Download Let be a binary operation on a set S. a (b c)=(a b) c for a,b,c S

Chapter 6 Abstract algebra
 Groups 

 Field

 Lattics and Boolean algebra
 Rings
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(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
 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
Inverse
elements
element
+
√
√
0
-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;]. 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;].  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
 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 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
 [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}.

1
-1
i
-i
1
-1
i
-i
1
-1
i
-i
-1
1
-i
i
i
-i
-1
1
-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,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  ai
n
i 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
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. 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.
 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
 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)