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...

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...

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...

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

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...

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...

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...

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

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...

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...

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...

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...

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

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

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...

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

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...

of group actions on these three surfaces. The quotient of a free action of a discrete group G on a simply connected space Y has fundamental group π 1...

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...

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

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

Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection...

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...

the techniques of computational physics. A lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice...

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...

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