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Inmathematics, atopological groupG is called adiscrete group if there is nolimit point in it (i.e., for each element inG, there is a neighborhood which only contains that element). Equivalently, the groupG is discrete if and only if itsidentity isisolated.^[1]

AsubgroupH of a topological groupG is adiscrete subgroup ifH is discrete when endowed with thesubspace topology fromG. In other words there is a neighbourhood of the identity inG containing no other element ofH. For example, theintegers,Z, form a discrete subgroup of thereals,R (with the standardmetric topology), but therational numbers,Q, do not.

Any group can be endowed with thediscrete topology, making it a discrete topological group. Since every map from a discrete space iscontinuous, the topological homomorphisms between discrete groups are exactly thegroup homomorphisms between the underlying groups. Hence, there is anisomorphism between thecategory 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 atopological group orLie group is usefully endowed with the discrete topology, 'against nature'. This happens for example in the theory of theBohr compactification, and ingroup cohomology theory of Lie groups.

A discreteisometry 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 adiscrete set. A discretesymmetry group is a symmetry group that is a discrete isometry group.

Since topological groups arehomogeneous, 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 thesingleton containing the identity is anopen set.

A discrete group is the same thing as a zero-dimensionalLie group (uncountable discrete groups are notsecond-countable, so authors who require Lie groups to have this property do not regard these groups as Lie groups). Theidentity component of a discrete group is just thetrivial subgroup while thegroup of components is isomorphic to the group itself.

Since the onlyHausdorff 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 subgroupH ofG iscocompact if there is acompact subsetK ofG such thatHK =G.

Discretenormal subgroups play an important role in the theory ofcovering groups andlocally isomorphic groups. A discrete normal subgroup of aconnected groupG necessarily lies in thecenter ofG and is thereforeabelian.

Other properties:

• every discrete group istotally disconnected
• every subgroup of a discrete group is discrete.
• everyquotient of a discrete group is discrete.
• the product of a finite number of discrete groups is discrete.
• a discrete group iscompact if and only if it is finite.
• every discrete group islocally compact.
• every discrete subgroup of a Hausdorff group is closed.
• every discrete subgroup of a compact Hausdorff group is finite.

• Frieze groups andwallpaper groups are discrete subgroups of theisometry group of the Euclidean plane. Wallpaper groups are cocompact, but Frieze groups are not.
• Acrystallographic group usually means a cocompact, discrete subgroup of the isometries of some Euclidean space. Sometimes, however, acrystallographic group can be a cocompact discrete subgroup of a nilpotent orsolvable Lie group.
• Everytriangle groupT is a discrete subgroup of the isometry group of the sphere (whenT is finite), the Euclidean plane (whenT has aZ + Z subgroup of finiteindex), or thehyperbolic plane.
• Fuchsian groups 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 theupper half-plane model of the hyperbolic plane.
  • A Fuchsian group is sometimes considered as a special case of aKleinian group, by embedding the hyperbolic plane isometrically into three-dimensional hyperbolic space and extending the group action on the plane to the whole space.
  • Themodular 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 groups are, by definition, discrete subgroups of the isometry group ofhyperbolic 3-space. These includequasi-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 theupper half-space model of hyperbolic 3-space.
• Alattice in aLie group is a discrete subgroup such that theHaar measure of the quotient space is finite.

• crystallographic point group
• congruence subgroup
• arithmetic group
• geometric group theory
• computational group theory
• freely discontinuous
• free regular set

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