
G=½µ a0
0b¶, a, b ∈R∗+¾
ϕR2
G(x,y)
G0=½µ α0
0α¶, α ∈R∗+¾
O+
2(R) = ½µ cos θ−sin θ
sin θcos θ¶, θ ∈R¾
(r, 0) r∈R+
H=½µ cos hω sin hω
sin hω cos hω ¶¾
V⊂R3(R3,+)
V×R3−→ R3
(−→ v , −→ x)7−→ −→ x+−→ v
V∀−→ x∈R3V−→
x={−→
0}

GL2(C)⊂ M2(C) 2 ×2
ϕ:GL2(C)× M2(C)−→ M2(C)
(P, A)7−→ P AP −1
z, z0∈Cµz0
0z0¶z6=z0
A{z}µz0
0z¶
GL2(C)×GL2(C)∗
(P, Q)∗(P0, Q0) = (P P 0, QQ0)
ϕ:(GL2(C)×GL2(C)) × M2(C)−→ M2(C)
((P, Q), A)7−→ P AQ−1
Oµµ 0 0
0 0 ¶¶ , O µµ 1 0 0 0 ¶¶ , O µµ 1 0 0 1 ¶¶
A
R3<−→ x , −→ y >=x1y1+x2y2+x3y3
S2={−→ x , < −→ x , −→ x >= 1} ⊂ R3(0,0,0)
O+
2(R)×S2−→ S2
<(0,0,1) >
VRs∈ L(V)
s◦s=IdV
V V +V−s
V=R2dim(V+) = dim(V−) =
1
R2P={x1=
(1,0), x2, x3}G
P
f f(P) = P

f, f0R2f|P=f0
|Pf=f0
GS3
S33
ϕ:S3×R3−→ R3
(σ, (x1, x2, x3)) 7−→ (xσ−1(1), xσ−1(2), xσ−1(3))
ϕ
G→ S(X)
g7→ ϕg
VR3
R3
(e1, e2, e3)R3R={ei−ej,1≤i6=j≤3}
S3S3R
W x1+x2+x3= 0 R3=V⊕W V W
S3R3
U V
R