# Order of an Element of a Group

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

If there exist no such integer, we say that $a$ is a finite order or zero order. We shall use the notation $O\left( a \right)$ for the order of $a$.

Note that the only element of order one in a  group is the identity element $e$.

Important Note: If there exist a positive integer $m$ such that ${a^m} = e$, then the order of $a$ is definitely finite. Also we must have $O\left( a \right) \leqslant m$. When ${a^m} = e$, then the question of order of a being greater than $m$ does not arise. At the most it can be equal to $m$. If $m$ itself is the least positive such that ${a^m} = e$, then we will have $O\left( a \right) = m$.

For Example:
Find the order of each element of the multiplicative group $G$, where $G = \left\{ {1, - 1,i, - i} \right\}$
Since 1 is the identity element, its order is 1.

Now

Hence order of -1 is 2.

Again

Therefore order of $i$ is 4.

Similarly,

Hence order of -1 is 4.