The group exponent of a group 
is the least positive integer 
, if one exists, such that 
for every group element 
, where 
is the identity element.
If no such 
exists, the group has infinite exponent. A finite group
exponent is the least common multiple of
the orders of all group
elements (Vaughan-Lee 1993).
For a finite group, the group exponent divides the group order by Lagrange's
group theorem. A cyclic group of group
order 
has group exponent 
.
The trivial group has group exponent 1. Having every
group element of finite order
does not by itself imply finite group exponent.
