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 [math]\displaystyle{ G }[/math] is
- [math]\displaystyle{ \sup\{d(H)|H\leq G\} }[/math]
where [math]\displaystyle{ d(H) }[/math] is the rank of the abelian group
- [math]\displaystyle{ H/\Phi(H) }[/math],
where [math]\displaystyle{ \Phi(H) }[/math] is the Frattini subgroup of [math]\displaystyle{ H }[/math].
As the Frattini subgroup of [math]\displaystyle{ H }[/math] can be thought of as the group of non-generating elements of [math]\displaystyle{ H }[/math], it can be seen that [math]\displaystyle{ d(H) }[/math] will be equal to the size of any minimal generating set of [math]\displaystyle{ 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.
References
- ↑ 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.
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