free product

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).