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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.{{citation
Definition
The Prüfer rank of pro-p-group 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.
References
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
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