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Let H be a normal subgroup of G, and let G/H={Hg|g  G} For  Hg 1 and Hg 2  G/H,

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. Let H be a normal subgroup of G, and let G/H={Hg|g  G}

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Let H be a normal subgroup of G, and let G/H={Hg|g  G} For  Hg 1 and Hg 2  G/H,

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

  2. 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.

  3. 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.

  4. 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|

  5. 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).

  6. 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).

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

  8. 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={eG} • (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)=?eG‘ • Inverse element:For aKer,a-1?Ker • ii)For gG,aKer, g-1*a*g?Ker

  9. 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

  10. 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

  11. 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={ei|R}. Then [R/Z;][W;*]. • Let (x)=e2ix •  is a homomorphism function from [R;+] to [W;*], •  is onto • Ker={x|(x)=1}=Z

  12. 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.

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

  14. 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.

  15. 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] = {a0 + a1x + a2x2 + …+ anxn|nZ+, ai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.

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

  17. Quiz:1.Let G be a cyclic group generated by the element g, where |G|=18. Then g6 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={ei|R}. Then [C*/W;][R+;*]. • 3.Let X be any non-empty set. Show that • [P(X); ∪, ∩] is not a ring.

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