Récapitulatif sur les structures

∗

(G, ∗)G∗G
∗
∗
G
∗
(G, ∗) (G, ∗) (H, ∗
H)
H G ∗;
∗
H∗H;
(H, ∗
H)
H
(G, ∗)H G H (G, ∗)
H
H∗
H
(G, ∗)
(G, ∗)
(G1,∗
1) (G2,∗ 2) (G1,∗ 1)
(G2,∗
2)ϕ G1G2x y G
ϕ(x∗
1y) = ϕ(x)∗ 2ϕ(y).
ϕ(G1,∗
1) (G2,∗ 2)
ϕ(eG1) = eG2.
x G1ϕ(x−1)=(ϕ(x))−1.
H1(G1,∗
1)ϕ(H1) (G2,∗ 2)
H2(G2,∗
2)ϕ−1(H2) (G1,∗ 1)
ϕ−1({eG2}) (G1,∗
1) ; ϕ
Ker(ϕ)
ϕ(G1,∗
1) (G2,∗ 2)ϕ
Ker(ϕ) = {eG1}

(A, +,×)A+×G
(A, +)
×
×
×+
×
(A, +,×) (A, +,×) (B, +
B
,×
B)
B A +×;
1AB;
+
B×
B+×B;
(B, +
B
,×
B)
B
(A, +,×)B A (A, +,×)
1A
B+
B
B×
(A1,+
1
,×
1) (A2,+
2
,×
2)ϕ
A1A2
ϕ(1A1) = 1A2
x y A1ϕ(x+
1
y) = ϕ(x) +
2
ϕ(y)
x y A1ϕ(x×
1
y) = ϕ(x)×
2
ϕ(y)
ϕ A1A2ϕ
ϕ(A1,+
1
,×
1) (A2,+
2
,×
2)
ϕ(A1,+
1) (A2,+ 2)
x1A1ϕ(x−1
1) = (ϕ(x1))−1
(A1,+
1
,×
1)ϕ
(A2,+
2
,×
2)ϕ

(A, +,×)
(A, +,×)
(A, +,×)
a b A a ×b= 0Aa= 0Ab= 0A
(K, +,×)
(K, +,×)
(K, +,×)
K0K
(K, +,×) (K, +,×)
(K, +,×)k K k (K, +,×)
1.1K∈k, k
2. k +,
3. k ,
4. k ×,
5. k
(K1,+
1
,×
1) (K2,+
2
,×
2)ϕ
K1K2
ϕ(1A1) = 1A2ϕ
x y K1ϕ(x+
1
y) = ϕ(x) +
2
ϕ(y)
x y K1ϕ(x×
1
y) = ϕ(x)×
2
ϕ(y)
ϕ ϕ
ϕ(K1,+
1
,×
1) (K2,+
2
,×
2)
x1K1ϕ(x−1
1) = (ϕ(x1))−1
(K1,+
1
,×
1)ϕ
(K2,+
2
,×
2)ϕ
(A, +,×) (Z,+,×) (K[X],+,×)
(A, +,×)=0A
(=0,+) (A, +)
x0=0a A a ×x0∈ =0

(K,+,×) (K=R,C, . . . ) (E,+
E
,·)E
+
E
E·K E K ×E→
E; (α, x)7→ α·x,(E,+
E)
αK x y E
α·(x+
E
y) = (α·x) +
E(α·y).
x E α β K
(α+β)·x= (α·x) +
E(β·x).
x E α β K
(α×β)·x=α·(β·x).
x E 1·x=x.
+
E+· · + E
(E,+, .)K(E,+,·) (H,+
H
,·
H,)
HE+·(H+H⊂H,K·H⊂H) ;
+
H
H+·
HK H ·;
(H,+
H
,·
H,)K
H
(E,+, .)K H E H (E,+, .)
H
H
K(E,+,·)K(F,+,·)
f E F (x,y)∈E2(α, β)∈K2f(α·x+β·y) =
α·f(x) + β·f(y).
f(E,+,·) (F,+,·)
f(0E) = 0F
H(E,+,·)f(H) (F,+,·)
H0(F,+,·)f−1(H0) (E,+,·)
f−1({0F}) (E,+,·) ; f
Ker(f)fKer(f) = {0E}