2.  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}.

3.  [E;+]  Example:S 3 ={e,  1,  2,  3,  4,  5 }  H 1 ={e,  1 }; H 2 ={e,  2 }; H 3 ={e,  3 };  H 4 ={e,  4,  5 } 。 H1H1

4.  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

5. 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 -1 H 。 (1)  is an everywhere function. for Ha=Hb, a -1 H?=b -1 H [a]  [b] iff [a] ∩[b]=  (2)  is one-to-one 。 For Ha,Hb,suppose that Ha  Hb ， and  (Ha)=  (Hb) (3)Onto

6.  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|.

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

9. 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  S 3 ={e,  1,  2,  3,  4,  5 } :  H 1 ={e,  1 }; H 2 ={e,  2 }; H 3 ={e,  3 }; H 4 ={e,  4,  5 } are subgroups of S 3.  H 4 is a normal subgroup

10.  (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

11.  Theorem 6.21: Let H be a subgroup of G. H is a normal subgroup of G iff g -1 hg  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?

12.  Let H be a normal subgroup of G, and let G/H={Hg|g  G}  For  Hg 1 and Hg 2  G/H,  Let Hg 1  Hg 2 =H(g 1 *g 2 )  Lemma 3: Let H be a normal subgroup of G. Then [G/H;  ] is a algebraic system.  Proof:  is a binary operation on G/H.  For  Hg 1 =Hg 3 and Hg 2 =Hg 4  G/H,  Hg 1  Hg 2 =H(g 1 *g 2 ), Hg 3  Hg 4 =H(g 3 *g 4 ),  Hg 1  Hg 2 ?=Hg 3  Hg 4 ?  H(g 1 *g 2 )=?H(g 3 *g 4 )  g 3 *g 4  ?H(g 1 *g 2 ), i.e. (g 3  g 4 )  (g 1 *g 2 ) -1  ?H.

13.  Theorem 6.22: Let [H;  ] be a normal subgroup of the group [G;  ]. Then [G/H;  ] is a group.  Proof: associative  Identity element: Let e be identity element of G.  He=H  G/H is identity element of G/H  Inverse element: For  Ha  G/H, Ha -1  G/H is inverse element of Ha, where a -1  G is inverse element of a.

14.  Definition 19: Let [H;*] be a normal subgroup of the group [G;*]. [G/H;  ] is called quotient group, where the operation  is defined on G/H by Hg 1  Hg 2 = H(g 1 *g 2 ).  If G is a finite group, then G/H is also a finite group, and |G/H|=|G|/|H|

15.  Next: quotient group The fundamental theorem of homomorphism for groups  Exercise: P362 21, 22,23, 26,28,33,34