The pro-p completion of an abstract group is the inverse limit over its finite p-group quotients

Fix a prime $p$, and let $G$ be an abstract group. Its pro-$p$ completion is

where the indexing subgroups are ordered by reverse inclusion and the transition map $G\mathrm{/}{N}^{\mathrm{\prime }}\to G\mathrm{/}N$ is the natural quotient homomorphism whenever $N^{\mathrm{\prime }}\subseteq N$. This is a directed system: it contains $G$, and $N_1\cap N_2$ is a common upper bound in the reverse-inclusion order because $G\mathrm{/}(N_1\cap N_2)$ embeds in the finite $p$-group $G\mathrm{/}{N}_1\times G\mathrm{/}{N}_2$. Give every quotient the discrete topology and the limit its inverse-limit topology.

This is the $p$-primary analogue of The profinite completion is the inverse limit of the finite quotients G over N: only finite quotients whose order is a power of $p$ are retained, and the resulting topological group is pro-$p$ by A pro-p group is a profinite group that is an inverse limit of finite p-groups.