Let h be a normal subgroup of g, and let G/HHgIgEG ◆ For VHg1 and Hg2∈G/H, ◆ Let Hg oHg2=H(g1*g2 Lemma 3: Let H be a normal subgroup of G. Then G/H; e is a algebraic system a+ Proof: is a binary operation on G/H ◆ For VHg1=Hg3 and Hg2=Hg4∈G/H, ◆Hg1②Hg2=H(g12g2),Hg3SHg=H(g32g4) ◆Hg1Hg2=Hg3Hg? ◆H(g1*g2)=2H(g3g4 ?H(g g2),le (g3*g4)*(g1g2)∈?H
Let H be a normal subgroup of G, and let G/H={Hg|gG} For Hg1 and Hg2G/H, Let Hg1Hg2=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=Hg4G/H, Hg1Hg2=H(g1*g2 ), Hg3Hg4=H(g3*g4 ), Hg1Hg2?=Hg3Hg4? H(g1*g2 )=?H(g3*g4 ) g3*g4?H(g1*g2 ), i.e. (g3g4 )(g1*g2 ) -1?H
Theorem 6.22: Let H; be a normal subgroup of the group G; * Then g/H; o 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 vha∈G/H,HaeG/H is inverse element of a where aleg is inverse element of a
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=HG/H is identity element of G/H Inverse element: For HaG/H, Ha-1G/H is inverse element of Ha, where a-1G is inverse element of a
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 Hg1Hg2 = 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 aG such that x=(a)
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’. {xG| (x)= e'} is called the kernel of homomorphism function . We denoted by Ker( K(),or K)
. 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, xR}
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,bKer, a*b?Ker, i.e.(a*b)=?eG‘ Inverse element: For aKer, a -1?Ker ii)For gG,aKer, 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 (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 KaG/K，f(Ka)=(a) f is an isomorphism function。 Proof: For KaG/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,KbG/K, f(KaKb)=?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)=e2ix is a homomorphism function from [R;+] to [W;*], is onto Ker={x|(x)=1}=Z