# free product with amalgamated subgroup

###### Definition 1.

Let $G_{k}$, $k=0,1,2$ be groups and $i_{k}\colon\thinspace G_{0}\to G_{i}$, $k=1,2$ be monomorphisms. The free product of $G_{1}$ and $G_{2}$ with amalgamated subgroup $G_{0}$, is defined to be a group $G$ that has the following two properties

1. 1.

there are homomorphisms $j_{k}\colon\thinspace G_{k}\to G$, $k=1,2$ that make the following diagram commute