1. 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

2. 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|.

3. 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.

5. 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

6. (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

7. 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?

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