复旦大学：《离散数学》PPT教学课件（赵一鸣）25/28

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|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

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=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;.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)

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)