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.

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.

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|

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).

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).

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

(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

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

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

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

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)

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

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