If is a group and is an element of group , the order (or period) of is the least positive integer , such that
If there exists no such integer, we say that is a finite order or zero order. We shall use the notation for the order of .
Note that the only element of order one in a group is the identity element .
Important Note: If there exists a positive integer such that , then the order of is definitely finite. Also we must have . When , then the question of order of a being greater than does not arise. At the most it can be equal to . If itself is the least positive such that , then we will have .
Find the order of each element of the multiplicative group , where
Since 1 is the identity element, its order is 1.
Hence the order of -1 is 2.
Therefore the order of is 4.
Hence the order of -1 is 4.