Pro-p group

Summary

For a prime $p$, a pro-$p$ group is a profinite group that is an inverse limit of finite p-groups. The $p$-adic integers are the basic example. Pro-$p$ groups occur as Sylow subgroups of profinite groups and as Galois groups of maximal $p$-extensions of fields. Every $p$-adic analytic group has an open subgroup that is a pro-$p$ group. Much of the theory of finite $p$-groups carries over to them.