Free product

The free product of two groups G and H, denoted G * H, is the group generated by the disjoint union of their generators with no relations imposed between the generators of G and those of H. It is the group-theoretic analogue of the disjoint union of spaces: each group sits inside the free product as a subgroup, and the only relations are those already present in G and H separately. The free product is the coproduct in the category of groups, characterized by a universal property dual to the direct product. Like the free group, the free product is the default construction before constraints are added; its geometric counterpart is the amalgamated product, which glues groups along a shared subgroup.

The free product is the sound of two groups speaking past each other — each maintaining its own grammar, neither forced to translate. It is the group theory of coexistence without integration.