In mathematics, specifically in
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 ''p''-group or the ''p''-quasicyclic group or ''p''
∞-group, Z(''p''
∞), for a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
''p'' is the unique
''p''-group in which every element has ''p'' different ''p''-th roots.
The Prüfer ''p''-groups are
countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
abelian groups that are important in the classification of infinite abelian groups: they (along with the group of
rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...
) form the smallest building blocks of all
divisible groups.
The groups are named after
Heinz Prüfer, a German mathematician of the early 20th century.
Constructions of Z(''p''∞)
The Prüfer ''p''-group may be identified with the subgroup of the
circle group, U(1), consisting of all ''p''
''n''-th
roots of unity
In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group char ...
as ''n'' ranges over all non-negative integers:
:
The group operation here is the multiplication of
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
.
There is a
presentation
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
:
Here, the group operation in Z(''p''
∞) is written as multiplication.
Alternatively and equivalently, the Prüfer ''p''-group may be defined as the
Sylow ''p''-subgroup of the
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
Q''/''Z, consisting of those elements whose order is a power of ''p'':
:
(where Z
/''p''denotes the group of all rational numbers whose denominator is a power of ''p'', using addition of rational numbers as group operation).
For each natural number ''n'', consider the
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
Z/''p''
''n''Z and the embedding Z/''p''
''n''Z → Z/''p''
''n''+1Z induced by multiplication by ''p''. The
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
of this system is Z(''p''
∞):
:
If we perform the direct limit in the
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
of
topological group
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
s, then we need to impose a topology on each of the
, and take the
final topology on
. If we wish for
to be
Hausdorff, we must impose the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
on each of the
, resulting in
to have the discrete topology.
We can also write
:
where Q
''p'' denotes the additive group of
''p''-adic numbers and Z
''p'' is the subgroup of ''p''-adic integers.
Properties
The complete list of subgroups of the Prüfer ''p''-group Z(''p''
∞) = Z
/''p''Z is:
:
Here, each
is a cyclic subgroup of Z(''p''
∞) with ''p''
''n'' elements; it contains precisely those elements of Z(''p''
∞) whose
order divides ''p''
''n'' and corresponds to the set of ''p
n''-th roots of unity.
The Prüfer ''p''-groups are the only infinite groups whose subgroups are
totally ordered
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( r ...
by inclusion. This sequence of inclusions expresses the Prüfer ''p''-group as the
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
of its finite subgroups. As there is no
maximal subgroup of a Prüfer ''p''-group, it is its own
Frattini subgroup.
Given this list of subgroups, it is clear that the Prüfer ''p''-groups are
indecomposable (cannot be written as a
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of proper subgroups). More is true: the Prüfer ''p''-groups are
subdirectly irreducible. An abelian group is subdirectly irreducible if and only if it is isomorphic to a finite cyclic ''p''-group or to a Prüfer group.
The Prüfer ''p''-group is the unique infinite
''p''-group that is
locally cyclic (every finite set of elements generates a cyclic group). As seen above, all proper subgroups of Z(''p''
∞) are finite. The Prüfer ''p''-groups are the only infinite abelian groups with this property.
The Prüfer ''p''-groups are
divisible
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
. They play an important role in the classification of divisible groups; along with the rational numbers they are the simplest divisible groups. More precisely: an abelian group is divisible if and only if it is the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of a (possibly infinite) number of copies of Q and (possibly infinite) numbers of copies of Z(''p''
∞) for every prime ''p''. The (
cardinal
Cardinal or The Cardinal most commonly refers to
* Cardinalidae, a family of North and South American birds
**''Cardinalis'', genus of three species in the family Cardinalidae
***Northern cardinal, ''Cardinalis cardinalis'', the common cardinal of ...
) numbers of copies of Q and Z(''p''
∞) that are used in this direct sum determine the divisible group up to isomorphism.
As an abelian group (that is, as a
Z-module), Z(''p''
∞) is
Artinian but not
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
. It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every
Artinian ''ring'' is Noetherian).
The
endomorphism ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in ...
of Z(''p''
∞) is isomorphic to the ring of
''p''-adic integers Z
''p''.
In the theory of
locally compact topological groups the Prüfer ''p''-group (endowed with the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
) is the
Pontryagin dual
In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
of the compact group of
''p''-adic integers, and the group of ''p''-adic integers is the Pontryagin dual of the Prüfer ''p''-group.
[D. L. Armacost and W. L. Armacost]
On ''p''-thetic groups
, ''Pacific J. Math.'', 41, no. 2 (1972), 295–301
See also
*
''p''-adic integers, which can be defined as the
inverse limit of the finite subgroups of the Prüfer ''p''-group.
*
Dyadic rational, rational numbers of the form ''a''/2
''b''. The Prüfer 2-group can be viewed as the dyadic rationals modulo 1.
*
Cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
(
finite analogue)
*
Circle group (
uncountably infinite analogue)
Notes
References
*
*
*
*
{{DEFAULTSORT:Prufer Group
Abelian group theory
Infinite group theory
P-groups