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Concept in algebraic topology
topology, a p-compact group is a homotopical version of a compact Lie group, but with all the local structure concentrated at a single prime p. This concept
P-compact_group
Topological group with compact topology
mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space. Compact groups are a natural
Compact_group
Type of topological group in mathematics
a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important
Locally_compact_group
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
Topological group structure arising in Fourier analysis
theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped
Locally_compact_abelian_group
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
Type of mathematical space
mathematics, especially general topology and mathematical analysis, compactness is a property of a space that makes it behave in many ways like a finite
Compact_space
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
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
each non-compact simple Lie group G, one compact and one non-compact. The non-compact one is a cover of the quotient of G by a maximal compact subgroup
Simple_Lie_group
Catalogue of 100 galaxy groups made by Paul Hickson
Hickson Compact Group (abbreviation: HCG) is a collection of galaxies designated as published by Paul Hickson in 1982. The most famous group on Hickson's
Hickson_Compact_Group
N\triangleleft G} the quotient group G / N {\displaystyle G/N} is a p-group. Note that, as profinite groups are compact, the open subgroups are exactly
Pro-p_group
Type of topological space in mathematics
locally compact groups. Compact group – Topological group with compact topology F. Riesz's theorem Locally compact field Locally compact quantum group Locally
Locally_compact_space
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
Aggregation of galaxies with 50 or fewer members
Astronomer Paul Hickson created a catalogue of such groups in 1982, the Hickson Compact Groups. Compact groups of galaxies readily show the effect of dark matter
Galaxy_group
Type of topological space
said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it is both σ-compact and (weakly)
Σ-compact_space
Mathematical group
different family is the compact symplectic group, usually denoted Sp ( n ) {\displaystyle \operatorname {Sp} (n)} or U S p ( n ) {\displaystyle \mathrm
Symplectic_group
locally compact field is a topological field whose topology forms a locally compact Hausdorff space. These kinds of fields were originally introduced in p-adic
Locally_compact_field
Topological group that is in a certain sense assembled from a system of finite groups
profinite group is compact and totally disconnected. A non-compact generalization of the concept is that of locally profinite groups. Even more general
Profinite_group
Digital optical disc data storage format
The compact disc (CD) is a digital optical disc data storage format co-developed by Philips and Sony to store and play digital audio recordings. It employs
Compact_disc
Mathematical term in group theory
in group theory, the Prüfer p-group or the p-quasicyclic group or p ∞ {\displaystyle p^{\infty }} -group, Z ( p ∞ ) {\displaystyle \mathbb {Z} (p^{\infty
Prüfer_group
Concept in topology
In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst
Maximal_compact_subgroup
Group that is a topological space with continuous group operations
compact group always contains a compact open subgroup, which is necessarily a profinite group. (For example, the locally compact group GL ( n , Q p )
Topological_group
profinite group is a topological group that is Hausdorff, locally compact, and totally disconnected. Moreover, a locally profinite group is compact if and
Locally_profinite_group
U.S. agreement on presidential elections
The National Popular Vote Interstate Compact (NPVIC) is an agreement among a group of U.S. states and the District of Columbia to award all their electoral
National Popular Vote Interstate Compact
National_Popular_Vote_Interstate_Compact
compact Lie algebra), and that the complexification of u0 is g. Therefore, u0 is a compact form of g. Complexification (Lie group) Helgason 1978, p. 426
Real_form_(Lie_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
Universal construction of a complex Lie group from a real Lie group
original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear. For compact Lie groups, the complexification
Complexification_(Lie_group)
Mathematical group of the homotopy classes of loops in a topological space
cardinals and fundamental groups of compact spaces, Fundamenta Mathematicae 192 (2006), 87-92 [1] Hatcher (2002, §4.1) Adams (1978, p. 5) Brown (2006, §6.1)
Fundamental_group
Set with associative invertible operation
theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many
Group_(mathematics)
Group of 𝑛 × 𝑛 invertible matrices
\operatorname {GL} (n,\mathbb {C} )} is not compact; rather its maximal compact subgroup is the unitary group U ( n ) {\displaystyle \operatorname {U}
General_linear_group
(pseudo-)Riemannian manifold whose geodesics are reversible
In both the compact and the non-compact cases it turns out that there is exactly one for each complex simple Lie group, namely AI with p = 2 or q = 2
Symmetric_space
disconnected group is a topological group that is totally disconnected. Such topological groups are necessarily Hausdorff. Interest centres on locally compact totally
Totally_disconnected_group
Orthogonal group of an indefinite quadratic form
mathematics, the indefinite orthogonal group, O ( p , q ) {\displaystyle \operatorname {O} (p,q)} is the Lie group of all linear transformations of an
Indefinite_orthogonal_group
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
Concept in number theory
quotient A K / K {\displaystyle \mathbb {A} _{K}/K} is compact. As an additive locally compact abelian group, the adele ring is self-dual, making it a natural
Adele_ring
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
Differentiable manifold
\Gamma )} is a p-dimensional compact torus. Since Z is central in N, the group G acts on the compact nilmanifold P = N / Γ {\displaystyle P=N/\Gamma } with
Nilmanifold
Type of group in mathematics
(p+q,\mathbb {H} )\mid g^{*}I_{p,q}g=I_{p,q}\}.} When q = 0 {\displaystyle q=0} this is the compact group usually written S p ( n ) {\displaystyle \mathrm
Classical_group
Left-invariant (or right-invariant) measure on locally compact topological group
volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced
Haar_measure
Magnetic audio tape recording format
The cassette tape, officially named the Compact Cassette, and also known as audio cassette, or simply tape or cassette, is an analog magnetic tape recording
Cassette_tape
Discrete subgroup in a locally compact topological group
Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has
Lattice_(discrete_subgroup)
Mathematical theory
definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition
Compact_Lie_algebra
Mutual aid agreement between states and territories of the US
The Emergency Management Assistance Compact (EMAC) is a mutual aid agreement among states and territories of the United States. It enables states to share
Emergency Management Assistance Compact
Emergency_Management_Assistance_Compact
Semi-automatic pistol made by ČZUB
available chambered for the .40 S&W cartridge. CZ 75 D PČR Compact Very compact – similar to the P-01 in size, with an aluminum alloy frame also but lacks
CZ_75
Marketing term for a highly comfortable vehicle
cars span from sports cars to large saloons and sport utility vehicles. "Compact" luxury vehicles also fill a niche. Several car classification schemes
Luxury_car
Small construction excavator
are accomplished through the transfer of hydraulic fluid. The compact excavator's work group and blade are activated by hydraulic fluid acting upon hydraulic
Compact_excavator
Galaxy group from Hickson Compact Group
HCG 2 is one of the compact groups of galaxies in the Hickson Compact Group of galaxies. It is located in Pisces east of HD 2679 and is 192 million light-years
HCG_2
Formal agreement between two or more U.S. states
In the United States, an interstate compact is a legally binding pact or agreement between two or more states, or between states and any foreign sub-national
Interstate_compact
Motor vehicle platform
The Volkswagen Group A platform is an automobile platform shared among compact and mid-size cars of the Volkswagen Group. The first version debuted in
Volkswagen_Group_A_platform
Firearm
The KelTec P11 is a compact, semi-automatic, short-recoil operated pistol chambered in 9 mm Luger. Discontinued in 2019, the P11 was manufactured by Kel-Tec
KelTec_P11
HCG 1 is a compact group of galaxy in the Hickson Compact Group catalogue identified using a systematic search by the Palomar Observatory Sky Survey lead
HCG_1
Borel–de Siebenthal theory describes the closed connected subgroups of a compact Lie group that have maximal rank, i.e. contain a maximal torus. It is named
Borel–de_Siebenthal_theory
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
Compact semi-automatic striker-fired pistol by CZ
2014. The C in the name stands for compact. The weapon is designed for self-defense and use in the armed forces. The CZ P-10 C has been awarded the Pistol
CZ_P-10
Theorem about projections of coadjoint orbits of a connected compact Lie group
states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set. It is a special
Kostant's_convexity_theorem
First governing document of Plymouth Colony
The Mayflower Compact, originally titled Agreement Between the Settlers of New Plymouth, was the first governing document of Plymouth Colony. It was written
Mayflower_Compact
Manifold with Riemannian, complex and symplectic structure
compact Kähler manifold. Every Riemannian metric on an oriented 2-manifold is Kähler. (Indeed, its holonomy group is contained in the rotation group S
Kähler_manifold
Representations of finite groups, particularly on vector spaces
, {\displaystyle V,} is for compact groups given by p τ ( v ) = n τ ∫ G χ τ ( t ) ¯ ρ ( t ) ( v ) d t , {\displaystyle p_{\tau }(v)=n_{\tau }\int _{G}{\overline
Representation theory of finite groups
Representation_theory_of_finite_groups
Group homomorphism into the general linear group over a vector space
groups. Compact groups or locally compact groups — Many of the results of finite group representation theory are proved by averaging over the group.
Group_representation
Group that is also a differentiable manifold with group operations that are smooth
spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights. In 1900 David Hilbert challenged Lie
Lie_group
Conjecture in topology
Restricting to groups G which are locally compact and have a continuous, faithful group action on M, the conjecture states that G must be a Lie group. Because
Hilbert–Smith_conjecture
Mathematics term
In mathematics, a locally compact topological group G has property (T) if the trivial representation is an isolated point in its unitary dual equipped
Kazhdan's_property_(T)
Japanese mathematician
homotopy theory; it continues today in (for example) the theory of p-compact groups. "西田吾郎氏死去（京都大名誉教授・代数トポロジー、京都大元副学長）". Jiji Press. 2014-06-03. Retrieved
Goro_Nishida
Inputs for which a function's value is non-zero
compactly supported smooth functions are dense in the space of Schwartz functions. They are also dense in every Lp space for p < ∞ {\displaystyle p<\infty
Support_(mathematics)
(There are other notions of compactness in mathematics.) In a partially ordered set (P,≤) an element c is called compact (or finite) if it satisfies one
Compact_element
importance are for Γ a finite group (in the theory of Molien series, in particular), a compact group, a Lie group or algebraic group. For a basis-independent
Invariant_polynomial
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)
spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient
Zonal_spherical_function
and analysis, an exhaustion by compact sets of a topological space X {\displaystyle X} is a nested sequence of compact subsets K i {\displaystyle K_{i}}
Exhaustion_by_compact_sets
Branch of number theory
functions on the p-adic numbers is part of the theory of locally compact groups (abstract harmonic analysis). The usual meaning taken for p-adic analysis
P-adic_analysis
locally compact or reductive Lie groups is an algebra of measures under convolution. It can also be defined for a pair (g, K) of a maximal compact subgroup
Hecke_algebra_of_a_pair
2025 US national security scandal
government group chat leaks took place from March 11 to 15, 2025, when a group of U.S. national security leaders were observed on a group chat using the
United States government group chat leaks
United_States_government_group_chat_leaks
Classification in astronomy
In astronomy, the term compact object (or compact star) refers collectively to white dwarfs, neutron stars, and black holes. It could also include exotic
Compact_object
On when a family of real, continuous functions has a uniformly convergent subsequence
with domain a compact metric space (Dunford & Schwartz 1958, p. 382). Modern formulations of the theorem allow for the domain to be compact Hausdorff and
Arzelà–Ascoli_theorem
North American car classification
Subcompact car is a North American classification for cars smaller than a compact car. It is broadly equivalent to the B-segment (Europe), supermini (Great
Subcompact_car
Number system extending the rational numbers
the theory of locally compact groups (abstract harmonic analysis). The usual meaning taken for p-adic analysis is the theory of p-adic-valued functions
P-adic_number
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
Problem in Lie group theory
Neumann in 1933, giving an affirmative answer for compact groups. The locally compact abelian group case was solved in 1934 by Lev Pontryagin. The final
Hilbert's_fifth_problem
Small galaxy composed of up to several billion stars
Blue Compact Dwarf Galaxies". ASP Conference Series. 421: 65. arXiv:0909.5500. Bibcode:2010ASPC..421...65L. Papaderos, P. (7 May 2010). "Blue Compact Dwarf
Dwarf_galaxy
2008 agreement between US states
Great Lakes–St. Lawrence River Basin Water Resources Compact is a legally binding interstate compact among the U.S. states of Illinois, Indiana, Michigan
Great_Lakes_Compact
necessarily compact Lie group is given by the action a Lie subgroup H ⊆ G {\displaystyle H\subseteq G} on G {\displaystyle G} . Given a Lie group action of
Lie_group_action
Type of semi-automatic pistol
differs from the standard compact pistol in the following ways: use of the improved Beretta plated competition hammer group for a lighter trigger pull
Beretta_Px4_Storm
Israeli rock band
Minimal Compact is an Israeli rock band associated with the post-punk movement of the 1980s. Between its foundation in 1980 and its dissolution seven
Minimal_Compact
mathematics, a convergence group or a discrete convergence group is a group Γ {\displaystyle \Gamma } acting by homeomorphisms on a compact metrizable space M
Convergence_group
Homology Theory in Algebraic Topology is Compactly Supported
homology theory in algebraic topology is compactly supported if, in every degree n, the relative homology group Hn(X, A) of every pair of spaces (X, A)
Compactly_supported_homology
Type of mathematical space
flag variety G/P is a compact homogeneous Riemannian manifold K/(K∩P) with isometry group K. Furthermore, if G is a complex Lie group, G/P is a homogeneous
Generalized_flag_variety
Tiling of hyperbolic 3-space by uniform polyhedra
cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
One-dimensional complex manifold
Important examples of non-compact Riemann surfaces are provided by analytic continuation. If P ( x , y ) {\displaystyle P(x,y)} is any complex polynomial
Riemann_surface
Representation of the symmetry group of spacetime in special relativity
framework that applies to simply connected, compact groups. Non-compactness implies, for a connected simple Lie group, that no nontrivial finite-dimensional
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
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
Regular tiling of hyperbolic 3-space
In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli
Icosahedral_honeycomb
Mathematical group of loops in a Lie group
multiplication defined pointwise. When G is a compact Lie group, LG is a basic example of an infinite-dimensional Lie group, with Lie algebra L𝔤 = C∞(S1, 𝔤).
Loop_group
Motor vehicle platform
The Volkswagen Group MQB platform is the company's strategy for shared modular design construction of its transverse, front-engine, front-wheel-drive
Volkswagen_Group_MQB_platform
Memory card format
CompactFlash (CF) is a flash memory mass storage device used mainly in portable electronic devices. The format was specified and the devices were first
CompactFlash
Galaxy cluster in the constellation Capricornus
HCG 87 is a compact group of galaxies listed in the Hickson Compact Group Catalogue. This group is about 400 million light-years away in the constellation
HCG_87
Application of Fourier analysis to non-abelian topological groups
Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin
Noncommutative harmonic analysis
Noncommutative_harmonic_analysis
Class of compact connected topological spaces
discusses a class of topological groups. For the wrapped loop of wire, see Solenoid. In mathematics, a solenoid is a compact connected topological space (i
Solenoid_(mathematics)
Regular tiling of hyperbolic 3-space
one of four regular compact honeycombs in 3D hyperbolic space: There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including the
Order-5_cubic_honeycomb
Characterizes sets of lattices that are bounded in a certain sense
In mathematics, Mahler's compactness theorem, proved by Kurt Mahler (1946), is a foundational result on lattices in Euclidean space, characterising sets
Mahler's_compactness_theorem
Basic result in harmonic analysis on compact topological groups
result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann
Peter–Weyl_theorem
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