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

Example:

1. The set of all integers Z is an abelian group with respect to addition.
2. 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.