In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially in the area of
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
known as
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, the Prüfer rank of a
pro-p group
In mathematics, a pro-''p'' group (for some prime number ''p'') is a profinite group G such that for any open normal subgroup N\triangleleft G the quotient group G/N is a ''p''-group. Note that, as profinite groups are compact, the open subgroups ...
measures the size of a group in terms of the ranks of its
elementary abelian
In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same order. This common order must be a prime number, and the elementary abelian groups in which ...
section
Section, Sectioning, or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
s.
[.] The rank is well behaved and helps to define analytic pro-p-groups. The term is named after
Heinz Prüfer
Ernst Paul Heinz Prüfer (10 November 1896 – 7 April 1934) was a German Jewish mathematician born in Wilhelmshaven. His major contributions were on abelian groups, graph theory, algebraic numbers, knot theory and Sturm–Liouville theory.
In 19 ...
.
Definition
The Prüfer rank of
pro-p-group is
::
where
is the
rank
A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial.
People Formal ranks
* Academic rank
* Corporate title
* Diplomatic rank
* Hierarchy ...
of the abelian group
:
,
where
is the
Frattini subgroup
In mathematics, particularly in group theory, the Frattini subgroup \Phi(G) of a group is the intersection of all maximal subgroups of . For the case that has no maximal subgroups, for example the trivial group or a Prüfer group, it is def ...
of
.
As the Frattini subgroup of
can be thought of as the group of non-generating elements of
, it can be seen that
will be equal to the ''size of any minimal generating set'' of
.
Properties
Those
profinite group
In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups.
The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...
s with finite Prüfer rank are more amenable to analysis.
Specifically in the case of finitely generated
pro-p group
In mathematics, a pro-''p'' group (for some prime number ''p'') is a profinite group G such that for any open normal subgroup N\triangleleft G the quotient group G/N is a ''p''-group. Note that, as profinite groups are compact, the open subgroups ...
s, having finite Prüfer rank is equivalent to having an
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979
* ''Open'' (Go ...
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
that is
powerful. In turn these are precisely the class of
pro-p group
In mathematics, a pro-''p'' group (for some prime number ''p'') is a profinite group G such that for any open normal subgroup N\triangleleft G the quotient group G/N is a ''p''-group. Note that, as profinite groups are compact, the open subgroups ...
s that are
p-adic
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infin ...
analytic – that is groups that can be imbued with a
p-adic
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infin ...
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
structure.
References
{{DEFAULTSORT:Prufer Rank
Infinite group theory