Chapter 6 Abstract algebra

1. Chapter 6 Abstract algebra • Groups • Rings • Field • Lattics and Boolean algebra

2. 6.1 Operations on the set • Definition 1：An unary operation on a nonempty set S is an everywhere function f from S into S; A binary operation on a nonempty set S is an everywhere function f from S×S into S; A n-ary operation on a nonempty set S is an everywhere function f from Sn into S. • closed

3. Associative law: Let be a binary operation on a set S. a(bc)=(ab)c for a,b,cS • Commutative law: Let be a binary operation on a set S. ab=ba for a,bS • Identity element: Let be a binary operation on a set S. An element e of S is an identity element if ae=ea=a for all a S. • Theorem 6.1: If  has an identity element, then it is unique.

4. Inverse element: Let  be a binary operation on a set S with identity element e. Let a S. Then b is an inverse of a if ab = ba = e. • Theorem 6.2: Let  be a binary operation on a set A with identity element e. If the operation is Associative, then inverse element of a is unique when a has its inverse

5. Distributive laws: Let  and  be two binary operations on nonempty S. For a,b,cS, • a(bc)=(ab)(ac), (bc)a=(ba)(ca)

6. Definition 2: An algebraic system is a nonempty set S in which at least one or more operations Q1,…,Qk(k1), are defined. We denoted by [S;Q1,…,Qk]. • [Z;+] • [Z;+,*] • [N;-] is not an algebraic system

7. Definition 3: Let [S;*] and [T;] are two algebraic system with a binary operation. A function  from S to T is called a homomorphism from [S;*] to [T;] if (a*b)=(a)(b) for a,bS.

8. Theorem 6.3 Let  be a homomorphism from [S;*] to [T;]. If  is onto, then the following results hold. • (1)If * is Associative on S, then  is also Associative on T. • (2)If * is commutative on S, then  is also commutation on T • (3)If there exist identity element e in [S;*],then (e) isidentity element of [T;] • (4) Let e be identity element of [S;*]. If there is the inverse element a-1 of aS, then (a-1) is the inverse element (a).

9. Definition 4: Let  be a homomorphism from [S;*] to [T;].  is called an isomorphism if  is also one-to-one correspondence. We say that two algebraic systems [S;*] and [T;] are isomorphism, if there exists an isomorphic function. We denoted by [S;*][T;](ST)

10. 6.2 Semigroups,monoids and groups • 6.2.1 Semigroups, monoids • Definition 5: A semigroup [S;] is a nonempty set together with a binary operation  satisfying associative law. • Definition 6: A monoid is a semigroup [S; ] that has an identity.

11. Let P be the set of all nonnegative real numbers. Define & on P by • a&b=(a+b)/(1+ab) • Prove［P;&］is a monoid.

12. 6.2.2 Groups • Definition 7: A group [S; ] is a monoid, and there exists inverse element for aS. (1)for a,b,cS,a (b c)=(a b) c; (2)eS,for aS,a e=e a=a; (3)for aS, a-1S, a a-1=a-1 a=e

13. [R-{0},] is a group • [R,] is a monoid, but is not a group • [R-{0},], for a,bR-{0},ab=ba • Abelian (or commutative) group • Definition 8: We say that a group [G;]is an Abelian (or commutative) group if ab=ba for a,bG. • [R-{0},],[Z;+],[R;+],[C;+]are Abelian (or commutative) group . • Example: Let [G; ] be a group with identity e. If xx=e for xG, then [G; ] is an Abelian group.

14. Example: Let G={1,-1,i,-i}.

15. G={1,-1,i,-i}, finite group • [R-{0},],[Z;+],[R;+],[C;+]，infinite group • |G|=n is called an order of the group G • 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. • Prove that (G; ●) is a group. • Is (G;●) an Abelian group?

16. [R-{0},] , [R;+] • a+b+c+d+e+f+…=(a+b)+c+d+(e+f)+…, • abcdef…=(ab)cd(ef)…, • Theorem 6.4: If a1,…,an(n3), are arbitrary elements of a semigroup, then all products of the elements a1,…,an that can be formed by inserting meaningful parentheses arbitrarily are equal.

17. a1  a2 …  an If ai=aj=a(i,j=1,…,n), then a1 a2 … an=an。 na

18. Theorem 6.5: Let [G;] be a group and let aiG(i=1…,n). Then • (a1…an)-1=an-1…a1-1

19. Theorem 6.6: Let [G;] be a group and let a and b be elements of G. Then • (1)ac=bc, implies that a=b(right cancellation property)。 • (2)ca=cb, implies that a=b。(left cancellation property) • S={a1,…,an}, al*aial*aj(ij)， • Thus there can be no repeats in any row or column

20. Theorem 6.7: Let [G;] be a group and let a,b, and c be elements of G. Then • (1)The equation ax=b has an unique solution in G. • (2)The equation ya=b has an unique solution in G.

21. Let [G;] be a group. We define a0=e， • a-k=(a-1)k，ak=a*ak-1(k≥1) • Theorem 6.8: Let [G;] be a group and a G, m,n Z. Then (1)am*an=am+n (2)(am)n=amn • a+a+…+a=ma, ma+na=(m+n)a n(ma)=(nm)a

22. Next: Permutation groups • Exercise P348 (Sixth) ORp333(Fifth) 9,10,11,18,19,22,23, 24; • P355 (Sixth) ORP340(Fifth) 5—7,13,14,19—22 • P371 (Sixth) ORP357(Fifth) 1,2,6-9,15, 20 • Prove Theorem 6.3 (2)(4)