1. Let H be a normal subgroup of G, and let G/H={Hg|gG}
For Hg1 and Hg2G/H,
Let Hg1Hg2=H(g1*g2)
Lemma 3: Let H be a normal subgroup of G. Then [G/H; ] is a algebraic system.
Proof:  is a binary operation on G/H.
For Hg1=Hg3 and Hg2=Hg4G/H,
Hg1Hg2=H(g1*g2), Hg3Hg4=H(g3*g4),
Hg1Hg2?=Hg3Hg4?
H(g1*g2)=?H(g3*g4)
g3*g4?H(g1*g2), i.e. (g3g4)(g1*g2)-1?H.

2. Identity element: Let e be identity element of G.
Theorem 6.22: Let [H;] be a normal subgroup of the group [G;]. Then [G/H;] is a group.
Proof: associative
Identity element: Let e be identity element of G.
He=HG/H is identity element of G/H
Inverse element: For HaG/H, Ha-1G/H is inverse element of Ha, where a-1G is inverse element of a.

3. Definition 19: Let [H;. ] be a normal subgroup of the group [G;. ]
Definition 19: Let [H;*] be a normal subgroup of the group [G;*]. [G/H;] is called quotient group, where the operation  is defined on G/H by Hg1Hg2= H(g1*g2).
If G is a finite group, then G/H is also a finite group, and |G/H|=|G|/|H|

4. 6.5 The fundamental theorem of homomorphism for groups
6.5.1.Homomorphism kernel and homomorphism image
Lemma 4: Let [G;*] and [G';] be groups, and  be a homomorphism function from G to G'. Then (eG) is identity element of [G';].
Proof: Let x(G)G'. Then  aG such that x=(a).

5. Definition 20: Let  be a homomorphism function from group G with identity element e to group G' with identity element e’. {xG| (x)= e'} is called the kernel of homomorphism function . We denoted by Ker( K(),or K).

6. Example: [R-{0};*] and [{-1,1};*] are groups.

7. (1)[Ker;*] is a normal subgroup of [G;*].
Theorem 6.23：Let  be a homomorphism function from group G to group G'. Then following results hold.
(1)[Ker;*] is a normal subgroup of [G;*].
(2) is one-to-one iff K={eG}
(3)[(G); ] is a subgroup of [G';].
proof:(1)i) Ker is a subgroup of G
For a,bKer, a*b?Ker,
i.e.(a*b)=?eG‘
Inverse element: For aKer, a-1?Ker
ii)For gG,aKer, g-1*a*g?Ker

8. 6.5.2 The fundamental theorem of homomorphism for groups
Theorem 6.24 Let H be a normal subgroup of group G, and let [G/H;] be quotient group. Then f: GG/H defined by f(g)=Hg is an onto homomorphism, called the natural homomorphism.
Proof: homomorphism
Onto

9. Theorem 6. 25：Let  be a homomorphism function from group [G;
Theorem 6.25：Let  be a homomorphism function from group [G;*] to group [G';]. Then [G/Ker();][(G);]
isomorphism function f:G/ Ker()(G).
Let K= Ker(). For KaG/K，f(Ka)=(a)
f is an isomorphism function。
Proof: For  KaG/K，let f(Ka)=(a)
(1)f is an everywhere function from G/K to (G)
For Ka=Kb,(a)=?(b)
(2)f is a homomorphism function
For  Ka,KbG/K, f(KaKb)=?f(Ka)f(Kb)
(3) f is a bijection
One-to-one
Onto

10. Corollary 6. 2: If  is a homomorphism function from group [G;
Corollary 6.2: If  is a homomorphism function from group [G;*] to group [G';], and it is onto, then [G/K;][G';]
Example: Let W={ei|R}. Then [R/Z;][W;*].
Let (x)=e2ix
 is a homomorphism function from [R;+] to [W;*],
 is onto
Ker={x|(x)=1}=Z

11. For g2G2, (n)(N), g1G1 s.t (g1)=g2 ? surjection homomorphism.
Let : G1→G2 be a surjection homomorphism between two groups, N be a normal subgroup of G1, and KerN. Proof: G2/(N) is a group.
Theorem 6.22: Let [H;] be a normal subgroup of the group [G;]. Then [G/H;] is a group.
(N) is a subgroup of G2. Normal subgroup?
For g2G2, (n)(N), g1G1 s.t (g1)=g2 ?
surjection homomorphism.
g2-1(n)g2?(N)

12. For g1G1, f (g1)=(N)(g1)
Let : G1→G2 be a surjection homomorphism between two groups, N be a normal subgroup of G1, and Ker N. Proof: G1/N≌G2/(N).
Corollary 6.2: If  is a homomorphism function from group [G;*] to group [G';], and it is onto, then [G/K;][G';]
Prove:f :G1→G2/(N)
For g1G1, f (g1)=(N)(g1)
1)f is a surjection homomorphism from G1 to G2/ (N).
2)Kerf =N

13. Next: Rings, Integral domains, division rings and fields,
Exercise:P376 7,8,31,32
1.Let W={ei|R}. Then [C*/W;][R+;*]