Pro-p group ($P=\underleftarrow{\lim }{P}_i$)

A pro-$p$ group is an inverse limit of finite $p$-groups. Every continuous action on a finite set or finite module factors through a finite $p$-group. In particular its action on ${\mu }_p$ is trivial when the field characteristic differs from $p$, because $\mathrm{Aut}({\mu }_p)$ has order $p-1$.

A Sylow pro-$p$ subgroup of a profinite group is a closed pro-$p$ subgroup whose images are Sylow $p$-subgroups in every finite quotient. Such subgroups exist and are conjugate. Every open subgroup containing one has index prime to $p$. They allow prime-to-$p$ finite extensions to detect $p$-primary Galois cohomology.