TD n°2 : Sous-groupes distingués, actions de groupes

SnGLn(R)Z/nZ
GL2(Z/2Z) (Z/2Z)2GL2(Z/2Z)∼S3
G n
G
Snn≥3
R C θ·z=eiθz
n≥1µnnC∗
z7→ znC∗/µn∼C∗
G N G π :G→G/N
(a)H G N π(H)G/N
H/N π
(b)π G
N G/N
(c)H G N H/N G/N
(G/N)/(H/N)∼
=G/H
(d)K G N ∩K K
NK G NK/N ∼
=K/N ∩K
p
E a Z/pZEp
k·(e1,· · · , ep)=(eω(k+1),· · · , eω(k+p)).
ω(i){1,· · · , p}i p
(a)ap∼
=amod p
(b) (a)G
p p
H={z∈C|Im z > 0}
(a)
a b
c d·z=az +b
cz +d
SL2(R)H
(b)i

p
p G p pα
(a)G
(b)G i ∈ {0,· · · , α}
pi
(c)p2
G X
g∈G f(g)X g
(a)
X
x∈X
|Stab(x)|=X
g∈G
f(g).
(b)
=1
|G|X
g∈G
f(g).
(c)G X g ∈G
G
G
(d)|X| ≥ 2
X
g∈G
f(g)2≥2|G|.
(e)G X
G0G X
|G| ≤ X
g∈G
(f(g)−1)(f(g)− |X|)≤ |X||G0|
|G0| ≥ |G|/|X|
G G D(G)
aba−1b−1, a, b ∈G
(a)D(G)G G/D(G)
G
(b)π:G→G/D(G)H
f:G→H f :G/D(G)→H
f=f◦π
G Gi, i ∈
{0,· · · , n}G
G0={1} ⊂ G1⊂ · · · ⊂ Gn=G
i∈ {0,· · · , n −1}GiGi+1 Gi+1/Gi
(a)
(b)H G G H
G/H
(c)p
(d)Dn(G)D0(G) = G Dn+1(G) = D(Dn(G)) D
G{1}