The construction of a free product is similar in spirit to the construction of a free group (the universal group with a given set of generators).
The free product is the coproduct in the category of groups.
That is, the free product plays the same role in group theory that disjoint union plays in set theory, or that the direct sum plays in module theory.
Therefore, the free product is not the coproduct in the category of abelian groups.
The free product is important in algebraic topology because of van Kampen's theorem, which states that the fundamental group of the union of two path-connected topological spaces whose intersection is also path-connected is always an amalgamated free product of the fundamental groups of the spaces.
Free products are also important in Bass–Serre theory, the study of groups acting by automorphisms on trees.
Specifically, any group acting with finite vertex stabilizers on a tree may be constructed from finite groups using amalgamated free products and HNN extensions.
The free product G ∗ H is the group whose elements are the reduced words in G and H, under the operation of concatenation followed by reduction.
In this case, G ∗ H is isomorphic to the free group generated by x and y.
Suppose that is a presentation for G (where SG is a set of generators and RG is a set of relations), and suppose that is a presentation for H. Then That is, G ∗ H is generated by the generators for G together with the generators for H, with relations consisting of the relations from G together with the relations from H (assume here no notational clashes so that these are in fact disjoint unions).
In particular, where Fn denotes the free group on n generators.
It is isomorphic to the free product of two cyclic groups:[1] The more general construction of free product with amalgamation is correspondingly a special kind of pushout in the same category.
are given as before, along with monomorphisms (i.e. injective group homomorphisms): where
In other words, take the smallest normal subgroup
containing all elements on the left-hand side of the above equation, which are tacitly being considered in
, is the quotient group The amalgamation has forced an identification between
This is the construction needed to compute the fundamental group of two connected spaces joined along a path-connected subspace, with
taking the role of the fundamental group of the subspace.
Karrass and Solitar have given a description of the subgroups of a free product with amalgamation.
Free products with amalgamation and a closely related notion of HNN extension are basic building blocks in Bass–Serre theory of groups acting on trees.
One may similarly define free products of other algebraic structures than groups, including algebras over a field.
Free products of algebras of random variables play the same role in defining "freeness" in the theory of free probability that Cartesian products play in defining statistical independence in classical probability theory.