Discrete group
Encyclopedia
In mathematics
, a discrete group is a group
G equipped with the discrete topology. With this topology G becomes a topological group
. A discrete subgroup of a topological group G is a subgroup
H whose relative topology is the discrete one. For example, the integer
s, Z, form a discrete subgroup of the reals, R (with the standard metric topology
), but the rational number
s, Q, do not.
Any group can be given the discrete topology. Since every map from a discrete space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphism
s between the underlying groups. Hence, there is an isomorphism
between the category of groups
and the category of discrete groups. Discrete groups can therefore be identified with their underlying (non-topological) groups. With this in mind, the term discrete group theory is used to refer to the study of groups without topological structure, in contradistinction to topological or Lie group theory. It is divided, logically but also technically, into finite group theory, and infinite group theory.
There are some occasions when a topological group
or Lie group
is usefully endowed with the discrete topology, 'against nature'. This happens for example in the theory of the Bohr compactification
, and in group cohomology
theory of Lie groups.
, one need only look at a single point to determine if the topological group is discrete. In particular, a topological group is discrete if and only if the singleton containing the identity is an open set
.
A discrete group is the same thing as a zero-dimensional Lie group
(uncountable discrete groups are not second-countable so authors who require Lie groups to satisfy this axiom do not regard these groups as Lie groups). The identity component
of a discrete group is just the trivial subgroup
while the group of components is isomorphic to the group itself.
Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete. It follows that every finite subgroup of a Hausdorff group is discrete.
A discrete subgroup H of G is cocompact if there is a compact subset K of G such that HK = G.
Discrete normal subgroup
s play an important role in the theory of covering groups and locally isomorphic groups. A discrete normal subgroup of a connected
group G necessarily lies in the center
of G and is therefore abelian
.
Other properties:
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...
, a discrete group is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G equipped with the discrete topology. With this topology G becomes a topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
. A discrete subgroup of a topological group G is a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
H whose relative topology is the discrete one. For example, the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s, Z, form a discrete subgroup of the reals, R (with the standard metric topology
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
), but the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, Q, do not.
Any group can be given the discrete topology. Since every map from a discrete space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
s between the underlying groups. Hence, there is an isomorphism
Isomorphism of categories
In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1D and GF = 1C. This means that both the objects and the morphisms of C and D stand in a one to one correspondence to each other...
between the category of groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
and the category of discrete groups. Discrete groups can therefore be identified with their underlying (non-topological) groups. With this in mind, the term discrete group theory is used to refer to the study of groups without topological structure, in contradistinction to topological or Lie group theory. It is divided, logically but also technically, into finite group theory, and infinite group theory.
There are some occasions when a topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
or Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
is usefully endowed with the discrete topology, 'against nature'. This happens for example in the theory of the Bohr compactification
Bohr compactification
In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H...
, and in group cohomology
Group cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
theory of Lie groups.
Properties
Since topological groups are homogeneousHomogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
, one need only look at a single point to determine if the topological group is discrete. In particular, a topological group is discrete if and only if the singleton containing the identity is an open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
.
A discrete group is the same thing as a zero-dimensional Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
(uncountable discrete groups are not second-countable so authors who require Lie groups to satisfy this axiom do not regard these groups as Lie groups). The identity component
Identity component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group...
of a discrete group is just the trivial subgroup
Trivial group
In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...
while the group of components is isomorphic to the group itself.
Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete. It follows that every finite subgroup of a Hausdorff group is discrete.
A discrete subgroup H of G is cocompact if there is a compact subset K of G such that HK = G.
Discrete normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
s play an important role in the theory of covering groups and locally isomorphic groups. A discrete normal subgroup of a connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
group G necessarily lies in the center
Center (group theory)
In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
of G and is therefore abelian
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...
.
Other properties:
- every discrete group is totally disconnected
- every subgroup of a discrete group is discrete.
- every quotientQuotient groupIn mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
of a discrete group is discrete. - the product of a finite number of discrete groups is discrete.
- a discrete group is compactCompact groupIn mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
if and only if it is finite. - every discrete group is locally compactLocally compact groupIn mathematics, a locally compact group is a topological group G which is locally compact as a topological space. Locally compact groups are important because they have a natural measure called the Haar measure. This allows one to define integrals of functions on G.Many of the results of finite...
. - every discrete subgroup of a Hausdorff group is closed.
- every discrete subgroup of a compact Hausdorff group is finite.
Examples
- Frieze groupFrieze groupA frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
s and wallpaper groupWallpaper groupA wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
s are discrete subgroups of the isometry group of the Euclidean plane. Wallpaper groups are cocompact, but Frieze groups are not. - A space groupSpace groupIn mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...
is a discrete subgroup of the isometry group of Euclidean space of some dimension. - A crystallographic group usually means a cocompact, discrete subgroup of the isometries of some Euclidean space. Sometimes, however, a crystallographic group can be a cocompact discrete subgroup of a nilpotent or solvable Lie group.
- Every triangle groupTriangle groupIn mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle...
T is a discrete subgroup of the isometry group of the sphere (when T is finite), the Euclidean plane (when T has a Z + Z subgroup of finite indexIndex of a subgroupIn mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
), or the hyperbolic planeHyperbolic spaceIn mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
. - Fuchsian groupFuchsian groupIn mathematics, a Fuchsian group is a discrete subgroup of PSL. The group PSL can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting...
s are, by definition, discrete subgroups of the isometry group of the hyperbolic plane.- A Fuchsian group that preserves orientation and acts on the upper half-plane model of the hyperbolic plane is a discrete subgroup of the Lie group PSL(2,R), the group of orientation preserving isometries of the upper half-plane model of the hyperbolic plane.
- A Fuchsian group is sometimes considered as a special case of a Kleinian groupKleinian groupIn mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
, by embedding the hyperbolic plane isometrically into three dimensional hyperbolic space and extending the group action on the plane to the whole space. - The modular groupModular groupIn mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
is PSL(2,Z), thought of as a discrete subgroup of PSL(2,R). The modular group is a lattice in PSL(2,R), but it is not cocompact.
- Kleinian groupKleinian groupIn mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
s are, by definition, discrete subgroups of the isometry group of hyperbolic 3-space. These include quasi-Fuchsian groupQuasi-Fuchsian groupIn the mathematical theory of Kleinian groups, a quasi-Fuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve. If the limit set is equal to the Jordan curve the quasi-Fuchsian group is said to be of type one, and otherwise it is said to be of type two...
s.- A Kleinian group that preserves orientation and acts on the upper half space model of hyperbolic 3-space is a discrete subgroup of the Lie group PSL(2,C), the group of orientation preserving isometries of the upper half-space model of hyperbolic 3-space.
- A latticeLattice (discrete subgroup)In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...
in a Lie groupLie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
is a discrete subgroup such that the Haar measureHaar measureIn mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
of the quotient space is finite.
See also
- crystallographic point groupCrystallographic point groupIn crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...
- congruence subgroupCongruence subgroupIn mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2x2 integer matrices of determinant 1, such that the off-diagonal entries are even.An importance class of congruence...
- arithmetic groupArithmetic groupIn mathematics, an arithmetic group in a linear algebraic group G defined over a number field K is a subgroup Γ of G that is commensurable with G, where O is the ring of integers of K. Here two subgroups A and B of a group are commensurable when their intersection has finite index in each of them...
- geometric group theoryGeometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
- computational group theoryComputational group theoryIn mathematics, computational group theory is the study ofgroups by means of computers. It is concernedwith designing and analysing algorithms anddata structures to compute information about groups...
- freely discontinuous
- free regular setFree regular setIn mathematics, a free regular set is a subset of a topological space that is acted upon disjointly under a given group action.To be more precise, let X be a topological space. Let G be a group of homeomorphisms from X to X...