a a a A aa a ∗, a (a, b) A a c ∈ A. a c = a ∗ b.
aa a a a A a aa a a A×A A.
1. aa, a ∗ a a a A, a , a, b ∈ A a ∗ b a a A.
2. aa, a ∗ a a A, a a a a, b ∈ A, a∗b , a, a A, a. a a a, a a b a A a a a, a aa∗b a a a a.
. a a a a , aa a ∗ a a, a a, a a a. aa, a a a.
A, a aaa aaa ∗, aa .
a a, , , a aa, a a A, a a a aa a ∗, aaa a A.
aa a aaa a a.
a a a a . a, a, aa a , a a, a , .
aa a ∗ a A a- aa, a ∀ a, b, c ∈ A (a ∗ b) ∗ c = a ∗ (b ∗ c).
aa a ∗ a A a- a, a ∀ a, b ∈ A a ∗ b = b ∗ a.
- , aa a ∗ a A aa a a, a ∃ a, b ∈ A a ∗ b 6= b ∗ a.
3. (, , , ).
(G, ∗), :
1. ∗ G , ..
∀ a, b, c ∈ G (a ∗ b) ∗ c = a ∗ (b ∗ c).
2. G e, .. e, ∀ a ∈ G a ∗ e = e ∗ a = a.
3. a ∈ G G a′, -
a, .. a′,
|
|
a ∗ a′ = a′ ∗ a = e.
:
1.
(Z, +), (Q, +), (R, +), (C, +).
2.
(Q \ {0}, ), (R \ {0}, ), (C \ {0}, ), (Q+, ), (R+, ).
3.
(Zm×n, +), (Qm×n, +), (Rm×n, +), (Cm×n, +).
4.
(GL(n,Q), ), (GL(n,R), ), (GL(n,C), ).
(G, ∗) (), ∗ G . ′
(G, ) , :
1. G , .. ∀ a, b, c ∈ G (a b) c = a (b c).
2. G e, .. e,
∀ a ∈ G a e = e a = a.
3. a ∈ G G a−1, -
a, .. a−1,
a a−1 = a−1 a = e.
1. (G, ) -
. a G a−1 ∈ G.
2. (G, ) , .. a, b, c ∈ G :
ab = ac ⇒ b = c,
ba = ca ⇒ b = c.
3. (G, )
a, b ∈ G ax = b ya = b G , x = a−1b, y = ba−1.
11.4. (G, ) , :
1) G ;
2) a, b ∈ G ax = b ya = b
G.
5. a b G
(ab)−1 = b−1a−1,
.. , , , .
6. a1, a2,..., an G
(a1a2... an)−1 = a−1n a−1n−1... a−11
1. (G, ) , a ∈ G.
k, l
akal = ak+l, (1)
(ak)l = akl.
a G , an = e n ∈ N. n ∈ N an = e, a. a ord(a) | a a G , an e n ∈ N.
(G, ) . a b G c ∈ G, a = bc.
4. (, , , )
(G, ∗) H G, ∗, G, .
.
(G, ∗) G, {e} G.
G . Z (R, +).
.
1. H (G, )
|
|
e G.
2. a H
(G, ) H a−1 (.. G ).
3. .
4. H
(G, ). a b H H G, .. a/b
1 ( ).
H G
G , :
∀ a, b ∈ H ab ∈ H (.. H ), (1)
∀ a ∈ H a−1 ∈ H. (2)
2 ( ).
H G
G ,
∀ a, b ∈ H ab−1 ∈ H.
11′ ( ).
H G +
G , :
∀ a, b ∈ H a + b ∈ H (.. H +), (1)
∀ a ∈ H − a ∈ H.
2′ ( ).
H G +
G ,
∀ a, b ∈ H a + (−b) ∈ H.
3. < a > G.
(G, ) , a ∈ G. < a >= {ak | k ∈ Z} G G, a.
(G, +) , a ∈ G. < a >= {ka | k ∈ Z} G G, a.
(K,+, ) + ,
1. (K, +) , ..
) + K , .. ∀ a, b, c ∈ K (a + b) + c = a + (b + c);
) K 0 +, .. 0, ∀ a ∈ K a + 0 = 0 + a = a;
) a ∈ K K −a, .. −a, a + (−a) = (−a) + a = 0;
) + K , .. ∀ a, b ∈ K a + b = b + a.
2. K , .. ∀ a, b, c ∈ K (ab)c = a(bc).
3. K +, .. ∀ a, b, c ∈ K ((a + b)c = ac + bc ∧ c(a + b) = ca + cb).
. , . , 2 . , (K,+, ) 2, .
.
1. (Z,+, ), (Q,+, ), (R,+, ), (C,+, ).
2. (Zn×n,+, ), (Qn×n,+, ), (Rn×n,+, ), (Cn×n,+, ).
3. (Z[x],+, ), (Q[x],+, ), (R[x],+, ), (C[x],+, ).
4. (FX,+, ), (C[a,b],+, ), (D[a,b],+, ).