Given presentations for the , it is straightforward to find a presentation of ; if , where each is the free group on the set and is the normal subgroup of generated by the subset , then the free product
is presented by the disjoint unions of the and the . As with anything satisfying a universal property, the result (up to a unique coherent isomorphism) does not depend on the presentations.
The fact that free products always exist now follows from the fact that any group has a presentation; we can always take to be the underlying set of and take to be the set of all words in that equal the identity in . The value of the more general construction above is that one often has much smaller and to work with. Even if is infinite, the and might be finite (in the strictest sense), making this part of finite mathematics and directly subject to the methods of combinatorial group theory.
Revised on October 8, 2010 15:17:46
by Tim Porter