Order of an element in a group
The order of an element a∈A in a group (A,*) is the least positive integer n for which an is egual to the identity element (u) of group $$ a^n = u $$
In a group, the order of an element may or may not exist.
If there exists no such integer, the element has infinite order.
Example
The group (Z4,+4) is composed of the set of integers Z4 = {0,1,2,3} and the addition operation (+4) mod4.
| a +4 b | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 |
| 1 | 1 | 2 | 3 | 0 |
| 2 | 2 | 3 | 0 | 1 |
| 3 | 3 | 0 | 1 | 2 |
The identity element of the group is zero.
$$ u = 0 $$
The order of element 0 is one because 01 is equal to the identity element (0).
$$ 0^1 = 0 $$
The order of element 1 is four because 14 is equal to the identity element (0).
$$ 1^4 = 1 + 1 + 1 + 1 = 0 $$
The order of element 2 is two because 22 is equal to the identity element (0).
$$ 2^2 = 2 + 2 = 0 $$
The order of element 3 is four because 34 is equal to the identity element (0).
$$ 3^4 = 3 + 3 + 3+ 3 = 0 $$
Example
The group (S,·) is a multiplicative where S={1,-1,i,-i} is a set with four complex number.
The identity element is 1
$$ u = 1 $$
The order of element 1 is one because 11 is equal to the identity element (u=1). The element 1 is the identity element.
$$ 1^1 = 1 $$
The order of element -1 is two because -12 is equal to the identity element (u=1).
$$ (-1)^2 = (-1) \cdot (-1) = 1 $$
The order of element i (imaginary unit) is four because i4 is equal to the identity element (u=1).
$$ i^4 = i \cdot i \cdot i \cdot i = $$
$$ = i^2 \cdot i \cdot i $$
$$ = -1 \cdot i \cdot i $$
$$ = -i \cdot i $$
$$ = -i^2 $$
$$ = -(-1) = 1 $$
The order of element -i is four because -i4 is equal to the identity element (u=1).
$$ -i^4 = (-i) \cdot (-i) \cdot (-i) \cdot (-i) = $$
$$= i^2 \cdot (-i) \cdot (-i) $$
$$= -1 \cdot (-i) \cdot (-i) $$
$$ = i \cdot (-i) = $$
$$ = -i^2 = -(-1) = 1 $$
Example
In the additive group (Z, +) each element a∈Z different from zero to ≠ 0 is an element of infinite order.
$$ (Z,+) $$
For example, given a=4 then na≠0 for all n>0
$$ 4 \ne 0 \\ 4 + 4 \ne 0 \\ 4 + 4 + 4 \ne 0 \\ \vdots \\ n \cdot 4 \ne 0 \ \ \ \ \forall \ n \ne 0 $$
