Pushout of groups
= Pushout of groups
{title2=$G_1*_{H}G_2$}
For homomorphisms $i_j:H\to G_j$, the <pushout of groups> is $(G_1*G_2)/\langle\!\langle i_1(h)i_2(h)^{-1}:h\in H\rangle\!\rangle$. It has the <universal property> for homomorphisms out of $G_1,G_2$ agreeing on $H$. When the maps $i_j$ are injective, it is the <amalgamated free product>, with the factors and common subgroup embedded.