Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts Given a pair of groups Grp, or more generally an -tuple of groups, what is traditionally called their free product and denoted or generally is really their coproduct in Grp: (Compare the dual notion of “direct product of groups”, which are really category theoretic products of groups.) Informally, the “free product” is the group whose elements are freely generated from those of the , subject only to the relations given by the group operations in each of these groups. More generally, if each of the given groups is equipped with a homomorphism from a fixed group (often taken to be monomorphisms, hence injections, hence subgroup-inclusions and reducing to the previous situation when is the trivial group) then what is traditionally called their amalgamated free product (or similar), and denoted is the corresponding pushout (or generally the colimit) in the category Grp, hence the unique group, up to isomorphism, which receives homomorphisms from the such that their precompositions with the respective all agree, is universal with this property in that for any other group receiving such homomorphisms these factor through the respective via a single and unique comparison homomorphism, shown as a dashed arrow in the following diagram: