Groupe opérant sur un ensemble

n d n
H G d
m n
f:Z→Z/mZ×Z/nZ
x7→ (xmod m;xmod n)
f mnZ
f
Z/mnZ Z/mZ×Z/nZf
f
x(mod 5)
x(mod 3)
Z/20Z×Z/15Z×Z/30Z Z/aZ×Z/bZ×Z/cZ
a|b|c
E#E
σSn
supp(σ) = {x∈ {1, ..., n}/σ(x)6=x}
G H G H
gHg−1=H g ∈G
{1}
A4
Sn
n

Sn
Sn(1,2),(1,3), ..., (1, n)
Sn(1,2),(2,3), ..., (n−1, n)
Sn(i;i+ 1) = (1,2, ..., n)i−1(1,2)(1,2, ..., n)1−i
i∈ {2; ...;n−1}
Sn(1,2, ..., n) (1,2)
Ann≥3
Ann≥5
A3A4
n
An
HAn
H
H#supp(g)≥4
σ∈An[g;σ] = gσg−1σ−1∈
H
#supp(g)=4 g= (a, b)(c, d)a, b, c, d
σ= (a, b, e)e a, b, c d
[g;σ]
#supp(g) = 5
#supp(g)≥6σ= (a, b, c)a∈supp(g), b =g(a)
c6=a, b, g(b) [g;σ]
n Dn
A0, A1, ..., An−1
Dn2n A0
Dn
r2π
ns(OA0)
Dn=< r, s >
srs−1
D3D4
E s E s ∈L(E)
s2=id
s
E= ker(s−id)⊕ker(s+id)
G A1A2A3A4A5A6A7A8
−−−→
A1A5=−−−→ A2A6=−−−→ A3A7=−−−→ A4A8O G+
G
f(O)f∈G G
O(R3)
G ϕ :G→
S4
ϕ
[A1A2]
ker ϕ={−id;id}
G
ϕ|G+
G'S4× {Id;−Id}
G

pn
p
p
G pnE
{x∈E/∀g∈G, g.x =x}
#≡#E(mod p)
Z G 6={1}p
p2
p3
M3(Z/pZ)
1a b
0 1 c
0 0 1
a, b, c ∈Z/pZ.
p p
p
G n =pr×m p r, m ∈N∗m p
G pr
p
p G p
spp p
p
m
pr=mmod p
(X+Y)nZ[X, Y ]Z/pZ[X, Y ]
Epr•:G× E → E
(g, E)7→ g.E
•GE
Σ
p
x∈ΣHxprHx
Hxpr
H p K p G G/H K
F#F=mmod p
g∈G∈gHg−1
Pp G
GP
spm
GPsp= 1
mod p
p×q p q