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In mathematics, specifically group theory, the free product is an operation that takes two groups and and constructs a new group . The result contains both and as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from and into a group factor uniquely through a homomorphism from to . Unless one of the groups and 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.
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