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 a 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. 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.
a b  a b
G  {
 0, a, b, c, d  R}
 |
c d  c d
a b  a b
 |
H  {
 0, a, b, c, d  Q}
c d  c d
 2 0
 2a



 0 1 H  { c



 2
H
 0
2b 
 | a, b, c, d  Q}
d 
0   2a b 
  {
 | a, b, c, d  Q}


1   2c d 
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.
H is a normal subgroup of G iff g-1hgH
for gG and hH.
 Exercise: P362 21, 22,23, 26,28,33,34