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Topic: Projective linear group

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Projective special linear group

Discrete group

Fuchsian group

SL(2,Z)

Kleinian group

Tessellation

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	PlanetMath: projective special linear group
		"projective special linear group" is owned by alozano.
		Cross-references: finite, finite field, simple group, root of unity, scalar, origin, projective space, group, quotient, matrices, linear transformations, dimension, finite dimensional, quotient group, center, special linear group, field, vector space
		This is version 1 of projective special linear group, born on 2005-03-28.
	planetmath.org /encyclopedia/ProjectiveSpecialLinearGroup.html (125 words)

	Monoids and Groups. Group Theory and Symmetries - Numericana
		The centralizer in a group G of a subset E consists of all the elements of G which commute with every element of E. It is a subgroup of G. The centralizer in G of G itself is the center of G (it's the intersection of all centralizers in G).
		The alternating group is the derived subgroup of the symmetric group: A
		The derived subgroup of the Quaternion group is {+1,-1}.
	home.att.net /~numericana/answer/groups.htm (5181 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-19)
		to be the composition of a linear representation of
		The most important examples of projective representations are: the spinor representation of an orthogonal group and the Weyl representation of a symplectic group.
		The classification of the irreducible projective representations of finite groups was obtained by I.
	eom.springer.de /p/p075320.htm (299 words)

	PlanetMath: special linear group
		is defined to be the subgroup of the general linear group
		Cross-references: standard basis, matrix, linear transformation, group, field, determinant, invertible linear transformations, general linear group, subgroup, vector space
		This is version 4 of special linear group, born on 2002-02-22, modified 2005-05-04.
	planetmath.org /encyclopedia/SL.html (108 words)

	Permutation Representations of Linear Groups
		An indexed set of affine or projective points on which M acts, such that the indexing gives the correspondence between this set and the G-set of M. Furthermore, most of the function in this family are parametrized by two objects: the degree and the coefficient field of the matrix group.
		Construct the projective general linear group G = PGL(n, q), i.e., the group corresponding to the action of GL(n, q) on the projective points of the n-dimensional vector space V over K = GF(q), where n >= 2 and q is a prime power.
		Construct the projective special linear group G = PSL(n, q), i.e., the group corresponding to the action of SL(n, q) on the projective points of the n-dimensional vector space V over K = GF(q), where n >= 2 and q is a prime power.
	www.umich.edu /~gpcc/scs/magma/text325.htm (2701 words)

	How to Make the Mathieu Group M(24)
		The group of orientation-preserving transformations which preserve the unmarked complex of triangles is isomorphic to the group
		This group G(X) acts transitively on each of the sets of 24 vertices, of 84 edges, and of 56 faces, but here we are particularly interested in the action on the vertices.
		However, the group generated by all three {t,j,k} is the whole Mathieu group.
	homepages.wmich.edu /~drichter/mathieu.htm (1988 words)

	Permutation Representations of Linear Groups
		Construct the automorphism group G = PGammaL(n, q) of the projective general linear group PGL(n, q) corresponding to the n-dimensional vector space V over the field K = GF(q), where n >= 2 and q is a prime power.
		Construct the automorphism group G = PSigmaL(n, q) of the projective special linear group PSL(n, q) corresponding to the n-dimensional vector space V over the field K = GF(q), where n >= 2 and q is a prime power.
		Construct the automorphism group G = PGammaU(n, q) of the projective general unitary group PGU(n, q) corresponding to the n-dimensional vector space V over the field K = GF(q^2), where n >= 2 and q is a prime power.
	www.math.uiuc.edu /Software/magma/text250.html (1680 words)

	Read This: Adventures in Group Theory
		Adventures in Group Theory is a tour through the algebra of several "permutation puzzles." Although the main focus is on the Rubik's Cube, several other puzzles are explored to a lesser degree.
		The quaternions, finite cyclic groups, dihedral groups, and symmetric groups are presented.
		For example, the structure of this group, the center of the Rubik's Cube Group, the structure of some of the subgroups of the Rubik's Cube Group, including an embedding of the quaternions into the group, and an example of two elements which generate the whole Rubik's Cube group.
	www.maa.org /reviews/joynergroups.html (960 words)

	[ref] 47 Group Libraries
		Two permutations groups of the same degree are considered to be equivalent, if there is a renumbering of points, which maps one group into the other one.
		Additionally to the catalogue of groups there exists an identification routine for groups of small order; that is, a function that returns the catalogue number of a given group.
		All groups in the library are primitive permutation groups of the indicated degree.
	www.math.temple.edu /computing/gap/ref/CHAP047.htm (5565 words)

	What Is K-Theory (Site not responding. Last check: 2007-10-19)
		An attempt to understand how this theorem may be generalized to arbitrary rings generated the notion of projective module (direct summand of free module).
		In the seventies, Quillen introduced his Plus-construction which consists of modifying a space by adding a few cells so that a perfect subgroup of the fundamental group is killed, though the homology of the space remains unchanged.
		The key point being that the higher homotopy groups of the new space differ completely from those of the former one.
	www.math.usf.edu /~emohamed/K_THEORY.htm (450 words)

	Cartan's Corner : Longitudinal, Transverse, and Torsion Waves.
		The linear Lorentz group, and the non-linear projective fractional transformations.
		It is remarkable, that the second non-linear projective group does not require a finite limiting speed for the propagation of the discontinuity, or defect.
		The solutions to to eikonal equation can be put into three equivalence classes defined by a certain group structures, but essentially related to the three types of waves: longitudinal, transverse, torsion.
	www22.pair.com /csdc/car/carfre41.htm (467 words)

	Group theory basics
		Such groups may still have nontrivial automorphisms (such as the automorphism of the additive group Z taking each a to -a).
		linear transformations), any subgroup H of a group G is the image of a homomorphism, namely the inclusion homomorphism from H to G obtained by sending each element of H to itself.
		Simple groups play a role in group theory analogous to that of primes in number theory; indeed one readily sees that the group Z/nZ is simple if and only if n is prime.
	www.math.harvard.edu /~elkies/M55a.99/group.html (910 words)

	Cohomology and Quadrics
		Note further that the Automorphism group of QP2 is the 35-dim simple group PGL(3,Q), the projective general linear group of 3-dim Quaternionic space.
		Yang and B. Lee in hep-th/9503204 describe the relationship between the cohomology of the compact Lie algebra of a Lagrangian gauge theory and the BRST cohomology.
		The Lie sphere map linearises the action of the Lie sphere group which is a group of contact transformations in E3 generated by conformal transformations and normal shifts.
	www.valdostamuseum.org /hamsmith/coquad.html (4239 words)

	Dodecahedral Faces of the Mathieu group of degree 12
		Groups are objects in mathematics that measure symmetry in nature.
		A group is a set with a binary operation that has an inverse, an identity and is associative.
		Automorphism: An isomorphism from a group G to itself is an automorphism.
	web.usna.navy.mil /~wdj/m_12.htm (3203 words)

	Quotients of a Universal Locally Projective Polytope of Type {5,3,5} (Auxiliary Information)
		In the article mentioned, it is discovered that the symmetry group W is finite, but very large.
		This means that the largest locally projective polytope of type {5,3,5} has 5003460 dodecahedra, arranged in groups of 10 around its 10006920 vertices.
		Using the subgroups of the symmetry group, and using earlier results by the first author, it is possible to characterise all the quotients of this polytope.
	cso.ulb.ac.be /~dleemans/abstracts/535 (738 words)

	[No title]
		This is equivalent to saying that G is a finite diagonalizable group scheme, or that G is the Cartier dual of a finite abelian group, considered as a group scheme over k.
		Cp ACpnTGLp as a homomorphism of abelian groups*.
		H3TPGLp= 0; and H 2Gmis the infinite cyclic group generated by the first Chern class t of the identity character Gm = Gm, while H 2GLpis the cyclic group generated by the first Chern class of the determinant GL p= Gm, whose image in H2Gm is pt.
	hopf.math.purdue.edu /Vistoli/PGL_p.txt (15156 words)

	GAP Manual: 48.12 CharTable
		The columns of the table will be sorted in the same order, as the classes of the group, thus allowing a bijection between group and table.
		The computation of character tables needs to identify the classes of group elements very often, so it can be helpful to store a class list of all group elements.
		for the Sylow 2 subgroup of the alternating group A_{11}.
	www-groups.dcs.st-and.ac.uk /~gap/Gap3/Manual3/C048S012.htm (868 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-19)
		One of the classical groups, defined as the group of automorphisms of a skew-symmetric bilinear form
		The other real forms of this group are also sometimes called symplectic groups.
		As a consequence, its tangent mapping at a fixed point belongs to the symplectic group of the tangent space.
	eom.springer.de /s/s091820.htm (301 words)

	ABSTRACT ALGEBRA ON LINE: Structure of Groups
		Then Aut(G) is a group under composition of functions, and Inn(G) is a normal subgroup of Aut(G).
		Let x be an element of the group G. Then the elements of the conjugacy class of x are in one-to-one correspondence with the left cosets of the centralizer C(x) of x in G. Example.
		Any group homomorphism from G into the group Sym(S) of all permutations of S defines an action of G on S. Conversely, every action of G on S arises in this way.
	www.math.niu.edu /~beachy/aaol/structure.html (1547 words)

	Geometry and Topology - Cambridge University Press
		The discussion moves from Euclidean to non-Euclidean geometries, including spherical and hyperbolic geometry, and then on to affine and projective linear geometries.
		Group theory is introduced to treat geometric symmetries, leading to the unification of geometry and group theory in the Erlangen program.
		Topology combines with group theory to yield the geometry of transformation groups,having applications to relativity theory and quantum mechanics.
	www.cambridge.org /catalogue/catalogue.asp?isbn=0521613256 (288 words)

	Correction (Site not responding. Last check: 2007-10-19)
		On page 292 line 3 we incorrectly state that that the automorphism group of the projective space PG m-1
		We do not need to know what the automorphism group is, but for the sake of completeness we mention that it is given by the fundamental theorem of projective geometry:
		(q) with the automorphism group of the field Aut(F
	www.its.caltech.edu /~keevash/papers/projective-turan-correction.html (123 words)

	Indecomposable Codes -- Isometry classes of linear codes
		In order to apply the results of the theory of finite group actions, this equivalence relation for linear (n,k)-codes is translated into an equivalence relation for generator matrices of linear codes, and these generator matrices are considered to be functions
		Determination of the cycle type of a linear map or of a projectivity respectively.
		Since normal forms of regular matrices are strongly connected with companion and hypercompanion matrices (see [6]) of monic, irreducible polynomials over GF(q) it is important to know the exponent or subexponent of such polynomials (see [11][6]).
	www.mathe2.uni-bayreuth.de /frib/html/art7/art7_2.html (887 words)

	Math 213a: Complex Analysis (Fall 2003)
		A d-dimensional subspace of P(V) is the subset obtained from a (d+1)-dimensional subspace of V. It is thus itself a projective space, of dimension d.
		The action is not faithful (unless k is the field of 2 elements): the kernel is the group of scalar matrices.
		We shall soon see that the fact that fractional linear transformations preserve angles between circles is a very special case of the fundamental result that analytic functions with nonzero derivative are conformal (angle-preserving).
	www.math.harvard.edu /~elkies/M213a.03/index.html (2377 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, Section 7.7 (Site not responding. Last check: 2007-10-19)
		Prove that there are no simple groups of order 200.
		Prove that there are no simple groups of order 132.
		Prove that there are no simple groups of order 160.
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/77.html (241 words)

	577
		The course will begin with an introduction to some standard topics in the representation theory of groups, focusing in particular on finite groups.
		Special attention will be given to the two smallest non-abelian simple groups, the alternating group of order 60 and the projective special linear group of order 168.
		A group-theoretic approach allows one to extend these ideas from finite groups to the broader class of compact groups, including orthogonal and unitary groups.
	www.math.washington.edu /Grads/Courses/1999-2000/577.html (233 words)

	[No title]
		The study of Projective Geometry has its roots in the art and science of the late middle ages.
		The basic objects of study in Projective Geometry is the same as in all geometries: points, lines, planes, conics, etc. However Projective Geometry systematically introduces the concept of infinity into the geometric universe, thereby ``compactifying'' the ordinary space R^n (or C^n) by adding points at infinity.
		The prerequisites are a good understanding of linear algebra and a little abstract algebra (notion of a ring and ideal is enough).
	www.math.colostate.edu /~miranda/courses.html (695 words)

	GAP Manual: 7.65. GroupId
		The function will work for all groups of order at most 100 or whose order is a product of at most three primes.
		If is a 2- or 3-group of order at most 100, its number in the appropriate p-group library is also returned.
		Note that this list of names is neither complete, i.e., most of the groups of order 64 do not have a name even if they are of one of the types described below, nor does it uniquely determine the group up to isomorphism in some cases.
	www.math.uiuc.edu /Software/GAP-Manual/GroupId.html (763 words)

	Computation of Partial Spreads
		The projective space PG(n,q) is the geometry whose elements are the subspaces of V, with two elements being incident if one is contained in the other.
		The associated group gamma.group of automorphisms of gamma is PGL(n+1,q) in its permutation action on the lines of PG(n,q).
		Partial spreads in PG(3,q) are studied in Section 17.6 of the book: J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Clarendon Press, Oxford, 1985, where, in PG(3,q), a partial spread of size k is called a k-span and a maximal partial spread of size k is called a complete k-span.
	www.maths.qmul.ac.uk /~leonard/partialspreads (1123 words)

	twim
		The plane curves of a given degree are parametrized by a projective space.
		The projective linear group, the group of automorphisms of the projective plane, acts on this projective space.
		We study the orbits of this action, and their closures in the projective space.
	www.math.fsu.edu /~smith/HTML_twim_past/1997.10.13-17.html (260 words)

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