Group Order
The order of a group (S,*) reflects the count of elements within set S, essentially measuring the set's size or cardinality, symbolized as $ |S| $.
Groups may possess either a finite or an infinite order, contingent upon the elements comprising set S.
An Illustrative Example
The group (Z8,+8) is formed from the integer set Z8={0,1,2,3,4,5,6,7}, employing addition modulo 8 (+8) as its operation.
$$ (Z_8,+_8) $$
This constitutes a finite group, as it encompasses a limited quantity of elements.
The set Z8 contains 8 elements.
Consequently, the group's order stands at 8.
Example 2
The group (Z,+) comprises the entire set of integers Z, utilizing addition as its operation.
$$ (Z,+) $$
This group extends into infinity, given the limitless nature of the integer set Z.
Thus, this group's order is deemed infinite.
And the list goes on.
