104: groupes finis. Exemples et applications.

G G
Card(G)/Card(H) = Card(G/H)
d d n
A4
an= 1
k ak= 1 k n
aϕ(n)≡1[n]
nSn
2n n
Z4
Z/nZ
Z
nZn∈N Z/nZ:= {a+nZ|a∈Z}=
{a+nZ|a∈ {0, . . . , n −1}} n n
a n Cn(a)
Z/nZ
n∈N Z/nZo(Cn(m)) = n
n∧mZ/nZ,+)
m n ∧m= 1
Cn(k) (Z/nZ,∗)⇔k∧n= 1 ⇔k(Z/nZ,+)
nZ/nZ Z/nZ
d n
Z/Z

E
E E Bij(E)E
{1,...,|E|} S|E|Sn
Sn
n−1
Sn
σ∈ Sni(σ)
σ(i, j)i < j σ(i)> sigma(j)
ε:Sn→({−1,1})
σ7→ (−1)i(σ)
ε(σ) = Q
i<j
σ(j)−σ(i)
j−i
Sn{−1,1}ε−1
a ε(a)=(−1)p+1
Kerε
SnAnn!/2
2Sn
Ann= 3 n>5A4
SnD(Sn) = D(An) = An
Dn
n n
Dn
r2π/n s s r s
Dn
n2
−Id
rdd|n Dd
< r2, s > < r2, rs >
Z/2Z
Dnn D2D2
D2nId −Id R2

H8
H8Id −Id
G X ω
X|X|=P
x∈ω
|G|
Stab(x)
G pα∀i6α
pipZ/pZ
G n p
nZ/pZGpp
G
k
k∗
G X g ∈G
fix(g)g G X
1
|GX
g∈G
|fix(G)|
4 3 2
76 4 6 4
7575
2
Dn/2A4A5S5
G pαm p ∧m= 1
pα
m1p
A5

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
p2Z/p2Z Z/pZ2
Z/pZ×Z/qZ
Z/pZ×Z/qZ
Z/pZoZ/qZ
S3Z/3Z
p3Z/p3Z Z/pZ2×
Z/Z2Z/p3Z Z/pZ2o Z/Z2Z/pZ2o Z/Z
Z/ZD4
H8
Z/nZ o Z/2Z
Ano Z/2Z
p1, . . . , pn
Z/p1p2. . . pn'Z/p1Z×Z/p2Z×. . . ×Z/pnZ Z/p1p2. . . pn'
Z/p1Z×Z/p2Z×. . . ×Z/pnZp1, . . . , pn
Z/6Z'Z/2Z×Z/3Z
p1, . . . , pn
(Z/p1p2. . . pn)∗'(Z/p1Z×Z/p2Z×. . . ×Z/pnZ)∗
x≡a[n]
x≡b[m]
n∧m x0
x0+nmt t ∈Z
u n m
a+n(b−a)u
x≡10[47]
x≡5[111]
4334 + 5217t

Z/nZ
d d
n
P SL2(F2)' S3
P SL2(F3)' A4
P SL2(F4)' A5
P SL2(F5)' A5
xn=Id
F2
GLn(C)
G O2(R)
G⊂SO(E)G2kπ/n G
n Z/nZ
G6⊂ So(E)G Z/2Z
Id
G X G
X ω X |X|=P
x∈ω
|G|
Stab(x)
u U
G SO(3, R)
n X G
X2(n−1)
G X g.x =g(x)
SO(3, R)