1. 6.3.2 Cyclic groups 1.Order of an element
Definition 13: Let G be a group with an identity element e. We say that a is of order n if an =e, and for any 0<m<n, ame. We say that the order of a is infinite if an e for any positive integer n.
Example:group[{1,-1,i.-i};],
i2=-1,i3=-i, i4=1
(-i)2=-1, (-i)3=i, (-i)4=1

2. Theorem 6.14: Let a is an element of the group G, and let its order be n. Then am=e for mZ iff n|m.
Example: Let the order of the element a of a group G be n. Then the order of ar is n/d, where d=(r,n) is maximum common factor of r and n.
Proof: (ar)n/d=e,
Let p be the order of ar.
p|n/d, n/d|p
p=n/d

3. 2. Cyclic groups
Definition 14: The group G is called a cyclic group if there exists gG such that h=gk for any hG , where kZ.We say that g is a generator of G. We denoted by G=(g).
Example:group[{1,-1,i.-i};],1=i0,-1=i2,-i=i3,
i and –i are generators of G.
[Z;+]

4. Example：Let the order of group G be n
Example：Let the order of group G be n. If there exists gG such that g is of order n，then G is a cyclic group, and G is generated by g.
Proof:

5. Theorem 6.15: Let [G; *] be a cyclic group, and let g be a generator of G. Then the following results hold.
(1)If the order of g is infinite, then [G;*] [Z;+]
(2)If the order of g is n, then [G;*][ Zn;]
Proof:(1)G={gk|kZ},
:GZ, (gk)=k
(2)G={e,g,g2,gn-1},
:GZn, (gk)=[k]

6. 6.4 Subgroups, normal subgroups and quotient groups
Definition 15: A subgroup of a group (G; *) is a nonempty subset H of G such that * is a group operation on H.
Example : [Z;+] is a subgroup of the group [R; +].
G and {e} are called trivial subgroups of G, other subgroups are called proper subgroups of G.

7. Theorem 6. 16: Let [G;·] be a group, and H be a nonempty subset of G
Theorem 6.16: Let [G;·] be a group, and H be a nonempty subset of G. Then H is a subgroup of G, iff
(1) for any x, y H, x·yH; and
(2) for any xH, x-1 H.
Proof: If H is a subgroup of G, then (1) and (2) hold.
(1) and (2) hold
eH
Associative Law
inverse

8. Theorem 6. 17: Let [G;·] be a group, and H be a nonempty subset of G
Theorem 6.17: Let [G;·] be a group, and H be a nonempty subset of G. Then H is a subgroup of G, iff a·b-1H for a,b H.
Example: Let [H1;·] and [H2;·] be subgroups of the group [G;·]，Then [H1∩H2;·] is also a subgroup of [G;·]
[H1∪H2;·] ?
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 subgroup of G? Why?

9. 6.4.2 Coset
Let [H;] is a subgroup of the group [G;]. We define a relation R on G, so that aRb iff for ab-1H for a,bG. The relation is called congruence relation on the subgroup [H;]. We denoted by ab(mod H)。
Theorem 6.18 ：Congruence relation on the subgroup [H;] of the group G is an equivalence relation

10. [a]={x|xG, and xa(mod H)}={x|xG, and xa-1H}
Let h=xa-1. Then x=ha，Thus
[a]={ha|hH}
Ha={ha|hH} is called right coset of the subgroup H
aH={ah|hH} is called left coset of the subgroup H
Let [H;] be a subgroup of the group [G;], and aG. Then
(1)bHa iff ba-1H
(2)baH iff a-1bH

11. Definition 16: Let H be a subgroup of a group G, and let aG
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}.

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

13. Lemma 2：Let H be a subgroup of the group G
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

14. P362 11,12 Exercise:Exercise:P357 22—26
1. Let G be a group. Suppose that a, and bG, ab=ba. If the order of a is n, and the order of b is m. Prove:
(1)The order of ab is mn if (n,m)=1
(2)The order of ab is LCM(m,n) if (n,m)1 and (a)∩(b)=. LCM(m,n) is lease common multiple of m and n