Examples
The order of the identity element.
The order of (1,2,3) in S3.
The order of i in the multiplicative group of C*.
The order of 2 in (Z/5Z)*.
The order is 1.
The order is 3. Indeed, (1,2,3)3 = e but (1,2,3)2 = (1,3,2)
which is not e.
The order is 4.
The order is 4, as follows from
22 = 4, 23 = 3, 24 = 1.