Discrete group - Citizendia

category of groups
Basic notions in group theory
subgroups, normal subgroups
quotient groups
group homomorphisms, kernel, image
(semi-)direct product, direct sum
types of groups
finite, infinite
discrete, continuous
multiplicative, additive
abelian, cyclic, simple, solvable

In mathematics, a discrete group is a group G equipped with the discrete topology. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, one can often define a direct product of objectsalready known giving a new one The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a finite group is a group which has finitely many elements Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " With this topology G becomes a 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 A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is For example, the integers, Z, form a discrete subgroup of the reals, R, but the rational numbers, Q, do not. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions

Any group can be given the discrete topology. Since every map from a discrete space is continuous, the topological homomorphisms of a discrete group are exactly the group homomorphisms of the underlying group. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function Hence, there is an isomorphism between the category of groups and the category of discrete groups. In Category theory, two categories C and D are isomorphic if there exist Functors F: C &rarr D and G In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such 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. In Mathematics, a finite group is a group which has finitely many elements Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups.

There are some occasions when a topological group or Lie group is usefully endowed with the discrete topology, 'against nature'. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group This happens for example in the theory of the Bohr compactification, and in group cohomology theory of Lie groups. In Mathematics, the Bohr compactification of a Topological group G is a compact Hausdorff topological group H that may be Canonically In Abstract algebra, Homological algebra, Algebraic topology and Algebraic number theory, as well as in applications to Group theory proper

1 Properties
2 Examples
3 Links to more examples
4 See also

Properties

Since topological groups are homogeneous, one need only look at a single point to determine if the group is discrete. In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group In particular, a topological group is discrete if and only if the singleton containing the identity is an open set. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in

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). In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Topology, a second-countable space is a Topological space satisfying the " second Axiom of countability " The identity component of a discrete group is just the trivial subgroup while the group of components is isomorphic to the group itself. In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity In Mathematics, a trivial group is a group consisting of a single element In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity

Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space 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 subgroups play an important role in the theory of covering groups and locally isomorphic groups. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, a covering group of a Topological group H is a Covering space G of H such that G is a topological A discrete normal subgroup of a connected group G necessarily lies in the center of G and is therefore abelian. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

Other properties:

every subgroup of a discrete group is discrete.
every quotient of a discrete group is discrete. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G
the product of a finite number of discrete groups is discrete.
a discrete group is compact if and only if it is finite. In Mathematics, a compact ( topological, often understood group is a Topological group whose Topology is Compact.
every discrete group is locally compact. In Mathematics, a locally compact group is a Topological group G which is locally compact as a Topological space.
every discrete subgroup of a Hausdorff group is closed.
every discrete subgroup of a compact Hausdorff group is finite.

Examples

Frieze groups and wallpaper groups are discrete subgroups of the isometry group of the Euclidean plane. A 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 A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the Wallpaper groups are cocompact, but Frieze groups are not.
A space group is a discrete subgroup of the isometry group of Euclidean space of some dimension. The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure
A crystallographic group usually means a cocompact, discrete subgroup of the isometries of some Euclidean space. The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure Sometimes, however, a crystallographic group can be a cocompact discrete subgroup of a nilpotent or solvable Lie group. The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure In Mathematics, a Lie algebra g is solvable if its derived series terminates in the zero subalgebra
Every triangle group 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 index), or the hyperbolic plane. In Mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a Triangle.
Fuchsian groups are, by definition, discrete subgroups of the isometry group of the hyperbolic plane. In Mathematics, a Fuchsian group is a particular type of group of isometries of the Hyperbolic plane. In
- 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. In Mathematics, the upper half-plane H is the set of Complex numbers \mathbb{H} = \{x + iy \| y > 0 x y \in \mathbb{R} \}
- A Fuchsian group is sometimes considered as a special case of a Kleinian group, by embedding the hyperbolic plane isometrically into three dimensional hyperbolic space and extending the group action on the plane to the whole space. In Mathematics, a Kleinian group, named after Felix Klein, is a finitely generated Discrete group &Gamma of orientation preserving conformal
- The modular group is PSL(2,Z), thought of as a discrete subgroup of PSL(2,R). In Mathematics, the modular group Γ is a fundamental object of study in Number theory, Geometry, algebra, and many other areas of advanced The modular group is a lattice in PSL(2,R), but it is not cocompact.
Kleinian groups are, by definition, discrete subgroups of the isometry group of hyperbolic 3-space. In Mathematics, a Kleinian group, named after Felix Klein, is a finitely generated Discrete group &Gamma of orientation preserving conformal In Mathematics, hyperbolic n -space, denoted H n, is the maximally symmetric Simply connected, n -dimensional These include quasi-Fuchsian groups.
- 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. In Geometry, a half-space is either of the two parts into which a plane divides the three-dimensional space
A lattice in a Lie group is a discrete subgroup such that the Haar measure of the quotient space is finite. In Lie theory and related areas of Mathematics, a lattice in a Locally compact Topological group is a Discrete subgroup with the property In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of Locally compact topological groups and subsequently define

Contents

Properties

Examples

Links to more examples

See also