复旦大学：《离散数学 Discrete Mathematics（上）》英文课件（赵一鸣）26/30

Definition 19: Let H; be a normal subgroup of the group G;.G/H; O is called quotient group where the operation e is defined on G/h by Hg oHg2' H(g *g2) o If G is a finite group, then G/H is aiso a finite group, and G/HFG H

 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 46.5.1.Homomorphism kernel and homomorphism image t Lemma 4: Let G; *I and IG; l be groups, and op be a homomorphism function from G to G. Then pleg) is identity element of IG;. that x=p(aP(GEG. Then 3 aEG such ◆ Proof:Letx

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)

. Example: [R-103, * and K-1, 13; *] are groups. 1x>0 p(x) Kerq={x|x>0,x∈R} -1x<0

Example: [R-{0};*] and [{-1,1};*] are groups.    −   = 1 0 1 0 ( ) x x  x Ker ={x | x  0, xR}

O Definition 20: Let o be a homomorphism function from group G with identity element e to group G with identity element e. XEG(x) e is called the kernel of homomorphism function (p We denoted by Kero(k(p),or k)

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)

Theorem 6.23: Let o be a homomorphism function from group G to group G. Then following results hold. ◆(1)Kerφ;] is a normal subgroup of[G; . (2)p is one-to-one iff K=(ed) ◆(3)|p(G;] is a subgroup of g';°l proof:(1)i Kero is a subgroup of g ◆ For va,b∈kerq,a2b∈?kerq, ◆ie.g(a2b)=?ec ◆ Inverse element: For va∈Kerq,al∈?Kerp ◆i) For GeC,2a∈Kerq,g1*a*g∈?Kerq

 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 Leth 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

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 op be a homomorphism function from group G; to group G; Then g/Ker(φ);|φ(G); e isomorphism function f: G/ Ker(o)-(p(G) Let K= Ker(). For VKaEG/K, f(Ka=op(a) Is an isomorphism iunction ◆ Proof: For v Ka∈G/kK,letf(Ka)=q(a) (fis an everywhere function from G/K to p(G) For Ka=kb, cp(a)=?p(b) +(2) is a homomorphism function ◆ For v Ka,kb∈G/K,f(Kakb)=?fka)efkb) ◆(3) fis a bijection ◆One-to-one ◆Onto

 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 (p is a homomorphism function from group [G; to group IG;1 and it is onto, then IG/K;8|[G";° ◆ Example:Letw=ee∈R}.Then R/; OEW; x ◆Letq(x)=e2x +p is a homomorphism function from R;+l to w; ◆ is onto ◆Kerp={x(x)=1}=Z

 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

Next: The fundamental theorem of homomorphism for groups; Rings eXercise 1. Prove Theorem 6.23(2)(3) 2.Let W=leeR. Then C*/; Ert;

 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+ ;*]