Nombre d`endomorphismes nilpotents sur un corps fini

Fq
EGL(E)
n≥1Fq
nMn(Fq) (qn−1)(qn−q). . . (qn−qn−2)
f
n
x y x f
f
f
n
Xn
SLn(C)
GLn(Fq) Mn(Fq)
n
N0=
0. . . . . . . . . 0
1 0
0 1 0
0
0. . . 0 1 0
N n Nn= 0 Nn−16= 0
xker Nn−1{1, x, N(x), . . . , Nn−1(x)}
(Fq)nN
|O(x)|=|G|/|Gx|
N0GLn(Fq)N0
N n N N0
f f

x6∈ ker Nn−1{1, x, N(x), . . . , Nn−1(x)}
y y =PaiNi(x)x N
P(N) = PaiNi
C N y =C(x)
P C(x) = P(N)(x)C=P(N)
y C(y) = P(N)(y)y=Q(N)(x)Q
C(y) = (C◦Q(N))(x)=(Q(N)◦C)(x)
C N N
· · · =Q(N)(C(x)) = Q(N)(P(N)(x)) = P(N)(Q(N)(x)) = P(N)(y).
C=P(N) = a0+a1N+· · · +adNdN
Fq[X]→Mn(Fq)X N
N Xn
Fq[N]⊂Mn(Fq)NFq[X]/XnnFq
C=a0+a1N+· · · +an−1Nn−1a0= 0 C
a06= 0 C a0Id
C
N
a06= 0 (q−1)qn−1
|GLn(Fq)|
(q−1)qn−1=(qn−1)(qn−q). . . (qn−qn−1) (q−1)qn−1= (qn−1)(qn−q). . . (qn−qn−2).