Groupes Exercice 1. (NEW) Exercice 2. (NEW) Exercice 3. (NEW

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G A B G A+B > G
AB =G
r∈Qr > 0rQ
Q2Q[√r]
Q[√r]
r6=sQ[√r]Q[√s]Q2
Q2
g5=e
ghg−1=h2e
h
2
(xy)2= (yx)2
x2y=yx2
xyx−1=x−1yx
G, H G ×H
∀g, g0∈G, ∀h, h0∈H, (g, h)·(g0, h0)=(gg0, hh0).
·G×H
a b c
a a a a
b a b b
c a b c
a b c
a b c a
b c a b
c a b c
a b c d
a a b c d
b b a d c
c d c b a
d c d a b
G·
∀a, b ∈G, ∃x, y ∈G a =x·b=b·y.

(G, ·)
G E φ :G−→ E
? E
∀x, y ∈E, x ? y =φφ−1(x)φ−1(y).
? G E
x, y ∈Rx?y=xp1 + y2+y√1 + x2
p1+(x?y)2=√1 + x2p1 + y2+xy
(R, ?)
(R,+) (R, ?)
x, y ∈Rx?y=3
px3+y3(R, ?) (R,+)
E∗
E
G H G
H G
G Z(G) = {a∈G∀b∈G, ab =ba}G
a∈G:C(a) = {b∈G ab =ba}a
Z(G)C(a)G
∆
E G =P(E)
(G, ∆)
a∈E
φa:G−→ Z/2Z
X7−→ {0a /∈X
1a∈X
φa
E={1, . . . , n}
Φ : G−→ Z/2Zn
X7−→ (φ1(X), . . . , φn(X))
Φ
G H, K, L G H ⊂K H ∩L=K∩L
H+L=K+L H =K
(HK)

G H, K G φ:H×K−→ G
(h, k)7−→ hk
φ
z∈HK z =h0k0h0∈H k0∈K
z φ (h0t, t−1k0)t∈H∩K
(HK) (H∩K) = (H) (K)
HK G ⇐⇒ HK ⊂KH ⇐⇒ HK =KH
G(G)φ:G−→ G
(G)◦
(Z)
a∈Gφa:G−→ G
x7−→ axa−1
φa∈(G)a7→ φa
n∈N∗G=Z/nZk∈Zd=k∧n
˙
k G
˙
k˙ d G
G
G f :G−→ G0
H G f−1(f(H)) = H+f
H0G0f(f−1(H0)) = H0∩f
G a n G0a0∈G0
φ:G−→ G0φ(a) = a0a0
n
Z/nZ>Z Z/nZ>C∗Z/nZ>Z/pZ
Q
(Q,+) (Q,+)
(Q,+) (Z,+)
(Q,+) (Q∗,×)
C∗
(C∗,×)
G G0f G G0a∈G a
f(a)
a, b ∈G a bab−1
a, b ∈G ab ba
ab
a, b G 
a α
b β
α∧β= 1
ab =ba.

ab
G a ∈G np n ∧p= 1
b, c ∈G b n c p a =bc =cb
G G
G
n∈Nn≥3ω= exp 2iπ
n
fk:C−→ C
z7−→ ωkz
gk:C−→ C
z7−→ ωkz
(0 ≤k < n)
G={f0, . . . , fn−1, g0, . . . , gn−1}
a > 0AkaωkG
A0. . . An−1
G
G f1g0f1◦g0=g0◦f−1
1
H ρ σ 
ρ n
σ2
ρσ =σρ−1.
G H
G
x x26=e
G∀x∈G, ∃!y∈G x =y2
G∀x∈G, x2=e
G(xy)(xy)
H G x ∈G\H K H ∪ {x}
K= 2 H
G
a b
E=R\ {0,1}
f:E−→ E
x7−→ 1
x
g:E−→ E
x7−→ 1−x
f g f g ◦
α∈C∗β∈Cfα,β :C−→ C
z7−→ αz +β

fα,β ◦
α, β fα,β
G H G G
∀x, y ∈G, x ∼y⇐⇒ ∃ h∈H x =hy.
∼e
a∈G˙a H
H G
H, K G x, y ∈G
x∼y⇐⇒ ∃ h∈H, ∃k∈K y =hxk.
x∈G Gx={(h, k)∈H×K hxk−1=x}Gx
H×K
H K
(H) (K)
ab (a, b)=1
G n =ab a ∧b= 1
A={x∈G xa=e}B={x∈G xb=e}
A B G
A∩B={e}AB =G
Q
HQH
Q
(Q,+) Q+∗,×)
(Q,+) (Q+∗,×)√2
C∗
p G z ∈Cn∈N
zpn= 1
G
H G G
z0G\H H z0
f:G−→ G0G
(f)×(f) = (G)
p
G pkp k ∈N∗Z G
G(Z)≡
0 mod p