Prüfer rank

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

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 ${\displaystyle H}$ .

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.

References

1. Yamagishi, Masakazu (2007), "An analogue of the Nielsen-Schreier formula for pro-p-groups", Archiv der Mathematik, 88 (4): 304–315, doi:10.1007/s00013-006-1878-4, MR 2311837, S2CID 120424528, Zbl 1119.20035.