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.