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。  is an everywhere function. for Ha=Hb, a-1H?=b-1H [a][b] iff [a]∩[b]= (2)  is one-to-one。 For Ha,Hb,if HaHb，then (Ha)=a-1H?(Hb) =b-1H (3)Onto

Definition 17：Let H is a subgroup of the group 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 Example: Let G be a finite group and |G|=p. If p is prime, then G is a cyclic group.

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

(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 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?

Next: Quotient group The fundamental theorem of homomorphism for groups Exercise: P376 (Sixth) OR P362(Fifth) 22,23, 26,28,33,34