Prüfer rank

Template:One source In mathematics, especially in the area of algebra known as group theory, the Prüfer rank of a pro-p group measures the size of a group in terms of the ranks of its elementary abelian sections.<ref>Template:Citation.</ref> The rank is well behaved and helps to define analytic pro-p-groups. The term is named after Heinz Prüfer.

Definition

The Prüfer rank of pro-p-group <math>G</math> is

<math>\sup\{d(H)|H\leq G\}</math>

where <math>d(H)</math> is the rank of the abelian group

<math>H/\Phi(H)</math>,

where <math>\Phi(H)</math> is the Frattini subgroup of <math>H</math>.

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

Properties

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.