Past exam of the topology and groups course of the University of Oxford / 2023 / 2 / a / ii / Solution

Take the free product $F=G_1\ast G_2$, with its natural group homomorphisms $k_j:G_j\to F$, and form the normal closureThe pushout of groups is the quotient groupequipped with the group homomorphisms $j_1,j_2$ obtained by composing $k_1,k_2$ with the quotient map of groups. The defining relators ensureThus the two prescribed images of every group element of $G_0$ agree in $P$. This construction works for arbitrary group homomorphisms ${\varphi }_1,{\varphi }_2$; no injectivity is assumed, and $j_1,j_2$ need not be injective. Its universal property is proved in the next part.