Chapter 6 Abstract algebra Groups  Rings  Field  Lattics and Boolean algebra

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

Associative law: Let  be a binary operation on a set S 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.

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

a(bc)=(ab)(ac), (bc)a=(ba)(ca) 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) Associative law commutative law Identity elements Inverse element + √ -a for a  1 1/a for a0

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

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.

Theorem 6. 3 Let  be a homomorphism from [S;. ] to [T;] 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) is identity 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).

Definition 4: Let  be a homomorphism from [S;. ] to [T;] 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)

6.2 Semigroups,monoids and groups 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.

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.

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

Abelian (or commutative) group [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.

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

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?

[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.

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

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

Theorem 6. 6: Let [G;] be a group and let a and b be elements of G 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

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.

a-k=(a-1)k， ak=a*ak-1(k≥1) 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

Exercise P348 (Sixth) OR p333(Fifth) 9,10,11,18,19,22,23, 24; Next: Permutation groups Exercise P348 (Sixth) OR p333(Fifth) 9,10,11,18,19,22,23, 24; P355 (Sixth) OR P340(Fifth) 5—7,13,14,19—22 P371 (Sixth) OR P357(Fifth) 1,2,6-9,15, 20 Prove Theorem 6.3 (2)(4)