Prüfer rank

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In mathematics, the Prüfer rank of a pro-p group G is

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

where d(H) is the rank of the abelian group

H / Φ(H),

where Φ(H) is the Frattini subgroup of H.

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

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