Intoduction to Group and Some Related terms

Before discussing group let's first know some related terms:

1. Algebraic Structure:- A non-empty set S is called algebraic structure with respect to a binary operation ⊗   if all the elements of the set follow the closer property. 

What it means is: let us say there are two elements a and b such that a,b \varepsilon S , then a ⊗  b must belong to S.

Example:

1. S ={1,2,3,4,5 }

then (S, *) is not algebraic structure. Why??

If we take two elements like 4 and 5 and apply  * operation between them we get 4*5 =20 but 20 is not part of set S. It means closure property is not followed.

Some more Examples:

2. Set of all real numbers over operation + ie, (R,+). 

3. let A ={1,2,3,4,5}

and R is a reflexive relation then (R, ∪) is an algebraic structure.

 

2. Semigroup:

If an algebraic structure (S,⊗ ) follows associate property then it is called Semigroup. 

Associativity depends on the operator. Example: 

An algebraic structure (S, *) is called Semigroup if 

a *(b*c) = (a*b)*c

Examples:

1. Natural numbers over operation  '+'  are a semigroup.

2. Integers over '-' are not semigroup. They are just Algebraic structure.

 

3. Monoid:-

A semigroup (S, ⊗ ) becomes Monoid if there exists an identity element 'e'  such that  a ⊗  e =a: for all  'a' belongs to S.

Examples:

1. (N, +) is not Monoid because there is no e such that  a+e  = a .

2. (N,*) is monoid because 1 is 'e' here:  a*e= a*1 = a

                          N is set of all natural numbers.

 

4. Group:-

A monoid (S,⊗  )is called a group if there is an element 'b' such that a ⊗ ​ b = e  : where 'b' is called inverse of  'a'. Examples:-

 (Z, +): We already know it is a monoid and there is also 'b' such that

a + b = 0: why 0? because 0 is the identity element for this monoid.

and

a +(-a)=  0 so '-a' is acting as inverse of 'a'.

 

 

 

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