Groupes - Page Personnelle de Jérôme Von Buhren

E= [0,1] ? E
∀(x, y)∈E2, x ? y =x+y−xy.
? E
E
P(E)
E=F([0,1],[0,1]) ? E
∀x∈[0,1],(f ? g)(x) = f(x)g(1 −x).
? E
E=F([0,1],[0,1]) ? E
∀x∈[0,1],(f ? g)(x) = max(f(x), g(x)).
? E
G=] −1,1[ ? G
∀(x, y)∈G2, x ? y =x+y
1 + xy .
(G, ?)
G=R2? G
∀((x, y),(x0, y0)) ∈G2,(x, y)?(x0, y0)=(x+x0, yex0+y0e−x).
(G, ?)
f:R→R
G H K G
H∪K G H ⊂K K ⊂H
G H K G
HK G HK =KH
X G
G X G
(m, n)∈N2
mZ∩nZ= ppcm(m, n)ZmZ+nZ= pgcd(m, n)Z.

G
g∈G
Hg={gn∈G|n∈Z}
G
G
(G, +)
H={g∈G| ∃n∈N∗, n ·g= 0}
G
{x+y√3∈R|(x, y)∈Z2, x2−3y2= 1}
R∗
E=R\{0,1}
f, g :E→E, f :x7→ 1
x, g :x7→ 1−x
(Q,+) Q
H G
G H G
G
G
G n
Glog2(n)
(i) (Z,+) (Z,+),(ii) (Q,+) (Q,+),
(iii) (Q,+) (Z,+),(iv) (Q,+) (Q∗,×).
n∈Nn>2
(Sn,◦) (C∗,×)
ϕ:G→C∗
X
g∈G
ϕ(g).
G f :G→G
x7→ x3
(xy)2=y2x2(xy)4=x4y4x, y ∈G
G
(C∗,×)S1×R∗
+

H(R,+)
a= inf{x∈H|x > 0}.
a > 0a∈H H =aZ
a= 0 HR
∀x∈R,∀ε > 0,∃h∈H|x−h|< ε.
GAut(G)
G G
Z(G) = {g∈G| ∀h∈G, gh =hg}.
Z(G)G
h∈G σh:G→G g 7→ hgh−1
Aut(G)
σ:G→Aut(G)g7→ σg
Ker(σ) = Z(G)
(C∗,×)
G x, y ∈G m
n m ∧n= 1 xy mn
G a ∈G m k ∈Z
akm/(m∧k)
G n m ∈Nm∧n= 1
y∈G x ∈G y =xm
G H Card(G)∧Card(H)=1
G H
G g2=e g ∈G
G
G G 2
2p p > 2
p
p
G={z∈C| ∃k∈N, zpk= 1}.
G(C∗,×)
G
G
G pq p∧q= 1
A={g∈G|gp=e}B={g∈G|gq=e}.
A B G
A∩B
G A ×B
G λ :G→G
e
G
ϕ:G→G, g 7→ g−1λ(g)
λ(g) = g−1G

0?
E1x∈E\{1}
1+(x−1)−1
E
∅
P(E)∅
A E A
? f = 1
? f = 0
x∈H\K y ∈K\H x ·y∈H
y=x−1·(x·y)∈H x ·y /∈K x ·y /∈H∪K
H∪K G
x∈X x−1∈X
X n ∈N∗xn=e x−1=
xn−1∈X
HgHgZ
G G Hg
G
K K =G
H⊂K H 6=G a ∈G\H x ∈H a ·x
H a =x·(a·x)−1∈H a ∈G\H⊂K
a·x∈G\H⊂K x =a−1·(a·x)∈K
f:Z→Z, x 7→ n·x
f:Q→Q, x 7→ a·x
f:Q→Z, x 7→ 0
f:Q→Q∗, x 7→ 1f(1) n
n∈N∗
ϕ a ∈G ϕ(a)6= 1
X
g∈G
ϕ(g) = X
g∈G
ϕ(a·g) = ϕ(a)·X
g∈G
ϕ(g),
(yx)3=y3x3(xy)2=y2x2
(xy)4= ((xy)2)2= (y2x2)2=x4y4
y3x3= (yx)3=x3y3f G
G=[
g∈G\{e}{g, λ(g)}∪{e},
G
ϕ ϕ(g) = ϕ(h)λ(hg−1) = hg−1
hg−1=e h =g
g=h−1λ(h)λ(g) = λ(h−1λ(h)) = λ(h)−1h=g−1λ
G