The free product 
of groups 
and 
is the set of elements of the form
 |
where 
and 
, with 
and 
possibly equal to 
, the identity element
of 
and 
.
Free products of more than two groups are defined recursively, i.e.,
 |
The free group 
is the free product of 
with itself 
times.
The notion of free products can be generalized from groups to categories.
