2. 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 a n =e, and for any 0<m<n, a m  e. We say that the order of a is infinite if a n  e for any positive integer n. §Example:group[{1,-1,i.-i};  ], §i 2 =-1,i 3 =-i, i 4 =1 §(-i) 2 =-1, (-i) 3 =i, (-i) 4 =1

3. §Theorem 6.14: Let a is an element of the group G, and let its order be n. Then a m =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 a r is n/d, where d=(r,n) is maximum common factor of r and n. §Proof: (a r ) n/d =e, §Let p be the order of a r. §p|n/d, n/d|p §p=n/d

4. §2. Cyclic groups §Definition 14: The group G is called a cyclic group if there exists g  G such that h=g k 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=i 0,-1=i 2,-i=i 3, §i and –i are generators of G. §[Z;+]

5. §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:

6. §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;*]  [ Z n ;  ] §Proof:(1)G={g k |k  Z}, §  :G  Z,  (g k )=k §(2)G={e,g,g 2,  g n-1 }, §  :G  Z n,  (g k )=[k]

7. 6.4 Subgroups, Normal subgroups and Quotient groups §6.4.1 Subgroups  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.

8. §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§eH §Associative Law §inverse

9. §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 [H 1 ;·] and [H 2 ;·] be subgroups of the group [G;·] ， Then [H 1 ∩H 2 ;·] is also a subgroup of [G;·] §[H 1 ∪ H 2 ;·] ? §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?

10. 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

11. §[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

12. §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}.

13. §[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

14. §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

15. §NEXT : Lagrange's Theorem, Normal subgroups and Quotient groups §Exercise:P371 (Sixth) OR P357 (Fifth) 22— 26 §P376 10,12,21 §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