 Pro-p group

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 **Pro-p group** is a type of mathematical object in the field of group theory, particularly in the area of profinite groups. More specifically, a Pro-p group is a topological group that is both locally compact and totally disconnected, and it is defined as an inverse limit of finite groups whose orders are powers of a prime \( p \).