Question

Show that there exists infinite number of commutative group is an infinite order

Answer 1

Solution:

A group is said to be Commutative , if elements of  group follow this Property:

That is if A and B are two elements of group, then

⇒A + B= B +A→→With respect to addition

⇒ A × B=B ×A→→→With respect to Multiplication

So, there exists infinite number of groups , which follows commutative property called Abelian group and have infinite order.

A group is said to be have infinite order , if number of elements in it are infinite.

For example,

1. Set of integers having addition as an Operation= I(+)={.......-5,-4,-3....0,1,2,3,.......}→→Infinite Abelian group

2. Set of Rational numbers having addition as well as multiplication as an Operation→$Q_{+}, [Q_{*}-(0)]$

3. Set of Real numbers having addition as well as multiplication as an Operation→$R_{+}, [R_{*}-(0)]$