Properties of Group and Abelian group
  1. The identity element for a group is unique.
  2. The inverse of any element in a group is unique. 
  3. The inverse of the identity element 'e' is 'e' itself.

Cancellation laws: 

(a*b) = (a*c)  then  b=c 

(a*c) =(b*c)  then   a=c

(a*b)^{^{-1}} = b^{-1} a^{-1} for all a,b  belongs to  group 

 

Abelian group:-

If a group follows commutative property then that group is called abelian group.

A group (S, ⊗) is said to be abelian if a⊗ b= b⊗a for all a,b ∈ Z.

Example:

1. (Z, +) is an abelian group.   // Z is set of all integer.

Check for Algebraic structure:-  Yes, you take any a, b ∈ Z and apply the  '+' operation. It follows closure property.

2. Check for Semigroup: Yes, because '+' follows the associative property.

3. Check for Monoid: Yes, because '0' is the identity element here. 

4. Check for the group: Yes, for every a ∈ Z, inverse exists. a +(-a) = 0 

5. Check for Abelian:- Yes because  ' +' follows the commutative property.

 

2. (M, *): M is set of all non-singular matrix 

It is just a group, not an abelian group. 

 

 

 

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