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...

integrated circuit Discrete group, a group with the discrete topology Discrete category, category whose only arrows are identity arrows Discrete mathematics...

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric...

group (the images of a given point under all group elements) forms a discrete set. All finite symmetry groups are discrete. Discrete symmetry groups come...

theoretical physics, a discrete symmetry is a symmetry under the transformations of a discrete group—e.g. a topological group with a discrete topology whose elements...

In particular, this is equivalent to proper discontinuity if G is a discrete group. It is said to be locally free if there exists a neighbourhood U of...

In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable...

from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups...

"continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group"). The continuous Heisenberg group arises in the description...

in any group G {\displaystyle G} , powers b k {\displaystyle b^{k}} can be defined for all integers k {\displaystyle k} , and the discrete logarithm...

representations. In discrete group theory, where G has the discrete topology, a simpler definition is used. In this setting, a group is amenable if one...

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous...

one), the finite abelian groups (with the discrete topology), and the additive group of the integers (also with the discrete topology), the real numbers...

In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space...

representations of the group. As such, they are similar to the group ring associated to a discrete group. If G is a locally compact Hausdorff group, G carries an...

mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving...

and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant...

reflection. Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane...

mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding...

The group cohomology of a discrete group G is the singular cohomology of a suitable space having G as its fundamental group, namely the corresponding...

specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space. In geometric group theory, a geometry...

presentation of a group can be used to construct the Cayley graph, a graphical depiction of a discrete group. Examples and applications of groups abound. A starting...

Noncompact solvable groups. Nontrivial free groups and free abelian groups. Historically property (T) was established for discrete groups Γ by embedding them...

topological groups. For example, finite groups correspond to finite groups, compact groups correspond to discrete groups, and metrisable groups correspond...

profinite group is a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups. In this context, an inverse...

space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of...

space D {\displaystyle \mathbb {D} } . Let G {\displaystyle G} be a discrete group and μ {\displaystyle \mu } a probability measure on G {\displaystyle...

of discrete two-dimensional point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group. Intl...

generated discrete groups are geometrically finite, but Greenberg (1966) showed that there are examples of finitely generated discrete groups in dimension...

modular group can be generalized to the Hecke groups, named for Erich Hecke, and defined as follows. The Hecke group Hq with q ≥ 3, is the discrete group generated...