In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a profinite group is a
topological group
In mathematics, topological groups are logically 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 st ...
that is in a certain sense assembled from a system of
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or ma ...
s.
The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists
such that every group in the system can be generated by
elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are
Lagrange's theorem and the
Sylow theorems.
To construct a profinite group one needs a system of finite groups and
group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
, in which case the finite groups will appear as
quotient groups of the resulting profinite group; in a sense, these quotients approximate the profinite group.
Important examples of profinite groups are the
additive groups of
''p''-adic integers and the
Galois groups of infinite-degree
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s.
Every profinite group is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
and
totally disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
. A non-compact generalization of the concept is that of
locally profinite groups. Even more general are the
totally disconnected groups.
Definition
Profinite groups can be defined in either of two equivalent ways.
First definition (constructive)
A profinite group is a topological group that is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the
inverse limit of an
inverse system of
discrete finite groups. In this context, an inverse system consists of a
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
, a collection of finite groups
, each having the
discrete topology, and a collection of
homomorphisms
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same ...
such that
is the identity on
and the collection satisfies the composition property
. The inverse limit is the set:
:
equipped with the
relative product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
.
One can also define the inverse limit in terms of a
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
. In
categorical terms, this is a special case of a
cofiltered limit construction.
Second definition (axiomatic)
A profinite group is a
Hausdorff,
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
, and
totally disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
topological group:
that is, a topological group that is also a
Stone space.
Profinite completion
Given an arbitrary group
, there is a related profinite group
, the profinite completion of
.
It is defined as the inverse limit of the groups
, where
runs through the
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 G ...
s in
of finite
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
(these normal subgroups are
partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients).
There is a natural homomorphism
, and the image of
under this homomorphism is
dense in
. The homomorphism
is injective if and only if the group
is
residually finite {{unsourced, date=September 2022
In the mathematical field of group theory, a group ''G'' is residually finite or finitely approximable if for every element ''g'' that is not the identity in ''G'' there is a homomorphism ''h'' from ''G'' to a finit ...
(i.e.,
, where the intersection runs through all normal subgroups of finite index).
The homomorphism
is characterized by the following
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
: given any profinite group
and any continuous group homomorphism
where
is given the smallest topology compatible with group operations in which its normal subgroups of finite index are open, there exists a unique
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
group homomorphism
with
.
Equivalence
Any group constructed by the first definition satisfies the axioms in the second definition.
Conversely, any group satisfying these axioms can be constructed as an inverse limit according to the first definition using the inverse limit
where
ranges through the open
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 G ...
s of
ordered by (reverse) inclusion. In other words,
is its own profinite completion.
Surjective systems
In practice, the inverse system of finite groups is almost always ''surjective'', that is, all its maps are surjective. Without loss of generality, we may consider only surjective systems, since given any inverse system, we can first construct its profinite group
, then ''reconstruct'' it as its own profinite completion.
Examples
* Finite groups are profinite, if given the
discrete topology.
* The group of
''p''-adic integers under addition is profinite (in fact
procyclic). It is the inverse limit of the finite groups
where ''n'' ranges over all
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s and the natural maps
for
. The topology on this profinite group is the same as the topology arising from the ''p''-adic valuation on
.
* The group of
profinite integers is the profinite completion of
. In detail, it is the inverse limit of the finite groups
where
with the modulo maps
for
. This group is the product of all the groups
, and it is the
absolute Galois group of any
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
.
* The
Galois theory of
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s of infinite degree gives rise naturally to Galois groups that are profinite. Specifically, if ''L''/''K'' is a
Galois extension, we consider the group ''G'' = Gal(''L''/''K'') consisting of all
field automorphisms of ''L'' that keep all elements of ''K'' fixed. This group is the inverse limit of the finite groups Gal(''F''/''K''), where ''F'' ranges over all intermediate fields such that ''F''/''K'' is a ''finite'' Galois extension. For the limit process, we use the restriction homomorphisms Gal(''F''
1/''K'') → Gal(''F''
2/''K''), where ''F''
2 ⊆ ''F''
1. The topology we obtain on Gal(''L''/''K'') is known as the ''Krull topology'' after
Wolfgang Krull. showed that ''every'' profinite group is isomorphic to one arising from the Galois theory of ''some'' field ''K'', but one cannot (yet) control which field ''K'' will be in this case. In fact, for many fields ''K'' one does not know in general precisely which
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or ma ...
s occur as Galois groups over ''K''. This is the
inverse Galois problem
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved.
There ...
for a field ''K''. (For some fields ''K'' the inverse Galois problem is settled, such as the field of
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s in one variable over the complex numbers.) Not every profinite group occurs as an
absolute Galois group of a field.
[Fried & Jarden (2008) p. 497]
* The
étale fundamental groups considered in algebraic geometry are also profinite groups, roughly speaking because the algebra can only 'see' finite coverings of an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
. The
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
s of
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, however, are in general not profinite: for any prescribed group, there is a 2-dimensional
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
whose fundamental group equals it.
* The automorphism group of a
locally finite rooted tree is profinite.
Properties and facts
*Every
product of (arbitrarily many) profinite groups is profinite; the topology arising from the profiniteness agrees with the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
. The inverse limit of an inverse system of profinite groups with continuous transition maps is profinite and the inverse limit functor is
exact on the category of profinite groups. Further, being profinite is an extension property.
*Every
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
subgroup of a profinite group is itself profinite; the topology arising from the profiniteness agrees with the
subspace topology. If ''N'' is a closed normal subgroup of a profinite group ''G'', then the
factor group
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, s ...
''G''/''N'' is profinite; the topology arising from the profiniteness agrees with the
quotient topology.
*Since every profinite group ''G'' is compact Hausdorff, we have a
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, thou ...
on ''G'', which allows us to measure the "size" of subsets of ''G'', compute certain
probabilities
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
, and
integrate functions on ''G''.
* A subgroup of a profinite group is open if and only if it is closed and has finite
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
.
*According to a theorem of
Nikolay Nikolov and
Dan Segal, in any topologically finitely generated profinite group (that is, a profinite group that has a
dense finitely generated subgroup
In algebra, a finitely generated group is a group (mathematics), group ''G'' that has some Finite set, finite generating set of a group, generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operatio ...
) the subgroups of finite index are open. This generalizes an earlier analogous result of
Jean-Pierre Serre for topologically finitely generated
pro-''p'' groups. The proof uses the
classification of finite simple groups.
*As an easy corollary of the Nikolov–Segal result above, ''any'' surjective discrete group homomorphism φ: ''G'' → ''H'' between profinite groups ''G'' and ''H'' is continuous as long as ''G'' is topologically finitely generated. Indeed, any open subgroup of ''H'' is of finite index, so its preimage in ''G'' is also of finite index, and hence it must be open.
*Suppose ''G'' and ''H'' are topologically finitely generated profinite groups that are isomorphic as discrete groups by an isomorphism ''ι''. Then ''ι'' is bijective and continuous by the above result. Furthermore, ''ι''
−1 is also continuous, so ''ι'' is a homeomorphism. Therefore the topology on a topologically finitely generated profinite group is uniquely determined by its ''algebraic'' structure.
Ind-finite groups
There is a notion of ind-finite group, which is the conceptual
dual to profinite groups; i.e. a group ''G'' is ind-finite if it is 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 an
inductive system of finite groups. (In particular, it is an
ind-group.) The usual terminology is different: a group ''G'' is called
locally finite if every
finitely generated subgroup is finite. This is equivalent, in fact, to being 'ind-finite'.
By applying
Pontryagin duality, one can see that
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
profinite groups are in duality with locally finite discrete abelian groups. The latter are just the abelian
torsion groups.
Projective profinite groups
A profinite group is projective if it has the
lifting property
In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly give ...
for every extension. This is equivalent to saying that ''G'' is projective if for every surjective morphism from a profinite ''H'' → ''G'' there is a
section ''G'' → ''H''.
[Serre (1997) p. 58][Fried & Jarden (2008) p. 207]
Projectivity for a profinite group ''G'' is equivalent to either of the two properties:
[
* the ]cohomological dimension In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory.
Cohomologic ...
cd(''G'') ≤ 1;
* for every prime ''p'' the Sylow ''p''-subgroups of ''G'' are free pro-''p''-groups.
Every projective profinite group can be realized as an absolute Galois group of a pseudo algebraically closed field. This result is due to Alexander Lubotzky and Lou van den Dries
Laurentius Petrus Dignus "Lou" van den Dries (born May 26, 1951) is a Dutch mathematician working in model theory. He is a professor emeritus of mathematics at the University of Illinois at Urbana–Champaign.
Education
Van den Dries began his u ...
.
Procyclic group
A profinite group is ''procyclic'' if it is topologically generated by a single element i.e., , the closure of the subgroup .
A topological group is procyclic if and only if where ranges over all 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 ...
s and is isomorphic to either or .
See also
*Locally cyclic group
In mathematics, a locally cyclic group is a group (''G'', *) in which every finitely generated subgroup is cyclic group, cyclic.
Some facts
* Every cyclic group is locally cyclic, and every locally cyclic group is abelian group, abelian.
* Every f ...
* Pro-''p'' group
* Profinite integer
*Residual property (mathematics) In the mathematical field of group theory, a group is residually ''X'' (where ''X'' is some property of groups) if it "can be recovered from groups with property ''X''".
Formally, a group ''G'' is residually ''X'' if for every non-trivial element ' ...
* Residually finite group
* Hausdorff completion
References
*
*.
*.
*.
*. Review of several books about profinite groups.
*.
*{{citation
, last = Waterhouse , first = William C. , author-link = William C. Waterhouse
, doi = 10.1090/S0002-9939-1974-0325587-3 , doi-access = free
, issue = 2
, journal = Proceedings of the American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages.
According to the ' ...
, pages = 639–640
, title = Profinite groups are Galois groups
, volume = 42
, year = 1974
, jstor = 2039560 , zbl=0281.20031
, publisher = American Mathematical Society .
Infinite group theory
Topological groups