4. Étude de PSL2(K) Exercice 1. Soit K un corps. On se propose de

P SL2(K)
K2(K)
1t
0 1 1 0 t1M=a b c d∈2(K)
c6= 0 s, t, u ∈K M =1s
0 11 0 t11u 0 1.
b
b=c= 0
K2(K)
B=a b
0a−1;a6= 0, b ∈K
M=a b
c d∈2(K)c6= 0 P1, P2∈B
M=P1wP2w=0 1
−1 0
2(K)B BwB
GB G 2(K)
B∩wBw−12(K)
∩g∈2(K)gBg−12(K)
g=1 0
1 1
K
2(K)D(2(K))
a, t ∈K∗a26= 1 a0
0a−1 1t 0 1
K N C2(K)
H=π−1(N)C2(K)N π :2(K)→
2(K)
HB B 2(K)
HB=BN
HB=2(K)UB
1a
0 1

u∈U wuw−1∈HU
HU =2(K)
2(K)/H U/(U∩H)
H D(2(K))
2(K)
(Z/3Z)2
2(Z/3Z)S4
2(Z/3Z)A4
SL2(Z/3Z)0−1
1 1 
SL2(Z/3Z)S4
SL2(Z/3Z)NoH N  SL2(Z/3Z)
H
K n D
n(K)H M ∈n(K)
H D oW W
K H
Dn(K)g∈n(K)gDg−1=D
K=F2