Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G {\displaystyle G} and H {\displaystyle H} and constructs a new group G ∗ H {\displaystyle G*H} . The result contains both G {\displaystyle G} and H {\displaystyle H} 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 G {\displaystyle G} and H {\displaystyle H} into a group K {\displaystyle K} factor uniquely through a homomorphism from G ∗ H {\displaystyle G*H} to K {\displaystyle K} .