# 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:

1. $G = \left\{ {1,\omega ,{\omega ^2}} \right\}$ ${\omega^2} = 1$ is the example of a finite group with order 3.
2. $G = \left\{ { \pm 1, \pm i} \right\}$ ${i^2} = - 1$ is the another example of a finite group with order 4.