In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties. Unless one of the groups G and H is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group (the most general group that can be made from 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. Even if the groups are commutative, their free product is not, unless one of the two groups is the trivial group. 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 is always an amalgamated free product of the fundamental groups of the spaces. In particular, the fundamental group of the wedge sum of two spaces (i.e. the space obtained by joining two spaces together at a single point) is simply the 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. Using the action of the modular group on a certain tessellation of the hyperbolic plane, it follows from this theory that the modular group is isomorphic to the free product of cyclic groups of orders 4 and 6 amalgamated over a cyclic group of order 2.
The free product (= coproduct) of groups is nicely set in the context of Categories and Groupoids in Philip Higgins 1971 book referenced below. The point is that a disjoint union of groups is not a group but it is a groupoid. A groupoid has a universal group and the universal group of a disjoint union of groups is the free (= coproduct) of the groups.
If G and H are groups, a word in G and H is a product of the form
where each si is either an element of G or an element of H. Such a word may be reduced using the following operations:
- Remove an instance of the identity element (of either G or H).
- Replace a pair of the form g1g2 by its product in G, or a pair h1h2 by its product in H.
Every reduced word is an alternating product of elements of G and elements of H, e.g.
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.
For example, if G is the infinite cyclic group <x>, and H is the infinite cyclic group <y>, then every element of G ∗ H is an alternating product of powers of x with powers of y. In this case, G ∗ H is isomorphic to the free group generated by x and y.
is a presentation for G (where RG is a set of generators and SG 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).
For example, suppose that G is a cyclic group of order 4,
and H is a cyclic group of order 5
Then G ∗ H is the infinite group
Because there are no relations in a free group, the free product of free groups is always a free group. In particular,
where Fn denotes the free group on n generators.
Generalization: Free product with amalgamation
where F is some arbitrary group. Start with the free product G ∗ H and adjoin as relations
for every f in F. In other words take the smallest normal subgroup N of G ∗ H containing all elements on the left-hand side of the above equation, which are tacitly being considered in G ∗ H by means of the inclusions of G and H in their free product. The free product with amalgamation of G and H, with respect to φ and ψ, is the quotient group
The amalgamation has forced an identification between φ(F) in G with ψ(F) in H, element by element. This is the construction needed to compute the fundamental group of two connected spaces joined along a connected subspace, with F taking the role of the fundamental group of the subspace. See: Seifert–van Kampen theorem.
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.
In other branches
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.