##### 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 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'.**