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|