Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 19I / a / i / Solution

The pushout of groups isIts maps from $G_1,G_2$ agree on $H$, and every other pair of group homomorphisms agreeing on $H$ factors through one unique homomorphism from $G$. This is the universal property. When the $i_j$ are injective, it is the amalgamated free product, and the two factors and the identified copy of $H$ embed in it. Some conventions use this name for the quotient even without injectivity; the embedding assertion requires injectivity.