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. a(bc)=(ab)c for a,b,cS

Commutative law: Let be a binary operation on a set S. ab=ba for a,bS

Identity element: Let be a binary operation on a set S. An element e of S is an identity element if ae=ea=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 ab = ba = 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

Distributive laws: Let and be two binary operations on nonempty S. For a,b,cS,

a(bc)=(ab)(ac), (bc)a=(ba)(ca)

Associative law commutative law

Identity elements

Inverse element

+ √ √ 0 -a for a

√ √ 1 1/a for a0

Definition 2: An algebraic system is a nonempty set S in which at least one or more operations Q1,…,Qk(k1), 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. An everywhere function from S to T is called a homomorphism from [S;*] to [T;] if (a*b)=(a)(b) for a,bS.

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 aS, then (a-1) is the inverse element (a).

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;](ST)

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.

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 aS.

(1)for a,b,cS,a (b c)=(a b) c;

(2)eS,for aS,a e=e a=a;

(3)for aS, a-1S, a a-1=a-1 a=e

[R-{0},] is a group [R,] is a monoid, but is not a group [R-{0},], for a,bR-{0},ab=baAbelian (or commutative) groupDefinition 8: We say that a group

[G;]is an Abelian (or commutative) group if ab=ba for a,bG.

[R-{0},],[Z;+],[R;+],[C;+] are Abelian (or commutative) group .

Example: Let [G; ] be a group with identity e. If x x=e for xG, then [G; ] is an Abelian group.

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

1 -1 i -i

1 1 -1 i -i

-1 -1 1 -i i

i i -i -1 1

-i -i i 1 -1

Multiplication table Abelian group

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,yR 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)+…, abcdef…

=(ab)cd(ef)…, Theorem 6.4: If a1,…,an(n3), 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

n

iia

1

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

Theorem 6.5: Let [G;] be a group and let aiG(i=1…,n). Then

(a1…an)-1=an-1…a1

-1

Next: Permutation groups and cyclic groups

Exercise p333 9,11,18,19,22,23; P340 5—7,13,14,19—21P357 2,6-9,Prove Theorem 6.3 (2)(4)