复旦大学：《离散数学 Discrete Mathematics（上）》英文课件（赵一鸣）25/30

O Definition 16: Let h be a subgroup of a group G, and let aeG. we define the left coset of H in G containing g, written gH, by gH={g制h∈H. Similarity we define the right coset of H in G containing g, written Hg, by Hg h gl h ehy G a ah a∈G a∈G

 Definition 16: Let H be a subgroup of a group G, and let aG. We define the left coset of H in G containing g,written gH, by gH ={g*h| h H}. Similarity we define the right coset of H in G containing g,written Hg, by Hg ={h*g| h H}.   a G a G G H a a H   = =

E;+ s Example: S3=e, 01, 02, 03, 04,05 H1={e,o1;H2={e,o2;H3={e,o3}; I. H(e, c4, os)o

 [E;+]  Example:S3={e,1 , 2 , 3 , 4 , 5 }  H1={e, 1 }; H2={e, 2 }; H3={e, 3 };  H4={e, 4 , 5 }。  H1

Lemma 2: Let h be a subgroup of the group G. Then gH=H and Hg=H for Vg∈G. Proof: (p: H>Hg, o (h)=h*g

 Lemma 2：Let H be a subgroup of the group G. Then |gH|=|H| and |Hg|=|H| for gG.  Proof: :H→Hg, (h)=hg

6.4.3 Lagrange's Theorem Theorem 6.19: Let H be a subgroup of the group G. Then ghgeg and Hglgeg have the same cardinal number ◆ Proof:LetS={Hgg∈G}andT=gHg∈G q:S→T,φ(Ha)=alHl 1)p is an everywhere function. for ha=hb a-H?=b-lH a≠blif|a]n[b]= 2)(p is one-to-one For Ha, Hb, suppose that Ha*Hb, and cp(Ha=(p(Hb) oNto

6.4.3 Lagrange's Theorem  Theorem 6.19: Let H be a subgroup of the group G. Then {gH|gG} and {Hg|gG} have the same cardinal number  Proof：Let S={Hg|gG} and T={gH|gG}  : S→T, (Ha)=a-1H。 (1)  is an everywhere function. for Ha=Hb, a -1H?=b-1H [a][b] iff [a]∩[b]= (2)  is one-to-one。 For Ha,Hb,suppose that HaHb，and (Ha)=(Hb) (3)Onto

Definition 17: Let H is a subgroup of the group G. The number of all right cosets(left cofets) of H is called index of H n g E;+ is a subgroup of z;+ K Es index?? Theorem 6.20: Let g be a finite group and let h be a subgroup of G. Then G is a multiple of H Example: Let g be a finite group and let the order of a in G be n Then n G

 Definition 17：Let H is a subgroup of the group G. The number of all right cosets(left cofets) of H is called index of H in G.  [E;+] is a subgroup of [Z;+].  E’s index?？  Theorem 6.20: Let G be a finite group and let H be a subgroup of G. Then |G| is a multiple of |H|.  Example: Let G be a finite group and let the order of a in G be n. Then n| |G|

Example: Let g be a finite group and G=p. If p is prime, then G is a cyclic group

 Example: Let G be a finite group and |G|=p. If p is prime, then G is a cyclic group

G ≠0,a,b,c,d∈R C H ≠0,a,b,c2d∈} c d)c a 2a√2b a,b,c,d∈Q 01 √20 a a,b,c,d∈Q 01

{ | 0,a,b,c,d R} c d a b c d a b G           = { | 0,a,b,c,d Q} c d a b c d a b H           = | a,b, c,d Q} c d 2a 2b H { 0 1 2 0          =         | a,b, c,d Q} 2c d 2a b { 0 1 2 0 H          =        

O 6.4.4 Normal subgroups Definition 18: A subgroup H of a group is a normal subgroup if gh=g for VgEG. 4 Example: Any subgroups of Abelian group are normal subgroups ◆S3={e o5: 90192,03,04905 ◆H1={e,o};H2={e,a2};H3={e,3};H4={e, 4, 05 are subgroups of s3. Hg is a normal subgroup

6.4.4 Normal subgroups  Definition 18：A subgroup H of a group is a normal subgroup if gH=Hg for gG.  Example: Any subgroups of Abelian group are normal subgroups  S3={e,1 , 2 , 3 , 4 , 5 } :  H1={e, 1 }; H2={e, 2 }; H3={e, 3 }; H4={e, 4 , 5 } are subgroups of S3 .  H4 is a normal subgroup

4(1)If H is a normal subgroup of G, then Hg= gH for Wg∈G (2)H is a subgroup of g. .3)Hg=gH, it does not imply hg=gh. ◆(4)IfHg=gH, then there exists h'∈ H such that hg= gh' for vh∈H

 (1) If H is a normal subgroup of G, then Hg=gH for gG  (2)H is a subgroup of G.  (3)Hg=gH, it does not imply hg=gh.  (4) If Hg=gH, then there exists h'H such that hg=gh' for hH

tHeorem 6.21: Let H be a subgroup of G. H .is a normal subgroup of g iff ghgEH for yg∈ G and h∈H Example:LetG={(X;y川x,y∈ R with x≠0}, and consider the binary operation o introduced by (x, y)o(z,w)=(xz, xw+) for (x,y),(z,w)∈G. Let H=l(, y yER. Is H a normal subgroup of g? Why? ◆1. H is a subgroup of c ◆2. normal?

 Theorem 6.21: Let H be a subgroup of G. H is a normal subgroup of G iff g-1hgH for gG and hH.  Example:Let G ={ (x; y)| x,yR with x 0} , and consider the binary operation ● introduced by (x, y) ● (z,w) = (xz, xw + y) for (x, y), (z, w) G. Let H ={(1, y)| yR}.Is H a normal subgroup of G? Why?  1. H is a subgroup of G  2. normal?