- The identity element for a group is unique.
- The inverse of any element in a group is unique.
- The inverse of the identity element 'e' is 'e' itself.
(a*b) = (a*c) then b=c
(a*c) =(b*c) then a=c
for all a,b belongs to 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.
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.