Basic Definitions : Group Theory #group theory
Basic Definitions :
* Binary Operation
* Algebraic Structure
* Quasi- Group or Groupoid
* Semi Group
* Monoid
* Group
* Abelian Group
Binary Operation :
Let S be a non-empty set . If f : SxS→R is a mapping , then f is called binary operation or binary composition on S.
Thus
- If a relation in S is such that every pair ( distinct or equal ) of elements of S taken in definite order is associated with a unique element of S then it is called a binary operation in S. Otherwise the relationis not binary operation in S and the relation is simply an operation in S.
- (a,b) 𝟄 SxS , ∃ a unique element f(a,b) 𝟄 S. We observe that addition, multiplication, subtraction are binary operations in R and division is not a binary operation in R why because division by 0 is not defined.
- In N the operation o defined by aob = (a+b)/ab is not a binary operation.
For a non-empty set S and for a,b in S ⇒ a * b in S
⇒ * is called a binary operation.
This is called closure Property.
For example :
1. consider the set of natural numbers N and the binary operation is +.
then a+b 𝟄 N for every natural numbers a and b.
Here N is said to hold the closure property under the binary operation +.
2. consider the set of integers Z and the binary operation is multiplication (x).
then axb 𝟄 Z for every integers a and b.
Here Z is said to hold the closure property under the binary operation x.
Counter Example :
Consider the set N of natural numbers and the binary operation division ÷ .
Here N is not holding closure property because for the integers 2 & 3, 2 ÷ 3 is not a natural number.
Algebraic Structure :
A non-empty set G equipped with one or more binary operations is called an algebraic structure or an algebraic system.
If o is a binary operation on G, then the algebraic structure is written as (G , o ).
Quasi Group :
A non-empty set S is said to be Quasi- Group or Groupoid under the binary operation * if S satisfies the following property
for a,b 𝟄 S ⇒ a*b 𝟄 S ( Closure Property )
For example :
The set Q under the binary opetation o defined by aob = ( a+ b ) / 2 is not a semi group
For any two rationals a and b , ( a + b ) / 2 is rational, closure property is clear.
checking for associative law:
For a,b,c in Q,
( a o b ) o c = ((a o b) + c ) / 2
= ( (a + b ) / 2 ) + c ) / 2
= ( a + b + 2c ) / 4
Now a o ( b o c ) = ( a + ( b o c ) ) /2
= ( a + ( b + c ) / 2 ) / 2
= ( 2a + b + c ) / 4
Since ( a o b ) o c ≠ a o ( b o c ) , Q does not holds associative law.
Semi Group :
A non-empty set S is said to be a Semi-Group under the binary operation * if S satisfies the following properties.
1. for a,b 𝟄 S ⇒ a*b 𝟄 S ( Closure Property )
2. for a,b,c 𝟄 S ⇒ (a*b)*c = a*(b*c) ( Associative Property )
Or
A Quasi- Group with the associative law is a semi group.
For example :
1. The sets N & Z of natural numbers and integers are semi groups under the binary operation addition (+).
2. The set (Q,-) is not a semi group . For the rational numbers 5,3/2, 1 ,
(5-3/2)-1 ≠ 5-(3/2-1) i.e. the associative law is failed.
3. The set Q under the binary operation o defined by aob = (a+b)/2 is not a semi group.
Monoid :
A non-empty set S is said to be a Semi-Group under the binary operation * if S satisfies the following properties.
1. for a,b 𝟄 S ⇒ a*b 𝟄 S ( Closure Property )
2. for a,b,c 𝟄 S ⇒ (a*b)*c = a*(b*c) ( Associative Property )
3. for a,e in S, a*e=e*a =a ( Identity Property )
Here the element e is called the identiy element.
OR
A semi group with identity is a monoid.
For Example :
1. (Z,+) is a monoid and the identity is 0. Here 0 is called the additive identity
2. (Z, . ) is a monoid and the identity is 1. Here 1 is called the multiplicative identity or unit element.
Group :
A non-empty set S is said to be a Semi-Group under the binary operation * if S satisfies the following properties.
1. for a,b 𝟄 S ⇒ a*b 𝟄 S ( Closure Property )
2. for a,b,c 𝟄 S ⇒ (a*b)*c = a*(b*c) ( Associative Property )
3. for a,e in S, a*e=e*a =a ( Identity Property )
4. for a 𝟄 S, there exist b in S such that a * b = e = b * a ( Inverse Property )
Here b is called the inverse of a in S.
Or
A monoid with inverse property is a group.
Examples :
( Z , + ) , ( Q , + ) , ( R , + ) and ( C , + ) are all groups.
Counter Example :
For the set of Integers Z, ( Z , . ) is not a group
because for 2 in Z , the multiplicative inverse of 2 i.e. 1/2 is not an integer and hence 1/2 does not belongs to Z.
Abelian Group :
For a group G, for a , b 𝟄 G, if a * b = b * a ( Commutative Law ), then G is called an Abelian group or Commutative Group.
Or
A group with abelian property is an Abelian Group
Example :
( Z , + ) , ( Q , + ) , ( R , + ) and ( C , + ) are all abelian groups.
Counter Example :
Consider the set of all 2x2 matrices is not abelian under the binary operation multiplication
because the matrix multiplication is not commutative.
#group #theory #binary #operation #abstract
#algebra #syllabus #semi #groupoid #monoid #abelian #commutative
#group #theory #groupoid #Semi group #monoid #group #abelian #algebra
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