2.  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.  Proof: (1) H is a normal subgroup of G  (2) g -1 hg  H for  g  G and h  H  For  g  G, Hg?=gH

3.  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.

4.  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.

5.  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|

6. 6.5 The fundamental theorem of homomorphism for groups  6.5.1.Homomorphism kernel and homomorphism image  Lemma 4: Let [G;*] and [G';  ] be groups, and  be a homomorphism function from G to G'. Then  (e) is identity element of [G';  ].  Proof: Let x  (G)  G'. Then  a  G such that x=  (a).

7.  Definition 20: Let  be a homomorphism function from group G with identity element e to group G' with identity element e’. {x  G|  (x)= e'} is called the kernel of homomorphism function . We denoted by Ker  ( K(  ),or K).

8.  Example: [R-{0};*] and [{-1,1};*] are groups.

9.  Theorem 6.23 ： Let  be a homomorphism function from group G to group G'. Then following results hold.  (1)[Ker  ;*] is a normal subgroup of [G;*].  (2)  is one-to-one iff K={e G }  (3)[  (G);  ] is a subgroup of [G';  ].  proof:(1)i) Ker  is a subgroup of G  For  a,b  Ker , a*b  ?Ker ,  i.e.  (a*b)=?e G‘  Inverse element: For  a  Ker , a -1  ?Ker   ii)For  g  G,a  Ker , g -1 *a*g  ?Ker 

10.  6.5.2 The fundamental theorem of homomorphism for groups  Theorem 6.24 Let H be a normal subgroup of group G, and let [G/H;  ] be quotient group. Then f: G  G/H defined by f(g)=Hg is an onto homomorphism, called the natural homomorphism.  Proof: homomorphism  Onto

11.  Theorem 6.25 ： Let  be a homomorphism function from group [G;*] to group [G';  ]. Then [G/Ker(  );  ]  [  (G);  ]  isomorphism function f:G/ Ker(  )  (G).  Let K= Ker(  ). For  Ka  G/K ， f(Ka)=  (a)  f is an isomorphism function 。  Proof: For  Ka  G/K ， let f(Ka)=  (a)  (1)f is a function from G/K to  (G)  For Ka=Kb,  (a)=?  (b)  (2)f is a homomorphism function  For  Ka,Kb  G/K, f(Ka  Kb)=?f(Ka)  f(Kb)  (3) f is a bijection  One-to-one  Onto

12.  Corollary 6.2: If  is a homomorphism function from group [G;*] to group [G';  ], and it is onto, then [G/K;  ]  [G';  ]  Example: Let W={e i  |  R}. Then [R/Z;  ]  [W;*].  Let  (x)=e 2  ix   is a homomorphism function from [R;+] to [W;*],   is onto  Ker  ={x|  (x)=1}=Z

13. 6.6 Rings and fields 6.6.1 Rings  Definition 21: A ring is an Abelian group [R, +] with an additional associative binary operation(denoted · such that for all a, b, c  R,  (1) a · (b + c) = a · b + a · c,  (2) (b + c) · a = b · a + c · a.  We write 0  R for the identity element of the group [R, +].  For a  R, we write -a for the additive inverse of a.  Remark: Observe that the addition operation is always commutative while the multiplication need not be.  Observe that there need not be inverses for multiplication.

14.  Example: The sets Z,Q, with the usual operations of multiplication and addition form rings,  [Z;+,  ],[Q;+,  ] are rings  Let M={(a ij )n  n|a ij is real number}, Then [M;+,  ]is a ring  Example: S ,[P(S); ,∩] ，  Commutative ring

15.  Definition 23: A ring R is a commutative ring if ab = ba for all a, b  R. A ring R is an unitary ring if there is 1  R such that 1a = a1 = a for all a  R. Such an element is called a multiplicative identity.

16.  Example: If R is a ring, then R[x] denotes the set of polynomials with coefficients in R. We shall not give a formal definition of this set, but it can be thought of as: R[x] = {a 0 + a 1 x + a 2 x 2 + …+ a n x n |n  Z +, a i  R}.  Multiplication and addition are defined in the usual manner; if then One then has to check that these operations define a ring. The ring is called polynomial ring.

17.  Theorem 6.26: Let R be a commutative ring. Then for all a,b  R,  where n  Z +.

18.  Quiz:1.Let G be a cyclic group generated by the element g, where |G|=18. Then g 6 is not a generator of G  2.Let Q be the set of all rational numbers. Define & on Q by  a&b=a+b+a  b  (1)Prove ［ Q;& ］ is a monoid.  (2)Is [Q;&] a group ？ Why?  Exercise:P367 6  1.Prove Theorem 6.23(2)(3)  2.Let W={e i  |  R}. Then [C * /W;  ]  [R + ;*].  3.Let X be any non-empty set. Show that  [P(X); ∪, ∩] is not a ring.