The group order of a group 
is its number of elements,
denoted 
.
If the order of a group is a finite
number, the group is said to be a finite
group.
The order of an element 
of a group 
is the smallest positive integer 
such that 
, where 
is the identity element,
or is infinite if no such 
exists. In general, finding the order of the element
of a group is at least as hard as factoring
(Meijer 1996). However, the problem becomes significantly easier if 
and the factorization
of 
are known. Under these circumstances,
efficient algorithms are known (Cohen 1993).
The group order of a permutation group or a named group can be computed in the Wolfram Language using the function GroupOrder[group].
