Prüfer rank

The Prüfer rank of pro-p-group ${\displaystyle G}$ is

${\displaystyle \sup\{d(H)|H\leq G\}}$

where ${\displaystyle d(H)}$ is the rank of the abelian group

${\displaystyle H/\Phi (H)}$,

where ${\displaystyle \Phi (H)}$ is the Frattini subgroup of ${\displaystyle H}$.

As the Frattini subgroup of ${\displaystyle H}$ can be thought of as the group of non-generating elements of ${\displaystyle H}$, it can be seen that ${\displaystyle d(H)}$ will be equal to the size of any minimal generating set of ${\displaystyle H}$.

Those profinite groups with finite Prüfer rank are more amenable to analysis.

Specifically in the case of finitely generated pro-p groups, having finite Prüfer rank is equivalent to having an open normal subgroup that is powerful. In turn these are precisely the class of pro-p groups that are p-adic analytic - that is groups that can be imbued with a p-adic manifold structure.