Finite pro-p presentation ($G=F_p(x_1,\dots ,x_d)/\overline{⟨r_1,\dots ,r_s{⟩}^{F_p}}$)

A finite generating set and finitely many relators in a free pro-p group, divided out by their closed normal closure. Relators are elements of the free pro-p group; they need not be finite ordinary words. A pro-p group with such a presentation is finitely presented in the pro-$p$ category.

Suppose $K$ is an open normal subgroup of a pro-p group $G$ and has a finite pro-p presentation. Choose lifts of generators of the finite quotient $G/K$. Add finitely many relators giving their conjugation and inverse conjugation on the generators of $K$, and replace each quotient relator by its value in $K$. The resulting finite finite pro-p presentation maps onto $G$. Its closed subgroup generated by the kernel generators is isomorphic to $K$, and its quotient by that subgroup is exactly $G/K$, so the map is also injective.