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mathematics, a compactly generated (topological) group is a topological group G which is algebraically generated by one of its compact subsets. This should
Compactly_generated_group
Topics referred to by the same term
mathematics, compactly generated can refer to: Compactly generated group, a topological group which is algebraically generated by one of its compact subsets
Compactly_generated
Property of topological spaces
space X {\displaystyle X} is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below
Compactly_generated_space
Group type in algebra
finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation)
Finitely_generated_group
group Commensurability (group theory) Compact group Compactly generated group Complete group Complex reflection group Congruence subgroup Continuous symmetry
List_of_group_theory_topics
Mathematical concept
condition of being compact is quite strong, so that most objects are not compact. A category C {\displaystyle C} is compactly generated if any object can
Compact_object_(mathematics)
boundedly generated, then so is G itself. Any quotient group of a boundedly generated group is also boundedly generated. A finitely generated torsion group must
Boundedly_generated_group
Topological algebra associated to continuous groups
related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally
Group algebra of a locally compact group
Group_algebra_of_a_locally_compact_group
Abstract structure in mathematics
In mathematics, compact quantum groups are generalisations of compact groups, where the commutative C ∗ {\displaystyle \mathrm {C} ^{*}} -algebra of continuous
Compact_quantum_group
Topological group that is in a certain sense assembled from a system of finite groups
in any topologically finitely generated profinite group (that is, a profinite group that has a dense finitely generated subgroup) the subgroups of finite
Profinite_group
Type of topological space in mathematics
locally compact Hausdorff spaces are compactly generated. Conversely, every compactly generated Hausdorff space is a quotient of some locally compact Hausdorff
Locally_compact_space
Locally compact topological group with an invariant averaging operation
In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant
Amenable_group
Type of mathematical space
theorem. A profinite group (e.g. Galois group) is compact. Compactly generated space Compactness theorem Continuous functions on a compact Hausdorff space
Compact_space
compact Lie group is a torus; i.e., a Lie group isomorphic to ( S 1 ) h {\displaystyle (S^{1})^{h}} . A connected complex Lie group that is a compact
Abelian_Lie_group
Category whose objects are topological spaces and whose morphisms are continuous maps
topological manifolds, with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak Hausdorff spaces
Category of topological spaces
Category_of_topological_spaces
Category in mathematics
explain the name: if a localizing subcategory S of a compactly generated triangulated category D is generated by a set of objects, then there is a Bousfield
Triangulated_category
Subgroup of a root system's isometry group
which is generated by reflections through the hyperplanes orthogonal to at least one of the roots, and as such is a finite reflection group. In fact it
Weyl_group
On the finite presentability of fundamental groups of 3-manifolds
necessarily compact) with finitely generated fundamental group, there is a compact three-dimensional submanifold, called the compact core or Scott core, such that
Scott_core_theorem
Digital optical disc data storage format
represented generated 12% each of the French music industry revenues. Sony and Philips received praise for the development of the compact disc from professional
Compact_disc
Left-invariant (or right-invariant) measure on locally compact topological group
{\displaystyle (G,\cdot )} be a locally compact Hausdorff topological group. The σ {\displaystyle \sigma } -algebra generated by all open subsets of G {\displaystyle
Haar_measure
Duality for locally compact abelian groups
locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex
Pontryagin_duality
The compact elements of Sub(A) are exactly the finitely generated substructures of A. Every substructure is the union of its finitely generated substructures;
Compact_element
Classification of a two-dimensional repetitive pattern
group: C1 The group p1 contains only translations; there are no rotations, reflections, or glide reflections. Examples of group p1 Computer generated
Wallpaper_group
Mathematical term in group theory
In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk
Grigorchuk_group
Mathematical group
subgroups they generate. The symplectic group Sp ( 2 n , F ) {\displaystyle \operatorname {Sp} (2n,\mathbb {F} )} can be generated by linear transformations
Symplectic_group
Tool in algebraic topology
E) is a finitely generated R-module. Then the cohomology groups Hj(X,E) are finitely generated R-modules. For example, for a compact Hausdorff space X
Sheaf_cohomology
Specification of a mathematical group by generators and relations
presentation. A group is finitely generated (respectively finitely related, finitely presented) if it has a presentation that is finitely generated (respectively
Presentation_of_a_group
Type of topological space
that X and Y are compactly generated Hausdorff spaces, so Hom(X,Y) is often taken with the compactly generated variant of the compact-open topology; the
CW_complex
Commutative group (mathematics)
study of finitely generated abelian groups is totally equivalent with the study of integer matrices. In particular, changing the generating set of A is equivalent
Abelian_group
Tiling of hyperbolic 3-space by uniform polyhedra
3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
Type of mathematical group
groups, polycyclic groups, finitely generated linear groups, and fundamental groups of compact 3-manifolds. A divisible group is a group G {\displaystyle
Residually_finite_group
Branch of mathematics that studies the properties of groups
I}} , the free group generated by F surjects onto the group G. The kernel of this map is called the subgroup of relations, generated by some subset D
Group_theory
Algebraic construct of interest in theoretical physics
Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their
Quantum_group
American carbon emission trading program
Analysis Group studied RGGI's first and second three-year compliance periods. They found that the effects of RGGI's first three years are generating in $1
Regional Greenhouse Gas Initiative
Regional_Greenhouse_Gas_Initiative
Locally compact group – Type of topological group in mathematics Ramification of local fields Topological abelian group Topological group – Group that is
Locally_compact_field
property Compactification Measure of non-compactness Paracompact space Locally compact space Compactly generated space Axiom of countability Sequential
List of general topology topics
List_of_general_topology_topics
Mathematical concept
a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted
Hyperbolic_group
On representability of a contravariant functor on the category of connected CW complexes
adjoint functor. Namely, if C and D are triangulated categories with C compactly generated and F a triangulated functor commuting with arbitrary direct sums
Brown's representability theorem
Brown's_representability_theorem
Lattice whose elements are the subgroups of a given group
subgroups are algebraic lattices. This means that they are complete and compactly generated. However in general, there is no restriction on the possible sublattices
Lattice_of_subgroups
finite algebra. A lattice is called algebraic if it is complete and compactly generated. In 1963, Grätzer and Schmidt proved that every algebraic lattice
Finite lattice representation problem
Finite_lattice_representation_problem
Correspondence between topics in Lie theory
correspondence for classical compact groups (cf. the table in "compact Lie groups" below.) If f : G → H {\displaystyle f:G\to H} is a Lie group homomorphism, then
Lie group–Lie algebra correspondence
Lie_group–Lie_algebra_correspondence
Universal construction of a complex Lie group from a real Lie group
larger compact group for which the irreducible representations stay irreducible and inequivalent when restricted to G. Since AN is generated by unitaries
Complexification_(Lie_group)
On when a family of real, continuous functions has a uniformly convergent subsequence
for a family of functions from a compactly generated Hausdorff space into a uniform space to be compact in the compact-open topology; see Kelley (1991
Arzelà–Ascoli_theorem
Mathematical theory
a finitely-generated Kleinian group with region of discontinuity Ω, then Ω/Γ has a finite number of components, each of which is a compact Riemann surface
Ahlfors_finiteness_theorem
Type of group in mathematics
corresponding spheres, and π1(KO) is generated by [LR] π2(KO) is generated by [LC] π4(KO) is generated by [LH] π8(KO) is generated by [LO] From the point of view
Orthogonal_group
Area in mathematics devoted to the study of finitely generated groups
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties
Geometric_group_theory
Concept in mathematics
mapping class group of a surface is a finitely generated group. The smallest number of Dehn twists that can generate the mapping class group of a closed
Mapping class group of a surface
Mapping_class_group_of_a_surface
Continuous deformation between two continuous functions
homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formally, a homotopy between two
Homotopy
Every subgroup of a cyclic group is cyclic, and if finite, its order divides its parent's
represented as the additive group on the integers, then the subgroup generated by d is a subgroup of the subgroup generated by e if and only if e is a
Subgroups_of_cyclic_groups
Chemical compound
assume the double-helix conformation, and also (in the eukaryotes) to be compactly coiled within the small cell nucleus. The double-stranded DNA molecules
Deoxyribose
topological generating set with no more than r {\displaystyle r} elements. More generally it was shown that a finitely generated profinite group is a compact p-adic
Pro-p_group
Representations of finite groups, particularly on vector spaces
\mu ^{2},\nu ,\mu \nu ,\mu ^{2}\nu \}} be the dihedral group of order 6 {\displaystyle 6} generated by μ , ν {\displaystyle \mu ,\nu } which fulfil the properties
Representation theory of finite groups
Representation_theory_of_finite_groups
Manifold with inversion symmetry
group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact
Hermitian_symmetric_space
Group that is also a differentiable manifold with group operations that are smooth
algebra infinitesimal generators of the group G {\displaystyle G} . The subgroup of G {\displaystyle G} generated by N {\displaystyle N} is the identity
Lie_group
In algebra, module with a finite generating set
and a finitely generated module over the integers is simply a finitely generated abelian group. The left R-module M is finitely generated if there exist
Finitely_generated_module
is compact. A finitely generated free group is compact. A trace monoid on a finite set of generators is compact. The bicyclic monoid is not compact. The
Compact_semigroup
Isometry group of Euclidean space
closed. Examples of such groups are, in 1D, the group generated by a translation of 1 and one of √2, and, in 2D, the group generated by a rotation about the
Euclidean_group
Set with associative invertible operation
classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects
Group_(mathematics)
Type of group used in topology and geometric group theory
In mathematics, a CAT(0) group is a finitely generated group with a group action on a CAT(0) space that is geometrically proper, cocompact, and isometric
CAT(0)_group
precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum
Zonal_spherical_function
Group of unitary complex matrices with determinant of 1
U(n), consisting of all n × n unitary matrices. As a compact classical group, U(n) is the group that preserves the standard inner product on C n {\displaystyle
Special_unitary_group
Construction in algebra
topology. The algebra of all continuous functions on a Lie group is a locally compact quantum group. Quasi-Hopf algebras are generalizations of Hopf algebras
Hopf_algebra
Discrete group of Möbius transformations
Kleinian group Γ has finite covolume if H3/Γ has finite volume. Any Kleinian group of finite covolume is finitely generated. A Kleinian group is called
Kleinian_group
Group in group theory and physics
semigroup is generated by − 1 2 L {\displaystyle -{\frac {1}{2}}{\mathcal {L}}} ; equivalently, with the opposite sign convention, it is generated by 1 2 ∑
Heisenberg_group
Equivalence relation of groups
H2. For example: A group is finite if and only if it is commensurable with the trivial group. Any two finitely generated free groups on at least 2 generators
Commensurability (group theory)
Commensurability_(group_theory)
Subject area in mathematics
says that all of the groups Gn(A) are finitely generated when A is a finitely generated Z-algebra. (The groups Gn(A) are the K-groups of the category of
Algebraic_K-theory
Semi-automatic pistol
continued to produce consistent groups when test fired for accuracy. The recoil reduction system is not present on the USP Compact models, which instead use
Heckler_&_Koch_USP
Mathematics term
is compact. Any countable discrete group with property (T) is finitely generated. An amenable group which has property (T) is necessarily compact. Amenability
Kazhdan's_property_(T)
Dutch-American music corporation
Universal Music Group N.V. (often abbreviated as UMG and referred to as Universal Music Group or Universal Music) is a Dutch-American music corporation
Universal_Music_Group
Smallest cardinality of a generating set for a group
=G\}.} If G is a finitely generated group, then the rank of G is a non-negative integer. The notion of rank of a group is a group-theoretic analog of the
Rank_of_a_group
locally compact abelian group G {\displaystyle G} , let A {\displaystyle A} be a compactly generated subgroup, and B {\displaystyle B} a compact subgroup
Schwartz–Bruhat_function
American multinational entertainment and record label conglomerate
Warner Music Group Corp., commonly abbreviated as WMG, is an American multinational entertainment and record label conglomerate headquartered in New York
Warner_Music_Group
Group in which each element has finite order
None of these examples has a finite generating set. Explicit examples of finitely generated infinite periodic groups were constructed by Golod, based on
Torsion_group
Topological concept in mathematics
and thus has finitely generated homology groups. If M {\displaystyle M} is a closed connected n-manifold, the n-th homology group H n ( M ; Z ) {\displaystyle
Closed_manifold
Conversion of character sequences into token sequences in computer science
into template code for compiling and executing). Regular expressions compactly represent patterns that the characters in lexemes might follow. For example
Lexical_analysis
Group making loans to developing countries
The World Bank Group (WBG) is a family of five international organizations that make leveraged loans to developing countries. The group is the largest
World_Bank_Group
Simple Lie group; the automorphism group of the octonions
In mathematics, G2 is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras g 2 , {\displaystyle {\mathfrak
G2_(mathematics)
the inside of Bi, then the group generated by these transformations is a Kleinian group. A Schottky group is any Kleinian group that can be constructed like
Schottky_group
Discrete subgroup in a locally compact topological group
sense, while both groups are essentially different: one is finitely generated and countable, while the other is not finitely generated and has the cardinality
Lattice_(discrete_subgroup)
Type of mathematical measure
theory for locally compact spaces from the functional-analytic viewpoint, it is necessary to extend measure (integral) from compactly supported continuous
Radon_measure
relation Descriptive set theory Finitely generated group Infinite-dimensional Lie group Locally compact group Polish group Gromov, Mikhael (1993). Gerl, Martin
Infinite_group
Topological structure in number theory
is additive for short exact sequences of finitely-generated modules. The rank of a finitely generated module is zero if and only if the module is a torsion
Iwasawa_algebra
Periodic set of points
than the lattice itself. As a group (dropping its geometric structure) a lattice is a finitely generated free abelian group, and thus isomorphic to Z
Lattice_(group)
Abelian group extending a commutative monoid
Grothendieck group G 0 ( Z ) {\displaystyle G_{0}(\mathbb {Z} )} is an abelian group generated by symbols [ A ] {\displaystyle [A]} for any finitely generated abelian
Grothendieck_group
Non-binding United Nations pact
SDG Publishers Compact, including 15 international and national publishers associations. An action group and signatory of the Compact, known as the SDG
SDG_Publishers_Compact
Classification system for symmetry groups in geometry
parentheses showing two adjacent [(3,3,3)] loops, and is also represented more compactly as [3[ ]×[ ]], representing the rhombic symmetry of the Coxeter diagram
Coxeter_notation
Theorem in algebraic geometry
theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating
Hurwitz's automorphisms theorem
Hurwitz's_automorphisms_theorem
Algebra of formal sums
generated by the relators of the presentation of G {\displaystyle G} . But since this subgroup is itself free abelian, it is also finitely generated,
Free_abelian_group
Italian electronics manufacturer
Bontempi group owns the rights for Farfisa keyboards. The Compact series contains four models – Combo Compact, Mini Compact, Compact Deluxe and Compact Duo
Farfisa
Group of rotations in 3 dimensions
rotating around y and then x. The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is
3D_rotation_group
Theorem in group theory
the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G {\displaystyle G} has more
Stallings theorem about ends of groups
Stallings_theorem_about_ends_of_groups
South Korean K-pop girl group
stylized in all caps) /ˈkɑːrə/ is a South Korean girl group formed by DSP Media in 2007. The group's current lineup is Gyuri, Seungyeon, Nicole, Jiyoung
Kara_(South_Korean_group)
Mathematical group of loops in a Lie group
bracket. For non-compact manifolds one often studies the compactly supported current group C∞c(M,G), or more generally section groups Γc(M,𝒢) of bundles
Loop_group
Double cover Lie group of the special orthogonal group
mathematics the spin group, denoted Spin(n), is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R)
Spin_group
Lie group homomorphism from the real numbers
one-parameter group that it generates. It is these infinitesimal transformations that generate a Lie algebra that is used to describe a Lie group of any dimension
One-parameter_group
of bandlimited functions. Another example are compactly supported shearlet systems, where a compactly supported function ψ ∈ L 2 ( R 2 ) {\displaystyle
Shearlet
{Z} ^{2n}} which is the expected result. Since every finitely generated abelian group is isomorphic to G ≅ Z n ⊕ ⨁ i = 1 m Z / a i {\displaystyle G\cong
Character_group
interim DC Attorney General Ed Martin and head of the Weaponization Working Group stated the Justice Department would publicly name and shame individuals
Targeting of political opponents and civil society under the second Trump administration
Targeting_of_political_opponents_and_civil_society_under_the_second_Trump_administration
Concept in mathematics
by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify
Reductive_group
Mathematical construction used in homotopy theory
a geometric realization functor which turns simplicial sets into compactly generated Hausdorff spaces. Most classical results on CW complexes in homotopy
Simplicial_set
COMPACTLY GENERATED-GROUP
COMPACTLY GENERATED-GROUP
Boy/Male
Indian, Telugu
Generated
Boy/Male
Indian, Sanskrit
Devoted; Venerated
Girl/Female
Indian
Who is to be Venerated and Respected
Girl/Female
Biblical
Penetrated.
Girl/Female
Tamil
Aninditha | அநிஂதிதா
Beautiful, Virtuous, Venerated
Aninditha | அநிஂதிதா
Boy/Male
Hindu
Compact, Massive, Widespread, Great, Large, Mighty, Powerful, Bright, Clear, Name of Vishnu
Girl/Female
Hindu, Indian, Marathi, Tamil
Compact; Promise
Boy/Male
Muslim
Honored, Venerated
Girl/Female
Hindu
Generates harmony in dance and music
Girl/Female
Indian
Beautiful, Virtuous, Venerated
Girl/Female
Tamil
Generates harmony in dance and music
Girl/Female
Tamil
Anindita | அநிஂதிதா
Beautiful, Virtuous, Venerated
Anindita | அநிஂதிதா
Boy/Male
Hindu, Indian
Compact; Safe; Secure
Biblical
penetrated
Boy/Male
Hindu, Indian
Self Generated; Lord Shiva
Boy/Male
Hindu, Indian
Compact; Firm; Solid
Boy/Male
Indian
Honored, Venerated
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Sindhi, Tamil, Telugu
Virtuous; Venerated
Girl/Female
Indian
Beautiful, Virtuous, Venerated
Boy/Male
Arabic, Muslim, Sindhi
Venerated; Honoured
COMPACTLY GENERATED-GROUP
COMPACTLY GENERATED-GROUP
Girl/Female
Hindu
Honest, Night
Girl/Female
Australian, German, Latin, Polish
Woods; Forest; From the Forest; From the Woods
Girl/Female
Hindu, Indian, Kannada, Malayalam, Marathi, Modern, Punjabi, Sikh, Tamil, Telugu
First Ray of the Sun; Careful
Boy/Male
Hindu, Indian, Rajasthani, Traditional
Servant of Lord Krishna
Boy/Male
Tamil
Lord Shiva
Boy/Male
Hindu
Girl/Female
Hindu
Intelligent
Girl/Female
American, Australian, Christian, Hebrew, Irish, Jain
The Lord is Gracious
Surname or Lastname
English (chiefly Devon)
English (chiefly Devon) : nickname for a thin or lean person, from Middle English lene ‘lean’ (Old English hlǣne).Irish : reduced Anglicized form of Gaelic Ó Liatháin (see Lehane).Reduced form of Scottish McLean.
Girl/Female
Hindu, Indian
Earth
COMPACTLY GENERATED-GROUP
COMPACTLY GENERATED-GROUP
COMPACTLY GENERATED-GROUP
COMPACTLY GENERATED-GROUP
COMPACTLY GENERATED-GROUP
imp. & p. p.
of Generate
v. t.
To beget; to procreate; to propagate; to produce (a being similar to the parent); to engender; as, every animal generates its own species.
p. pr. & vb. n.
of Generate
n.
Capability of being generated.
adv.
In a compact manner.
n.
One who makes a compact.
a.
Relating to autogenesis; self-generated.
a.
Generated by water.
v. t.
To regard with reverential respect; to honor with mingled respect and awe; to reverence; to revere; as, we venerate parents and elders.
imp. & p. p.
of Venerate
adv.
In a compact manner; with close union of parts; densely; tersely.
imp. & p. p.
of Compact
a.
Self-generated; produced independently.
n.
One who, or that which, generates, begets, causes, or produces.
a.
Compact; pressed close; concentrated; firmly united.
a.
Not begot; not yet generated; also, having never been generated; self-existent; eternal.
a.
Capable of being generated or produced.
a.
First formed or generated; original; primigenial.
n.
That which generates.
p. p. & a
Brief; close; pithy; not diffuse; not verbose; as, a compact discourse.