Abelian Group

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 ab is also in A.
Associativity:
For all a, b and c in A,
the equation (ab) • c = a • (bc) holds.
Identity element:
There exists an element e in A, such that for all elements a in A,
the equation ea = ae = a holds.
Inverse element:
For each a in A, there exists an element b in A
such that ab = ba = e, where e is the identity element.
Commutativity:
For all a, b in A, ab = ba.

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“.


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