[E;+] Example:S 3 ={e,  1 ,  2 ,  3 ,  4 ,  5 }

1. 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}.

2. [E;+] • Example:S3={e,1, 2, 3, 4, 5} • H1={e, 1}; H2={e, 2}; H3={e, 3}; • H4={e, 4, 5}。 • H1

3. 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

4. 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,suppose that HaHb，and (Ha)=(Hb) (3)Onto

5. 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|.

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

8. 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

9. (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

10. 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?

11. Let H be a normal subgroup of G, and let G/H={Hg|gG} • For Hg1 and Hg2G/H, • Let Hg1Hg2=H(g1*g2) • 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 Hg1=Hg3 and Hg2=Hg4G/H, • Hg1Hg2=H(g1*g2), Hg3Hg4=H(g3*g4), • Hg1Hg2?=Hg3Hg4? • H(g1*g2)=?H(g3*g4) • g3*g4?H(g1*g2), i.e. (g3g4)(g1*g2)-1?H.

12. 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.

13. 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 Hg1Hg2= H(g1*g2). • If G is a finite group, then G/H is also a finite group, and |G/H|=|G|/|H|

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