TD n°1 : Groupes

Z Z/nZn≥1
Z Z/nZ
d≥1nZ/nZd
Z Z/nZ
ϕ ϕ(m)=#{d∈N∗|d≤mpgcd(d, m) = 1}
n=X
d|n
ϕ(d)
G n d n G
d G
K G K∗
G
Q/Z(Q/Z,+)
n n ≥1
(Q,+)
(Q∗,×)
p≥3k≥1ak
p(1 + p)pk= 1 + akpk+1 (Z/pkZ)∗'Z/pk−1(p−1)Z
(Z/2kZ)∗
n≥1 (Z/nZ)∗
GAut(G)G
2
NCG n g ∈G gn∈N
N
GAut(G)G
a∈G ϕa:g7→ aga−1G
Inn(G)GInn(G)
Aut(G)
ϕ:a7→ ϕa

X=2 0
0 1
Y=1 1
0 1G GL2(R)
n∈ZX−nY XnHnGL2(R)
n∈ZHnHn+1
G
G
G x y
G
xy yx
x y G
ppcm x y
G H
G G G H
θ:G→S(G/H)g7→ {xH 7→ gxH}
Ker(θ)G H
|G|[G:H]! G
p|G|
[G:H] = p H G
G H G K
G
KH ={kh :k∈K, h ∈H}G
HK H
G
KH/H ∼
=K/(K∩H)
K →KH KH/H
G H CG1→H→G→G/H →1
1→H→G→K→1
H G K ∼
=G/H
0→2Z→Z→Z/2Z→0