Groupes : exemples et généralités (TD1)

G G/Z(G)G
5 6
Z/nZ,S3'PGL2(F2),A3,S4,A4'PGL2(F3).
G e e n ≥1gn= 1
g G
2
e e
G
G
G
G H G
K H G
K
H K G
F(F, +)
G H
G/H K/H K
G H
(G/H)/(K/H)'G/K
K G K ∩H K
HK ={hk |h∈H, k ∈K}G KH
H HK
K/(K∩H)'(KH)/H

G n
GSn
GAn+2
GGLn(k)k
GOn−1(R)
G n ≥2
G{a1, . . . , ak}i≥2ai
{a1, . . . , ai−1}
|Aut G| ≤ nlog2n
k n ≥1
1 1 1
1//µn(k)
//k×
det //k×n
//1
1//SLn(k)//
GLn(k)det //
k×//
1
1//PSLn(k)//
PGLn(k)det //
k×/k×n
//1
1 1 1
0→A→B→C→D→0
E
0→A→B→E→0 0 →E→C→D→0
Z/nZ
G G
G
p α ≥1 1 + p(Z/pαZ)×
(Z/pαZ)×'Z/pα−1(p−1)Z
(Z/2αZ)×α≥1
n≥1 (Z/nZ)×