A group G is said to be abelian group or commutative group if a•b = b•a, for all a,b ϵ G

Example:

- The set of all integers Z is an abelian group with respect to addition.
- If G = {1, -1, i, -i} the G is an abelian group with respect to multiplication.

The symbol • is a general placeholder for a concretely given operation.

To qualify as an **Abelian group**, the set and operation, (*A*, • ), must satisfy five requirements known as the * Abelian group axioms*:

**Closure:**- For all
*a*,*b*in*A*, - the result of the operation
*a*•*b*is also in*A*. **Associativity:**- For all
*a*,*b*and*c*in*A*, - the equation (
*a*•*b*) •*c*=*a*• (*b*•*c*) holds. **Identity element:**- There exists an element
*e*in*A*, such that for all elements*a*in*A*, - the equation
*e*•*a*=*a*•*e*=*a*holds. **Inverse element:**- For each
*a*in*A*, there exists an element*b*in*A* - such that
*a*•*b*=*b*•*a*=*e*, where*e*is the identity element. **Commutativity:**- For all
*a*,*b*in*A*,*a*•*b*=*b*•*a*.

More compactly, an Abelian group is a commutative group. A group in which the group operation is not commutative is called a “**non-Abelian group**” or “**non-commutative group**“.

- If G is a group and a, b are any two elements of G then i) (ab)-1 = b-1. a-1 ii) (a-1)-1 = a iii) If a, b, c are the elements of G then ab = bc ⇒ b = c and ba = ca ⇒ b = c
- Let G be a group and a^2 = e, for all a ϵG . Then prove that G is an abelian group.
- Prove that if (ab)^-1 = a^-1b^-1, for all a, b ϵG then G is abelian
- If G is a group that (ab)^n = a^n.b^n , for all a, b ϵG and for three consecutive integers n. Then prove that G is an abelian group.