Abelian Group

What is an Abelian Group?

An Abelian group is a group in which the operation is commutative.

For this reason, it is also called a commutative group.

It’s named after the Norwegian mathematician Niels Henrik Abel.

A Practical Example

The group consisting of the integers (Z) with addition (+) is an Abelian group, or commutative group.

$$ (Z,+) $$

This is because addition (+) satisfies the commutative property.

If we take any two integers, a and b, in the set Z, the following holds true:

$$ a+b = b+a \ \ \ \ \ \forall \ a,b \in Z $$

Note: Besides being commutative, the set of integers with addition (Z,+) forms a group because it satisfies the associative property $$ a + (b+c) = (a+b)+c $$ has an identity element $$ a+0 = 0+a = a $$ and each element has an inverse with respect to addition $$ a + (-a) = (-a)+a= 0 $$

Another Example

The set of integers with multiplication (Z,*) is not a group because 0 does not have an inverse (you cannot divide by zero).

$$ (Z,*) $$

Since it isn’t a group, the algebraic structure (Z,*) also cannot be an Abelian group.

And so on.