Order of a Group
Finite and Infinite Groups
If a group contains a finite number of distinct elements, it is called a finite group. Otherwise it is an infinite group.
In other words, a group $$\left( {G, * } \right)$$ is said to be finite or infinite according to whether the underlying set $$G$$ is finite or infinite.
Order of a Group
The number of elements in a finite group is called the order of the group. An infinite group is said to be of infinite order.
Note: It should be noted that the smallest group for a given composition is the set $$\left\{ e \right\}$$ consisting of the identity element $$e$$ alone.
Example:

$$G = \left\{ {1,\omega ,{\omega ^2}} \right\}$$ $${\omega^2} = 1$$ is the example of a finite group with order 3.

$$G = \left\{ { \pm 1, \pm i} \right\}$$ $${i^2} = – 1$$ is the another example of a finite group with order 4.