In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, specifically in group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, the Prüfer p-group or the p-quasicyclic group or p^∞-group, Z(p^∞), for a prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 p is the unique p-group

P-group

In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...

 in which every element has p pth roots. The group is named after Heinz Prüfer

Heinz Prüfer

Ernst Paul Heinz Prüfer was a German mathematician, who worked on abelian groups, algebraic numbers, knot theory and Sturm-Liouville theory. His advisor was Issai Schur.- See also :* Prüfer manifold...

. It is a countable

Countable set

In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...

 abelian group

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

 which helps taxonomize infinite abelian groups.

The Prüfer p-group may be represented

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

 as a subgroup of the circle group, U(1), as the set of pⁿth roots of unity

Root of unity

In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...

 as n ranges over all non-negative integers:


Alternatively, the Prüfer p-group may be seen as the Sylow p-subgroup of Q/Z, consisting of those elements whose order is a power of p:

There is a presentation

The Prüfer p-group is the unique infinite p-group

P-group

In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...

 which is locally cyclic

Locally cyclic group

In group theory, a locally cyclic group is a group in which every finitely generated subgroup is cyclic.-Some facts:*Every cyclic group is locally cyclic, and every locally cyclic group is abelian....

 (every finite set of elements generates a cyclic group).

The Prüfer p-group is divisible

Divisible group

In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n...

.

In the language of universal algebra

Universal algebra

Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....

, an abelian group is subdirectly irreducible if and only if it is isomorphic to a finite cyclic p-group or isomorphic to a Prüfer group.

In the theory of locally compact topological groups the Prüfer p-group (endowed with the discrete topology) is the Pontryagin dual of the compact group of p-adic integers, and the group of p-adic integers is the Pontryagin dual of the Prüfer p-group.

The Prüfer p-groups for all primes p are the only infinite groups whose subgroups are totally ordered by inclusion. As there is no maximal subgroup

Maximal subgroup

In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.In group theory, a maximal subgroup H of a group G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words H is a maximal element of the partially...

 of a Prüfer p-group, it is its own Frattini subgroup

Frattini subgroup

In mathematics, the Frattini subgroup Φ of a group G is the intersection of all maximal subgroups of G. For the case that G is the trivial group e, which has no maximal subgroups, it is defined by Φ = e...

.


This sequence of inclusions expresses the Prüfer p-group as the direct limit

Direct limit

In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...

 of its finite subgroups.

As a -module, the Prüfer p-group is Artinian

Artinian module

In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself...

, but not Noetherian

Noetherian module

In abstract algebra, an Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion....

, and likewise as a group, it is Artinian but not Noetherian. It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every Artinian ring

Artinian ring

In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...

 is Noetherian).

See also

• p-adic integers, which can be defined as the inverse limit
  Inverse limit
  In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
  
   of the finite subgroups of the Prüfer p-group.
• Dyadic rational
  Dyadic rational
  In mathematics, a dyadic fraction or dyadic rational is a rational number whose denominator is a power of two, i.e., a number of the form a/2b where a is an integer and b is a natural number; for example, 1/2 or 3/8, but not 1/3...
  
  , rational numbers of the form a/2^{b. The Prüfer 2-group can be viewed as the dyadic rationals modulo 1.}