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

   $$\xymatrix{&{G_{1}}\ar[dr]^{{j_{1}}}&\\ {G_{0}}\ar[ur]^{{i_{1}}}\ar[dr]_{{i_{2}}}&&{G}\\ &{G_{2}}\ar[ur]_{{j_{2}}}&}$$

2. 2.

   $G$ is universal with respect to the previous property, that is for any other group $G^{{\prime}}$ and homomorphisms $j_{k}^{{\prime}}\colon\thinspace G_{k}\to G^{{\prime}}$, $k=1,2$ that fit in such a commutative diagram there is a unique homomorphism $G\to G^{{\prime}}$ so that the following diagram commutes

   $$\xymatrix{&{G_{1}}\ar[dr]^{{j_{1}}}\ar@/^{1}pc/[drr]^{{j_{1}^{{\prime}}}}&&\\ {G_{0}}\ar[ur]^{{i_{1}}}\ar[dr]_{{i_{2}}}&&{G}\ar[r]^{{!}}&{G^{{\prime}}}\\ &{G_{2}}\ar[ur]_{{j_{2}}}\ar@/_{1}pc/[urr]_{{j_{2}^{{\prime}}}}&&}$$