Exercice 2. - Université de Rouen

◦
[•]
R
x∗y= ln(ex+ey)x, y ∈R.
[•]
∗R
x∗y:= 1 + xy x, y ∈R.
∗
∗
[•]
XP(X)P(X)
∆
[•]
R\{1}
x∗y=x+y−xy pour tous x, y ∈R\{1}.
∗
(R\ {1},∗)
[•]
n∈N∗nC(C∗,·)
[•]
(G, ?)G Z(G)G
G Z(G) (G, ?)
[•]
Z
ln(xy) = ln x+ ln y[•]
|zz0|=|z||z0|[•]
(xy)1
2=x1 2y1 2
ez+z0=ezez0
z+u=z+u
det(AB) = det Adet B
[•]
(G, ·) = e, a, . . . , a5,·
a, a2, . . . , a5

(G, ·) = e, a, . . . , an−1,·n n ≥2k∈
{1, . . . , n −1}akn/ (n, k)
d= (n, k)n=dn1k=dk1
(n1, k1) = 1
akn1
h∈N∗akh=e n1h
[•]
G= (Z/12Z,+)
H G 6 8
G.
9
[•]
f(Z/15Z,+) (Z/18Z,+) .
f f 1
f1
(Z/15Z,+) (Z/18Z,+)
(G, ·)
G
a, b ∈G(ab)2=a2b2
a, b ∈G(ab)−1=a−1b−1
f G G f(x) = x−1
x∈G x2=e G
U1U
(C∗,·)
E ?
a∈E daga)E E da(x) = a ? x ga(x) = x?a
? E
(E, ?)
(G, ?)G1G2(G, ?)G1∪G2
(G, ?)G1⊂G2G2⊂G1
(G, ?)e h G
hinf {n≥1 ; hn=e}

(G, ·)f: (Z/12Z)→(G, ·)
f112
(G1,+) (G2,·)f:G1→G2
x∈G1f(x)x
(G, ·)Z(G)G
G e G
Z(G) (G, ·)
x∈G x 2y∈G
yxy−16=e
yxy−1
G2
G
(G1,·) (G2,·)f:G1→G2e1
e2G1G2
f={x∈G1, f(x) = e2}
f f ={e1}
(G, ·)a∈G
fa:G→G
x7→ axa−1.
faG
fa
x, y ∈]−1,1[ 1 + xy > 0
]−1,1[
x?y:= x+y
1 + xy pour tous x, y ∈]−1,1[ .
?
(]−1,1[ , ?)
G a b ab
A G C(A) = {x∈G;∀a∈A, xa =ax}
A G C(A)G
A B G A ⊂BC(B)⊂ C(A)
X G X ⊂ C(C(X))
A G C(C(C(A))) = C(A)