Fiche résumée du cours d`algèbre 1 1 Groupes 1.1

(G, .)G
H G E H
E < E >
1G1G0
f:G→G0H G
f(H)G0H0G0
f−1(H0)G Im(f)Ker(f)
G0G
G H G x ∼
dy⇔
x−1.y ∈H x ∼
gy⇔x.y−1∈H
G/H H\G
a.H H.a a ∈G
G H |H| |G|
|G|=|H|.|G/H|
G E Π : G→E
Π
G H Π : G→G/H
g7→ g.H
G/H
Π
∀g∈G g.H.g−1=H
∀g∈G g.H. =H.g G/H =H\G
ϕ:G∈G0G0
H=Ker(ϕ)
{1}G

f:G→G0
e
f:G/Ker(f)→G0f=e f◦Π
Π : G→G/Ker(f)
e
f:G/Ker(f)→Im(f)
⇒
⇔
(x, y)∈G2x y
[x;y] = xyx−1y−1[x;y]=1⇔x y
D(G)
D(G)
G/D(G)H
G(H  G)G/H D(G)⊂H
Gab =G/D(G)G
G
G1
f1
→G2
f2
→G3...→Gn
fn
→Gn+1
∀i∈ {1..n}, Im(fi) = Ker(fi+1)
1→Hi →GΠ →N→1
H i(H) G N
G/H N H
G×X→X
(g, x)7→ g.x
∀x∈X1G.x =x
∀g g0∈G∀x∈X(g.g0).x =g.(g0.x)
G→X
ϕ:G→(S(X),◦)S(x)
X→X ϕ(g) : x7→ g.x
G X x ∈X
x O(x) = {g.x g ∈G}
X G
XΩ
ω∈Ωω xω∈ω
x G
stabG(x) = {g∈G|g.x =x}
stabG(x)G
G X x ∈X
Π : G/stabG(x)→O(x)
g.stabG(x)7→ g.x
|O(x)|=|G/stabG(x)| |G|
G
x X 1G∩
x∈XstabG(x) = {1G}G→S(X)
∀x∈X stabG(x) = {1G}

G→G G
n G →Sn
n Sn
G X R ={xωω∈Ω}
X
|X|=X
x∈R
|G/stabG(x)|
pkk∈N
|G|=pα.r p p∧r= 1
G pα
G pnn≥1
Z G
G p p2G
∀p|G| ∃ G G
G n G
GLn(Z
nZ
H G S G a ∈G
a.S.a−1∩H H
|G|=pα.m m ∧p= 1
H⊂G G
H
G
k k |m k ≡1 [p]
H N N H
ϕ:N→Aut(H)n.h =ϕ(n)(h)
H×N
(h, n)×(h0, n0)=(h(n.h0), nn0)=(hϕ(n)(h0), nn0)
H N ϕ
HoϕN H oN
N H
G=HoN
(∗) : 1 →Hi →GΠ →N→1

i(h) = (1, h) Π(h, n) = n i H
H=i(H)
(∗)S:
N→GΠ◦S=IdNSΠ
S
N=S(N)G
H  G H ∩N={1G}={(1,1)}H.N ={h.n |h∈H n ∈
N}=G N  G
H H N N N →H
n.h =n.h.n−1
G
H N H  G H ∩N={1}H.N =G
G=HoN
N H n.h =nhn−1NoG
1→Hi →GΠ →N→1
S:N→G, Π◦S=IdNN
S(N)G G =i(H)oS(N)'HoN
n.h =nhn−1⇔S(n).i(H) = S(n).i(h).S(n)−1
(Z
nZ)∗
n∈Ns∈Z
s∧n= 1
s(Z
nZ,+)
s∈(Z
nZ)∗
ϕ(n) = card(( Z
nZ)∗) = card{s∈ {1..n} | s∧n= 1}
p n ∈N∗ϕ(pn) = pn−1(p−1)
Aut(Z
nZ,+) '(Z nZ)∗
Z
nZ→QZ
pαi
iZn=
r
Q
i=1
pαi
i
(Z
nZ)∗'Y(Z pαi
iZ)∗
ϕ(n) =
r
Y
i=1
pαi−1
i(pi−1) = n
r
Y
i=1
(1 −1
pi
)
n∧m= 1 ϕ(nm) = ϕ(n).ϕ(m)
(Z
pZ)∗
Z
p−1Z∃a∈(Z nZ)∗(Z nZ)∗={1, a, a2...ap−1}
Kg⊂K K∗

P(X)K
d d K
∀n∈Nn=P
d|n
ϕ(d)
p > 2α∈N∗Z
pαZ
pα−1(p−1)
(Z
4Z)∗'Z 2Zα≥3 ( Z 2αZ)∗'Z 2Z×Z 2α−2Z
1 + p pα−1(Z
pαZ)∗
∀k∈N∗(1 + p)pk= 1 + λk.pk+1 λk∧p= 1
SnAn
σ Sn
< σ > {1..n}
σ supp(σ) = {a∈ {1..n} |
σ(a)6=a}
(a1..ak)
(a1..ak)=(a1a2)(a2a3)..(ak−1ak)
Sn
σ∈Snσ=τ1..τr=τ0
1..τ0
sτiτ0
i
s≡r[2]
ε:Sn→ {−1,1} ' Z
2Z
σ7→ (−1)r
An=Ker(ε)
SnSn(a1..ak)σ∈Sn
σ(a1..ak)σ−1= (σ(a1), σ(a2)..σ(ak))
(b1..bk) (a1..ak)
σ σ.(a1..ak) = σ(a1..ak)σ−1= (b1..bk)
σ∈Sntype(σ) = (en(σ), ...e2(σ)) ei(σ)
σ
σ
∀n∈N, n 6= 4 An
V44
A4S4V4
S4
n≥5SnH={1}
H=AnH=Sn