Abelian Group and Group Examples

In this class, we discussed the Abelian Group and group examples.

The reader should have prior knowledge of monoid and group. Click Here.

Abelian Group:

A group [G, *] is said to be an abelian or commutative group. if a*b = b*a ∀ a,b ∈ A.

Abelian group:

1) Closure

2) Associativity

3) Identity element

4) Inverse element

5) commutative

Example:

[Z, +]

Z is a set of integers.

1) closure satisfied. If we Add any two elements, we get the integer value.

Addition satisfies associativity.

The identity element with addition is 0.

The inverse of an element a is “-a”.

Addition satisfies the commutative property.

Group Examples:

1) the set of all cube roots of unity is an abelian group with multiplication operation.

The diagram below shows the table for cube roots of the unity multiplication operation.

The cube roots satisfy the closure property.

The cube roots are associative with multiplication operations.

The identity element is 1.

The inverse of w is w^2

The inverse of 1 is 1

The inverse of w^2 = w.

The cube roots satisfy commutative with multiplication.

Example 2:

Given operational table

Show that the operation “.” is a group on set A = {a, b, c, d}.

Solution:

The below diagram shows the table.

The operation “.” satisfies closure and associative property.

The identity element is a.

Inverse elements are (a, a), (b, d), (c, c)