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 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.

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