Free product

In abstract algebra, the free product of groups constructs a group from two or more given ones. Given for example groups G and H, the free product G*H could be constructed in this way: given presentations of G and of H, take the generators of G and of H, take the disjoint union of those, add the corresponding relations for G and for H. That is a presentation of G*H, the point being that there should be no interaction between G and H in the free product. If G and H are infinite cyclic groups, G*H is then a free group on two generators.

The free product applies to the theory of the fundamental group in topology. If connected spaces X and Y are joined at a single point, the fundamental group of the resulting space will be the free product of the fundamental groups of X and of Y. This is a special case of van Kampen's theorem. The modular group is a free product of cyclic groups of orders 2 and 3, up to a problem with defining it to within index 2. Groups can be shown to have free product structure by means of group actions on trees.

It may not look like an intrinsic definition. The dependence on the choice of presentation can be eliminated by showing that free product is the coproduct in the category of groups.

The more general construction of free product with amalgamation is correspondingly a pushout in the same category.