Prüfer rank explained

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.^[1] The rank is well behaved and helps to define analytic pro-p-groups. The term is named after Heinz Prüfer.

Definition

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

where

d(H) is the rank of the abelian group H/\Phi(H) ,

where

\Phi(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 .

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.

Notes and References

1. .