103: Exemples et applications des notions de sous

g−1Hg ⊂H g
S4
GL(E)
p−cycle
I
G
SnAn
Dnn{Id, −Id}
{1,−1}
D(G)G
G/D(G)
S3
{1,−1}
H K KH HK
K HK
H
G H
K H NG(H)K
NHH KH

AnSnDnDn
G
2
xy−1∈H
xH =Hx x
H
G/H
f G G0H
G f f G/H ¯ f f◦j=¯ f
j¯ f j(Ker(f)) ¯ f
f Im(f)G/Ker(f)
H G K G K/(H∩K)'HK/H
H G K G H ⊂K G/K '(G/H)/(K/H)
U V u v u(U∩v
G0
G
Z/pZp

An
A4
SnAn
D(Sn) = D(An) = An
P SL(2,F2)' S3
P SL(2,F3)' A4
P SLn(k)
|G|=pα∗m, p pgcd(p, m) = 1
pα
pα
H K (h, k)(h0, k0) =
(hh0, kk0)
H K G HK =G
G H ×K
N H ϕ
N×H(n, h)(n0, h0) = (nϕ(h)n0, hh0)
N×H N H N oϕH
N N oϕH H
NoϕH⇔ϕ⇔
N G H G NH
G H N N G NH =G N ∩H= 1
G
Z/ZD4
H8
SnAnoZ2
DnZ/nZ o Z2
8

p p2
p Zp
NA(1) = NA(3) = . . . =NA(59)
p2Z/p2Z Z/pZ2
Z/pZ×Z/qZ
Z/pZ×Z/qZ
Z/pZoZ/qZ
S3Z/3Z
p2q
p2q
C1p q −1C2p2q−1
C3q p −1C4q p + 1 C2C1
q= 2 Zp2oZ2Z2
po1Z2Z2 po2Z2
q>3
C1Zp2oZqZ2
poZq
C2ZqoZ2
pZqo1Zp2Zqo2Zp2
C2C3ZqoZp2Zqo1Z2
pZqo2Z2 pZp2oZq
C2C4ZqoZp2Zqo1Z2
pZqo2Z2 pZ2 poZq
C3Zp2oq
C4Z2
poq
p3
p3Z/p3Z Z/pZ2×
Z/Z2Z/p3Z Z/pZ2o Z/Z2Z/pZ2o Z/Z
G G0⊃G1. . . ⊃Gm
Gi+1 GiG Gm
Gm
GiG Gi/Gi+1

G Gm
f−1(G)f−1(Gm)
p−
G H H G/H
G=G1⊃G2⊃Gr=e G =H1⊃
H2⊃⊃ Hs=e r =s
Gi/Gi+1
L K
K L
K K k Gal(K/k)
F k
E k F k =E0⊂E1. . . ⊂
Em=E Ei+1/Ei
Xn−a a ∈Ein
Xp−X−a a ∈Eip > 0
F k
K k E
DecK(P)
Gal(DecK(P), K)