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Construction in group theory
especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced
Projective_linear_group
Group of 𝑛 × 𝑛 invertible matrices
the collineation group of projective space, for n > 2 {\displaystyle n>2} , and thus semilinear maps are of interest in projective geometry. If one removes
General_linear_group
Projective line over the real numbers
one-dimensional linear subspaces of a two-dimensional vector space over the reals. The automorphisms of a real projective line are called projective transformations
Real_projective_line
Group of matrices with determinant 1
SL(2, R) SL(2, C) Modular group (PSL(2, Z)) Projective linear group Conformal map Representations of classical Lie groups Hall 2015 Section 2.5 Hall
Special_linear_group
Map from algebra to geometric transforms
mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group P G L (
Projective_representation
Type of mathematical group
that the projective groups associated to these groups are also linear, though less obviously. For example, the group PSL2(R) is not a group of 2 × 2 matrices
Linear_group
In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space V = (V
Projective_orthogonal_group
Möbius transformation generalized to rings other than the complex numbers
a field, a linear fractional transformation is the restriction to the field of a projective transformation or homography of the projective line. When
Linear fractional transformation
Linear_fractional_transformation
Geometric concept of a 2D space with "points at infinity" adjoined
the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space. Not all projective planes can
Projective_plane
Projective space Projective transformation Projective geometry Projective linear group Quadric and conic section Glossary of linear algebra Glossary of
Outline_of_linear_algebra
Mathematical group
Jordan's theorem that the projective special linear group PSL(2, q) is simple for q ≠ 2, 3. This theorem generalizes to projective groups of higher dimensions
Group_of_Lie_type
Completion of the usual space with "points at infinity"
concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus
Projective_space
Mathematical group formed from the automorphisms of an object
the field extension. The automorphism group of the projective n-space over a field k is the projective linear group PGL n ( k ) . {\displaystyle \operatorname
Automorphism_group
Type of geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that
Projective_geometry
Isomorphism of projective spaces in geometry
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces
Homography
Topics referred to by the same term
mathematics, projective group may refer to: Projective linear group or one of the related linear groups Projective orthogonal group Projective unitary group Projective
Projective group (disambiguation)
Projective_group_(disambiguation)
Subgroup of the group of invertible n×n matrices
In mathematics, a linear algebraic group is a subgroup of the group of invertible n × n {\displaystyle n\times n} matrices (under matrix multiplication)
Linear_algebraic_group
Rational function of the form (az + b)/(cz + d)
are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2
Möbius_transformation
Algebraic variety with a group structure
orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally
Algebraic_group
In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up
Semilinear_map
Quotient of special unitary group by its center
holomorphic isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projective space. In terms of
Projective_unitary_group
Transformations induced by a mathematical group
the action of the general linear group on projective space. Particularly notable is PGL(2, K), the symmetries of the projective line, which is sharply 3-transitive
Group_action
Invariant in projective geometry
k-tuples of points are not in general position. While the projective linear group of the projective line is 3-transitive (any three distinct points can be
Cross-ratio
Abelian group related to division algebras
interpretation of the Brauer group of a field K is that it classifies the projective varieties over K that become isomorphic to projective space over an algebraic
Brauer_group
Orientation-preserving mapping class group of the torus
In mathematics, the modular group is the projective special linear group PSL ( 2 , Z ) {\displaystyle \operatorname {PSL} (2,\mathbb {Z} )} of 2 × 2
Modular_group
Topics referred to by the same term
variety Projective linear group Projective module Projective line Projective object Projective transformation Projective hierarchy Projective connection
Projective
Coordinate system used in projective geometry
general ring A, a projective line over A can be defined with homogeneous factors acting on the left and the projective linear group acting on the right
Homogeneous_coordinates
In projective geometry, a bijection between projective spaces that preserves collinearity
In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to
Collineation
Sporadic simple group
is a subgroup of the projective linear group of dimension 6 over the finite field with 3 elements. The outer automorphism group has order 2, and the full
Mathieu_group_M12
in projective n-space Pn. It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line
Rational_normal_curve
Topics referred to by the same term
continuous mathematics Projective linear group (AKA projective general linear group, PGL), the induced action of the general linear group of a vector space
Linear_(disambiguation)
Model of the extended complex plane plus a point at infinity
manifolds. In projective geometry, the sphere is an example of a complex projective space and can be thought of as the complex projective line P 1 ( C
Riemann_sphere
Aspect of mathematical group theory
are conjugate. The projective linear group of dimension two over the finite field with five elements, PGL(2, 5), acts on the projective line over the field
Automorphisms of the symmetric and alternating groups
Automorphisms_of_the_symmetric_and_alternating_groups
Group of real 2×2 matrices with unit determinant
by fractional linear transformations. The group action factors through the quotient PSL(2, R) (the 2 × 2 projective special linear group over R). More
SL2(R)
Concept in mathematics
In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates
Quaternionic_projective_space
Subgroup of GLn(k)
subgroup in the projective general linear group is a parabolic subgroup consisting of all elements fixing a given point of projective space. The word
Mirabolic_group
Class of mathematical groups
p≥5 the projective special linear groups PSL2(9) the projective special linear groups PSL2(2n) for n≥2 the projective special linear groups PSL3(2n)
C-group
Undeciphered writing system of ancient Crete
were four major branches of this group: Linear A, Linear B, Cypro-Minoan, and Cretan hieroglyphic. In the 1950s, Linear B was deciphered and its underlying
Linear_A
groups are necessarily connected. Products of special groups are special. The projective linear group is not special because there exist Azumaya algebras
Special group (algebraic group theory)
Special_group_(algebraic_group_theory)
Geometry with 7 points and 7 lines
constructed via linear algebra as the projective plane over the finite field with two elements. One can similarly construct projective planes over any
Fano_plane
Galois cohomology classes in H1(G(Ks/K),PGLn), where PGLn is the projective linear group, and n is one more than the dimension of the variety V. As usual
Severi–Brauer_variety
Group of all affine transformations of an affine space
translations, and the affine group of A can be described concretely as the semidirect product of V by GL(V), the general linear group of V: Aff ( V ) = V ⋊
Affine_group
Natural number
principal moduli for different genus zero congruence groups commensurable with the projective linear group P S L 2 ( Z ) {\displaystyle \operatorname {PSL_{2}}
171_(number)
Type of topological space
standard round metric, the measure of projective space is exactly half the measure of the sphere. Real projective spaces are smooth manifolds. On Sn, in
Real_projective_space
Algebraic variety in a projective space
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in
Projective_variety
Discrete group of Möbius transformations
of the projective linear group PGL(2,C). Thus, a Kleinian group can also be defined as a subgroup Γ of PGL(2,C). Classically, a Kleinian group was required
Kleinian_group
groups PSL2(q) The projective special linear groups PSL3(p) for p = 1 + 2a or p = 1 + 2a3, and PSL3(4) The projective special unitary groups PSU3(p) for p = 1 - 2a
Thin group (finite group theory)
Thin_group_(finite_group_theory)
Automorphism group of the Klein quartic
In mathematics, the projective special linear group PSL(2, 7), isomorphic to GL(3, 2), is a finite simple group that has important applications in algebra
PSL(2,7)
Mathematical concept
complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space
Complex_projective_space
Line with a point at infinity added
In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a point
Projective_line
Compact non-orientable two-dimensional manifold
real projective plane, denoted R P 2 {\displaystyle \mathbf {RP} ^{2}} or P 2 {\displaystyle \mathbb {P} _{2}} , is a two-dimensional projective space
Real_projective_plane
Undirected graph with 14 vertices
within an edge. The automorphism group of the Heawood graph is isomorphic to the projective linear group PGL2(7), a group of order 336. It acts transitively
Heawood_graph
linear group, while a projective representation is a homomorphism G → PGL(n, C) from G to a projective linear group. Projective representations of G correspond
Covering groups of the alternating and symmetric groups
Covering_groups_of_the_alternating_and_symmetric_groups
Group of unitary matrices
unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL ( n , C ) {\displaystyle
Unitary_group
Well studied projective geometries over finite fields
planes. In projective geometry, a specific instance of this problem that has numerous applications is determining whether, and how, a projective space can
Spread_(projective_geometry)
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
n > 1, when it is isomorphic to the projective special linear group. The first classification of simple Lie groups was by Wilhelm Killing, and this work
Simple_Lie_group
Mathematics procedure
linear group GL(V) acts on the projective space P(V) by automorphisms. A related procedure embeds a vector space V over a field K into the projective
Projectivization
Sporadic simple group
the projective special linear group of 3-dimensional space over the finite field with 4 elements (Dixon & Mortimer 1996, pp. 192–205). This group, sometimes
Mathieu_group_M24
Class of simple graphs defined from vector spaces
{\displaystyle \operatorname {Aut} (J_{q}(n,k))} isomorphic to the projective linear group P Γ L ( n , q ) {\displaystyle \operatorname {P\Gamma L} (n,q)}
Grassmann_graph
Generalizations of codimension-1 subvarieties of algebraic varieties
degree zero. As a result, for a projective curve X, the degree gives a homomorphism deg: Cl(X) → Z. For the projective line P1 over a field k, the degree
Divisor_(algebraic_geometry)
Group representation
Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible
Representation_of_a_Lie_group
Branch of mathematics
in terms of linear maps. This is also the case of homographies and Möbius transformations when considered as transformations of a projective space. Until
Linear_algebra
Mathematical operation
is a mapping between linear operators (Nikolski 1988). A simple example of a Cayley transform can be done on the real projective line. The Cayley transform
Cayley_transform
Mathematical object studied in the field of algebraic geometry
called a projective algebraic set if V = Z(S) for some S. An irreducible projective algebraic set is called a projective variety. Projective varieties
Algebraic_variety
Type of mathematical space
to mean a projective homogeneous variety, that is, a smooth projective variety X over a field F with a transitive action of a reductive group G (and smooth
Generalized_flag_variety
Projective construction in ring theory
mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring A (with 1), the projective line P1(A)
Projective_line_over_a_ring
In algebraic geometry, a weighted projective space P(a0,...,an) is the projective variety Proj(k[x0,...,xn]) associated to the graded ring k[x0,...,xn]
Weighted_projective_space
generating n-dimensional linear system of divisors on a line bundle on X. The choice of a projective embedding of X, modulo projective transformations is likewise
Algebraic geometry of projective spaces
Algebraic_geometry_of_projective_spaces
Five sporadic simple groups
the Zassenhaus groups. The Zassenhaus groups notably include the projective general linear group of a projective line over a finite field, PGL(2,Fq), which
Mathieu_group
Structure in combinatorial mathematics
the exceptional embedding of the projective special linear group PSL(2,5) in PSL(2,11) – see projective linear group: action on p points for details.
Block_design
Geometric transformation that preserves lines but not angles nor the origin
hyperplane at infinity of a projective space, the affine transformations are the projective transformations of that projective space that leave the hyperplane
Affine_transformation
Concept in mathematics
mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field
Reductive_group
Group homomorphism into the general linear group over a vector space
mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to
Group_representation
Generalization of an ordered basis of a vector space
of linear frames, for instance, any two frames are related by an element of the general linear group. Projective frames are related by the projective linear
Moving_frame
Ihara, Yasutaka (1966). "On discrete subgroups of the two by two projective linear group over p {\displaystyle {\mathfrak {p}}} -adic fields". Journal of
Ihara_zeta_function
Subspace defined by a polynomial of degree 2 over a field
by working in projective space rather than affine space. An example is the quadric surface x y = z w {\displaystyle xy=zw} in projective space P 3 {\displaystyle
Quadric_(algebraic_geometry)
Group in mathematical representation theory
discriminant group of an even lattice, equipped with its natural quadratic form. A modern point of view on the existence of the linear (not projective) Weil
Metaplectic_group
Result in linear algebra and projective geometry
In linear algebra and projective geometry, Gerbaldi's theorem, proved by Gerbaldi (1882), states that one can find six pairwise apolar linearly independent
Gerbaldi's_theorem
Group of mathematical theorems
second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting G = GL 2 (
Isomorphism_theorems
Vector satisfying some of the criteria of an eigenvector
In linear algebra, a generalized eigenvector of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} is a vector which satisfies certain criteria
Generalized_eigenvector
Concept in algebraic geometry
of the family. These arose first in the form of a linear system of algebraic curves in the projective plane. It assumed a more general form, through gradual
Linear_system_of_divisors
Mathematical set with some added structure
two-dimensional linear subspace of the (n+1)-dimensional linear space. More generally, a k-dimensional projective subspace of the projective space corresponds
Space_(mathematics)
Point found separated from another, given a point pair
In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following
Projective_harmonic_conjugate
Abstract regular 4-polytope
each edge. Its automorphism group has 660 elements. The automorphism group is isomorphic to the projective special linear group of the 2-dimensional vector
11-cell
American mathematician (born 1937)
scheme of the projective space of dimension n over the ring of integers. Construction of an "orbit scheme" M of the projective linear group PGLn acting
David_Mumford
Upper-half plane model of hyperbolic non-Euclidean geometry
a linear fractional transformation of complex numbers, and the hyperbolic motions are represented by elements of the projective special linear group
Poincaré_half-plane_model
Statistical modeling method
In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory
Linear_regression
Abstract ring with finite number of elements
from F is used in Galois geometry, with the projective linear group serving as the multiplicative group of A. Wedderburn's little theorem asserts that
Finite_ring
Concept in group theory
must be in general linear position. In other words, the n-dimensional projective transforms act transitively on the space of projective frames of R P n {\displaystyle
Multiply transitive group action
Multiply_transitive_group_action
Space with one dimension
one-parameter group under the Lie group–Lie algebra correspondence. More generally, a ring is a length-one module over itself. Similarly, the projective line over
One-dimensional_space
Method to solve optimization problems
interior of the feasible region. In 1984, N. Karmarkar proposed a projective method for linear programming. Karmarkar's algorithm improved on Khachiyan's worst-case
Linear_programming
Research program on the symmetries of geometry
example projective geometry rightly talked about conic sections, but not about circles or angles because those notions were not invariant under projective transformations
Erlangen_program
Topics referred to by the same term
11 in humans for the group P G L 2 {\displaystyle \mathrm {PGL} _{2}} in mathematics, see projective linear group and modular group This disambiguation
PGL2
Generalization of complex inner products
sesquilinear forms are antilinear (resp. linear) in their second (resp. first) argument. In a projective geometry G, a permutation δ of the subspaces
Sesquilinear_form
Euclidean space without distance and angles
affine linear transformation extends uniquely to a projective linear transformation, so the affine group is a subgroup of the projective group. For instance
Affine_space
Syllabic script used for writing Mycenaean Greek
contains Linear B Unicode characters. Without proper rendering support, you may see question marks, boxes, or other symbols instead of Linear B. Linear B is
Linear_B
Mathematical group based upon a finite number of elements
systematic exploration of finite groups of Lie type started with Camille Jordan's theorem that the projective special linear group PSL(2, q) is simple for q
Finite_group
Group without normal subgroups other than the trivial group and itself
second smallest nonabelian simple group is the projective special linear group PSL(2,7) of order 168, and every simple group of order 168 is isomorphic to
Simple_group
Specific algebraic group
\operatorname {GL} (1)} , is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional
Algebraic_torus
group Monster group Baby Monster group Bimonster Projective group Reductive group Simple group Quasisimple group Special linear group Symmetric group
List_of_group_theory_topics
PROJECTIVE LINEAR-GROUP
PROJECTIVE LINEAR-GROUP
Female
Scottish
Variant spelling of Scottish Lilias, LILEAS means "lily."
Girl/Female
Irish
Protective.
Girl/Female
Irish
Protective.
Girl/Female
Indian
Protective Angel
Girl/Female
Muslim
Protective Angel
Girl/Female
Muslim
Protective Angel
Surname or Lastname
English
English : variant of Lingard.French : occupational name for a maker of or dealer in linen goods, from Old French linge ‘linen (goods)’ (see Linge 1).
Surname or Lastname
English
English : metronymic from Line.
Boy/Male
German
Protective
Male
Yiddish
 Variant spelling of Yiddish Lieber, LIBER means "beloved." Compare with another form of Liber.
Girl/Female
Indian
Protective Angel
Male
English
Irish Anglicized form of Gaelic Fionnbarr, FINBAR means "fair-headed."
Boy/Male
Hindu
Lingam
Female
English
Variant spelling of English Linsey, LINSAY means "Lincoln's wetlands."
Girl/Female
German American
Protective.
Girl/Female
German, Swedish
Protective Victory
Boy/Male
British, English, Netherlands
Protective
Boy/Male
German
Protective
Male
Greek
(ΑἰνÎας) Variant spelling of Greek AineÃas, AINEAS means "praiseworthy."
Boy/Male
Polish
Protective shield.
PROJECTIVE LINEAR-GROUP
PROJECTIVE LINEAR-GROUP
Boy/Male
Hindu
Loved, Respected
Boy/Male
Hindu
A cavalier, A Hindu month, Medical God
Boy/Male
Hindu
Beautiful Sun
Girl/Female
Indian, Traditional
Fairy; Power
Boy/Male
Indian, Punjabi, Sikh
Protector of the Renowned
Male
Russian
(ÐрхиÌп) Russian form of Greek Archippos, ARKHIP means "master of horses."
Surname or Lastname
English
English : unexplained.
Boy/Male
Hindu
Girl/Female
American, French, German, Hebrew
Dear; Man; The Plain; Beloved Meadow
Boy/Male
Australian, Irish, Latin
Our Lord; Belonging to God
PROJECTIVE LINEAR-GROUP
PROJECTIVE LINEAR-GROUP
PROJECTIVE LINEAR-GROUP
PROJECTIVE LINEAR-GROUP
PROJECTIVE LINEAR-GROUP
a.
In the direction of a line; of or pertaining to a line; measured on, or ascertained by, a line; linear; as, lineal magnitude.
a.
Bringing into being; causing to exist; producing; originative; as, an age productive of great men; a spirit productive of heroic achievements.
adv.
In a linear manner; with lines.
a.
Of or pertaining to a line; consisting of lines; in a straight direction; lineal.
a.
Pertaining to projection, or to a projectile.
a.
Like a line; narrow; of the same breadth throughout, except at the extremities; as, a linear leaf.
a.
Of, pertaining to, or included by, two lines; as, bilinear coordinates.
a.
Composed of lines; delineated; as, lineal designs.
n.
One who lines, as, a liner of shoes.
n.
One who adjusts things to a line or lines or brings them into line.
v. t.
To mark with a line or lines; to cover with lines; as, to line a copy book.
n.
The quality or state of projecting, or being projected; projection; protrusion.
a.
Descending in a direct line from an ancestor; hereditary; derived from ancestors; -- opposed to collateral; as, a lineal descent or a lineal descendant.
a.
Linear.
n.
The representation of something; delineation; plan; especially, the representation of any object on a perspective plane, or such a delineation as would result were the chief points of the object thrown forward upon the plane, each in the direction of a line drawn through it from a given point of sight, or central point; as, the projection of a sphere. The several kinds of projection differ according to the assumed point of sight and plane of projection in each.
a.
Projecting or impelling forward; as, a projectile force.
a.
Of a linear shape.
n.
Being within view or consideration, as a future event or contingency; relating to the future: expected; as, a prospective benefit.
a.
Having the quality or power of producing; yielding or furnishing results; as, productive soil; productive enterprises; productive labor, that which increases the number or amount of products.
a.
Caused or imparted by impulse or projection; impelled forward; as, projectile motion.