# Order of an Element of a Group

If is a group and is an element of group , the order (or period) of is the least positive integer such that

If there exist 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 exist 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 .

__For Example__**:**

Find the order of each element of the multiplicative group , where

Since **1** is the identity element, its order is **1**.

Now

Hence order of

**-1**is

**2**.

Again

Therefore order of is

**4**.

Similarly,

Hence order of

**-1**is

**4.**