Internal amalgamated free product

Definition

Suppose $G$ is a group and $G_{1}$ and $G_{2}$ are two subgroups of $G$ with intersection equal to a subgroup $H$. $G$ is an internal free product of $G_{1}$ and $G_{2}$ amalgamated at $H$ if the following condition is satisfied: given any word with the property that its letters alternate between $G_{1}$ and $G_{2}$, such that the word equals the identity element, all the elements of the word are from $H$.

The internal amalgamated free product of $G_{1}$ and $G_{2}$ over the subgroup $H$ is isomorphic to their external amalgamated free product.