Free Product

In group theory, the free product is an operation that takes two groups G and H and builds a new group, written G star H, containing both as subgroups, generated by their combined elements, and universal in the sense that any two homomorphisms out of G and H into a third group factor uniquely through it. Unless one of the two groups is trivial the free product is always infinite, and its construction parallels that of a free group on a given set of generators; it serves as the coproduct in the category of groups, playing there the role that disjoint union plays for sets. The free product is important in algebraic topology through van Kampen's theorem, which identifies the fundamental group of a union of two suitably connected spaces as an amalgamated free product of their individual fundamental groups, and it also underlies Bass-Serre theory, the study of groups acting on trees.