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{{short description|Type of topological group}}
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[[File:Number-line.svg|right|thumb|300px|The integers with their usual topology are a discrete subgroup of the real numbers.]]

In [[mathematics]], a [[topological group]] ''G'' is called a '''discrete group''' if there is no [[limit point]] in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and only if its [[Identity element|identity]] is [[Isolated point|isolated]].{{sfn|Pontrjagin|1946|p=54}}

A [[subgroup]] ''H'' of a topological group ''G'' is a '''discrete subgroup''' if ''H'' is discrete when endowed with the [[induced topology|subspace topology]] from ''G''. In other words there is a neighbourhood of the identity in ''G'' containing no other element of ''H''. For example, the [[integer]]s, '''Z''', form a discrete subgroup of the [[real number|reals]], '''R''' (with the standard [[Metric space|metric topology]]), but the [[rational number]]s, '''Q''', do not.

Any group can be endowed with the [[discrete topology]], making it a discrete topological group. Since every map from a discrete space is [[Continuous (topology)|continuous]], the topological homomorphisms between discrete groups are exactly the [[group homomorphism]]s between the underlying groups. Hence, there is an [[Isomorphism of categories|isomorphism]] between the [[category of groups]] and the category of discrete groups. Discrete groups can therefore be identified with their underlying (non-topological) groups.

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.

A discrete [[isometry group]] is an isometry group such that for every point of the metric space the set of images of the point under the isometries is a [[discrete set]]. A discrete [[symmetry group]] is a symmetry group that is a discrete isometry group.

==Properties==
Since topological groups are [[homogeneous space|homogeneous]], one need only look at a single point to determine if the topological group is discrete. In particular, a topological group is discrete only if the [[singleton (mathematics)|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 have this property do not regard these groups as Lie groups). The [[identity component]] of a discrete group is just the [[trivial group|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 group]]s and [[locally isomorphic groups]]. A discrete normal subgroup of a [[connected space|connected]] group ''G'' necessarily lies in the [[center (group theory)|center]] of ''G'' and is therefore [[abelian group|abelian]].

''Other properties'':
*every discrete group is [[totally disconnected]]
*every subgroup of a discrete group is discrete.
*every [[quotient group|quotient]] of a discrete group is discrete.
*the product of a finite number of discrete groups is discrete.
*a discrete group is [[compact group|compact]] if and only if it is finite.
*every discrete group is [[locally compact group|locally compact]].
*every discrete subgroup of a Hausdorff group is closed.
*every discrete subgroup of a compact Hausdorff group is finite.

==Examples==
* [[Frieze group]]s and [[wallpaper group]]s are discrete subgroups of the [[isometry group]] of the Euclidean plane. Wallpaper groups are cocompact, but Frieze groups are not.
* 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 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 of a subgroup|index]]), or the [[Hyperbolic space|hyperbolic plane]].
* [[Fuchsian group]]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 group]], 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 group]] PSL(2,'''Z''') is 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 group]]s are, by definition, discrete subgroups of the isometry group of [[hyperbolic 3-space]]. These include [[quasi-Fuchsian group]]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 [[lattice (discrete subgroup)|lattice]] in a [[Lie group]] is a discrete subgroup such that the [[Haar measure]] of the quotient space is finite.

==See also==
*[[crystallographic point group]]
*[[congruence subgroup]]
*[[arithmetic group]]
*[[geometric group theory]]
*[[computational group theory]]
*[[freely discontinuous]]
*[[free regular set]]

== Citations ==
{{reflist}}

==References==
{{refbegin}}
*{{cite book|author-last=Pontrjagin|author-first=Leon|title=Topological Groups|date=1946|publisher=[[Princeton University Press]]|url=https://www.amazon.com/Topological-Groups-L-Pontrjagin/dp/B000PS6XVM/ref=sr_1_1?dchild=1&keywords=Topological+Groups&qid=1622710810&s=books&sr=1-1}}
*{{Springer|id=d/d033080|title=Discrete group of transformations}}
*{{Springer|id=d/d033150|title=Discrete subgroup}}
{{refend}}

==External links==
*{{Commonscatinline|Discrete groups}}

{{DEFAULTSORT:Discrete Group}}
[[Category:Discrete groups| ]]
[[Category:Geometric group theory]]

Discrete group - Revision history

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