Pro-p group

Remarks

Equivalently, $G$ is isomorphic as a topological group to an inverse limit of finite $p$-groups. Example: the p-adic integers $\Z_p = \varprojlim \Z/p^n\Z$. Its open subgroups are the subgroups $p^n \Z_p$, and each quotient $\Z_p / p^n \Z_p \cong \Z/p^n\Z$ has order $p^n$. Non-example: the profinite completion $\hat\Z$ of $\Z$ is profinite, but it has the open normal subgroup $6\hat\Z$ with quotient $\Z/6\Z$, which is not a $p$-group for any prime $p$. Every finite $p$-group with the discrete topology is a pro-$p$ group.