TD n°3 : Théorèmes de Sylow

G H G
NG(H) = {g∈G|H=gHg−1}NG(H)G
H
H NG(H)
G
NG(H)k G |G/NG(H)|=k
H G k
G S p G npp
G NG(S)G|G|=pam m ∧p= 1
np≡1p np|m
p p Sp
5 20
S3A4
G p q
p G q G
G H G P1p H
p P G P1=P∩H
G H p G G H
p G
G N G p G
N p G
G H G S p
H G =NG(S)H
G
S p G
S p G
S G
S G S
G
H G H
H G
H G S p G H ∩S
p H SH/H p G/H

H G S p G H ∩S
p H
S p G
S0p G S0⊂NG(S)S=S0
NG(NG(S)) = NG(S)
G p
G
An3
A5
n>5 3 AnAn
SnAn
p q p < q
pq q
p q −1pq
p2q
r > q pqr r
G p H G
H G
G12 G3G
A4
G56 G
G105 G5
7
G351 G
G340 G
85
G48 G8
16
G7
G60
G>5G
H G G G/H
2G5 15
G15 2 G2
H NH(G) 5 G
G5G
A5
6100