108: exemples de parties génératrices d`un groupe

G A G A
G G A < A >=G
n∈N Z/nZo(Cn(m)) = n
n∧mZ/nZ,+)
m n ∧m= 1 n
Z/nZ
ϕ(n)k6n k ∧n= 1
(Z/nZ,+)
p0p ϕ(p) = p−1
d n Z/nZ
d < Cn(n
d)>
ϕ
n=P
d|n
ϕ(d)
1
Aut(Z/nZ)'(Z/nZ)∗.
p α ∈N∗) (Z/pαZ)∗'Z/ϕ(pα)Z
(Z/2Z)∗'1α>2 (Z/2αZ)∗'Z/2Z×Z/2α−1Z
3Z/nZ
Aut(Z/nZ) (Z/nZ)∗n= 2 n= 4 n=pα
n= 2pαp α ∈N
k Xn−1n
k k
Z/dZ d|n
Kn
Z/nZ

QX−µiµi
m n
ϕ(n)
k0k ϕn
k
k0
Xn−1 = Qd|nϕd(X)
Sn
n−1
Sn
n
n6
(1 2),...,(1 n)Sn
(1 2),...,(n−1n) (1 2),...,((n−1) n)Sn
xk=xk+ 1 ⇒
f(x1, . . . , xp)=0
Sn
Snn
(1 2) (1 2 . . . n)Sn
n65An
Ann= 3 n>5A4
SnD(Sn) = D(An) = An

Dn
n n
Dn
r2π/n s s r s
Dn
n2
−Id
D4
a b
a n (ab)2=e Dn
SO3
SL(E)GL(E)
E f(x) = 0 u
H H
det(u)=1
Im(u−iD)⊂H
E/H
a H u(x) = x+f(x)a
E f(x) = 0 u
H H
1
Im(u−Id)6⊂ H
λ
x y u
v u(x) = y uv(x) = y

SL(E)GL(E)
SL(E)dim(E)>3n
n+ 1
P SL(n, k)k n >3
D(GL(n, k)) = D(SL(n, k)) = SL(n, k)n>3
n= 2 k6=F2, F3
n P SL SL
Sln
GLn(R)
GLn(R)
O(E)SO(E)
O(E)
O(E)n
n−dim {ptsfixesdeg}
n>3SO n
SO3
SO3(R)
P SOnn
SOn