6.5 The fundamental theorem of homomorphism for groups

1. 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|

2. 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).

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

4. 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).

5. 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. 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

7. 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

8. 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

9. Next: The fundamental theorem of homomorphism for groups; Rings • Exercise: • 1.Prove Theorem 6.23(2)(3) • 2.Let W={ei|R}. Then [C*/W;][R+;*].