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	General linear group - Wikipedia, the free encyclopedia
		In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication.
		The special linear group, written SL(n, F) or SL(n), is the subgroup of GL(n, F) consisting of matrices with determinant 1.
		These groups are important in the theory of group representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of polynomials.
	en.wikipedia.org /wiki/General_linear_group (1142 words)

	Projective linear group - Wikipedia, the free encyclopedia
		The projective linear group of a vector space V over a field F is the quotient group
		where GL(V) is the general linear group on V and Z(V) is the group of all nonzero scalar transformations of V.
		The projective linear groups therefore generalise the case PGL(2,C) of Möbius transformations (sometimes called the Möbius group), which acts on the projective line.
	en.wikipedia.org /wiki/Projective_linear_group (244 words)

	Group of Lie type - Wikipedia, the free encyclopedia
		In mathematics, a group of Lie type is a finite group related to the points of a simple algebraic group with values in a finite field.
		A classical group is, roughly speaking, a special linear, orthogonal, symplectic, or unitary group.
		The theory was clarified by the theory of algebraic groups, and the work of Claude Chevalley in the mid-1950s on the Lie algebras by means of which the Chevalley group concept was isolated.
	www.wikipedia.org /wiki/Ree_group (1239 words)

	Projective linear group -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		The projective linear group of a (additional info and facts about vector space) vector space V over a (A piece of land cleared of trees and usually enclosed) field F is the (additional info and facts about quotient group) quotient group
		Note that the groups Z(V) and SZ(V) are the (A point equidistant from the ends of a line or the extremities of a figure) centers of GL(V) and SL(V) respectively.
		The projective linear groups therefore generalise the case PGL(2) of (additional info and facts about Möbius transformation) Möbius transformations (sometimes called the (additional info and facts about Möbius group) Möbius group), which acts on the (additional info and facts about projective line) projective line.
	www.absoluteastronomy.com /encyclopedia/p/pr/projective_linear_group.htm (373 words)

	General linear group -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		The special linear group, written SL(n, F) or SL(n), is the ((mathematics) a subset (that is not empty) of a mathematical group) subgroup of GL(n, F) consisting of matrices with (A determining or causal element or factor) determinant 1.
		These groups are important in the theory of (additional info and facts about group representation) group representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of (A mathematical expression that is the sum of a number of terms) polynomials.
		The special linear group, SL(n, F), is the group of all matrices with (A determining or causal element or factor) determinant 1.
	www.absoluteastronomy.com /encyclopedia/G/Ge/General_linear_group.htm (1273 words)

	Permutation Representations of Linear Groups (Site not responding. Last check: 2007-10-08)
		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 parametrised by two objects: the degree and the coefficient field of the matrix group.
		Integers n and q corresponding to the degree and the field GF(q) of M (GF(q^2) in the case of the unitary groups).
		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.
	www.math.uga.edu /~matthews/DOCS/MAGMA/text299.html (2342 words)

	PlanetMath: projective special linear group (Site not responding. Last check: 2007-10-08)
		"projective special linear group" is owned by alozano.
		Cross-references: finite, finite field, simple group, theorem, root of unity, scalar, origin, projective space, group, 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/PSL.html (125 words)

	Encyclopedia: Zassenhaus group
		In mathematics, a Zassenhaus group is a certain sort of doubly transitive permutation group.
		In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M).
		In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g-1ng is still in N. The statement N is a normal subgroup of G is written:.
	www.nationmaster.com /encyclopedia/Zassenhaus-group (474 words)

	PlanetMath: special linear group (Site not responding. Last check: 2007-10-08)
		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/SpecialLinearGroup.html (108 words)

	[No title]
		Special attention has been set on the category of groups Gr, as the effect of a* * homotopi- cal localization on the fundamental group is often best described by a localiza* *tion functor L : Gr !
		The classical project* ive special* linear groups L2(q) = P SL2(q) of type A1(q), as well as the projective special* * unitary groups U3(q) = P SU3(q) of type 2A2(q), are almost all connected to an alternat* ing group* by a localization.
		The Conway groups are com* plete, the smaller ones are maximal simple subgroups of Co1 and there is a unique conj *ugacy class of each of them in Co1 as indicated in the ATLAS [4, p.180].
	hopf.math.purdue.edu /Rodriguez-Scherer-Thevenaz/simplegroups.txt (6910 words)

	[ref] 45 Group Libraries
		The first argument in such a pair is a function that can be applied to the groups in the library, and the second argument is either a single value that this function must return in order to have this group included in the selection, or a list of such values.
		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 any given group of suitable order.
	www.math.colostate.edu /manuals/gap/CHAP045.htm (3533 words)

	Articles - Lorentz group (Site not responding. Last check: 2007-10-08)
		Thus, the Lorentz group is an isotropy subgroup of the isometry group of Minkowski spacetime.
		The Lorentz group is a 6-dimensional noncompact Lie group which is not connected, and whose connected components are not simply connected.
		Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this MÃ¶bius transformation is a dilation from the origin.
	www.lastring.com /articles/Lorentz_group (3039 words)

	DISCRETA GAP interface
		Computes the Kramer-Mesner matrix for a given group and for integer parameters t and k (with $0 \le t \lt k \le d$, where $d$ is the degree of the permutation group).
		where [group] is the label of the group and [t] and [k] stand for the values of t and k, respectively.
		The file generators_fname contains generators for a permutation group on v points and is used to span the design (the degree of the group must be the same as the number of points specified in the base-block-file).
	www.mathe2.uni-bayreuth.de /betten/DISCRETA/discreta_gap.html (2689 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.math.uiuc.edu /Software/GAP-Manual/CharTable.html (881 words)

	Permutation Representations of Linear Groups
		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.math.ufl.edu /help/magma/text270.html (1680 words)

	default (Site not responding. Last check: 2007-10-08)
		Characters of the automorphism group of the group GL Algebra, 108 (1) 1987, 256-268, MR 88c: 20017, Zbl.
		With M.R. Pournaki, On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group, J. Linear and Multilinear Algebra, Vol.47, 137-149 (2000), MR 2001a: 20025, Zbl.
		With M.Ghorbany, A. Daneshkhah and H. Behravesh, Quasi-permutation representation of the group GL (q), J. Alg 243, 142-167 (2001).
	www.fos.ut.ac.ir /~darafsheh/pub.htm (1271 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)

	[No title]
		Many of the arguments are reminiscent of techniques, successfully used i* n the study of p-compact groups* and p-completed classifying spaces of finite groups.
		In particular al* l reduced homology groups* of Sm {d} are torsion groups and thus by [3] there is a homotopy equivalence of Sm {d} to the product of all its p-primary localizations for pri* *mes p 3 dividing d.
		The_automorphism group of _iis given* * by_Gi_for i =_1;_2 whereas the automorphism group of _0is trivial.
	hopf.math.purdue.edu /Broto-Levi/snk.txt (8766 words)

	Diamond Theory: Symmetry in Binary Spaces (Site not responding. Last check: 2007-10-08)
		For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4x4 array.
		In the 4x4 case, D is a four-diamond figure (left, below) and G is a group of 322,560 permutations generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2x2 quadrants.
		A table of the octahedral group O using the 24 patterns from the 2x2 case of the diamond theorem.
	m759.freeservers.com (1917 words)

	The Subgroups of #tex2html_wrap_inline2113#, or How to Compute the Table of Marks of a Finite Group.
		The concept of a table of marks of a finite group G was introduced by William Burnside in the second edition of his classic book Theory of Groups of Finite Order [Burnside 1911, chapter XII,].
		The purpose of this article is to introduce a method for the construction of the table of marks which is independent of the knowledge of the complete subgroup lattice of G and therefore can be applied to groups G which are too big to compute their complete subgroup lattice.
		Section 1 recalls basic properties of finite group actions, introduces the table of marks and describes its relation to the Burnside ring of a finite group.
	schmidt.ucg.ie /~goetz/pub/marks/marks.html (948 words)

	DISCRETA GAP interface
		The first string is a label for the group in ASCII, the second is the corresponding label in tex.
		Writes generators for the GAP permutation group grp in DISCRETA format into the file fname.
		Retrieves the generators for the group from the KM-file.
	www.mathe2.uni-bayreuth.de /discreta/GAP/gap_group.html (519 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, Section 7.7 (Site not responding. Last check: 2007-10-08)
		The set of all n × n matrices with entries in F and determinant 1 is called the special linear group over F, and is denoted by SL n
		Prove that there are no simple groups of order 200.
		Prove that there are no simple groups of order 132.
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/77.html (241 words)

	PSL(2,7) (Site not responding. Last check: 2007-10-08)
		Then G = PSL(2,7) is defined to be the quotient group SL(2,7) / obtained by identifying I and −I. In this article, we let G denote any group isomorphic to PSL(2,7).
		The Fano plane can be used to describe multiplication of octonions, so G acts on the set of octonion multiplication tables.
		Its group of conformal transformations is none other than G.
	www.worldhistory.com /wiki/P/PSL(2,7).htm (673 words)

	The Cross Group of the Rubik's Cube
		The cross group is the subgroup of the symmetric group on the set V of vertices of the cube generated by the moves XY, where X and Y are elements of the set of basic moves {R, L, U, D, F, B}.
		We will call this group C.) is the projective special linear group of degree 2 over F
		Those matrices define a Mobius transform that will permute one line to another as their related corners are permuted to another.
	web.usna.navy.mil /~wdj/rubik/crossgp.html (592 words)

	Encyclopedia: Projective linear group (Site not responding. Last check: 2007-10-08)
		People who viewed "Projective linear group" also viewed:
		The projective linear groups therefore generalise the case PGL(2) of Möbius transformations (sometimes called the Möbius group), which acts on the projective line.
		Click for other authoritative sources for this topic (summarised at Factbites.com).
	www.nationmaster.com /encyclopedia/Projective-linear-group (272 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Abstract:Varadhan's integration theorem, one of the corner stones of large-deviation theory, is generalized to the context of capacities.
		It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors.
		An overview of $t$-$(q+1,k,\lambda)$ designs with $PSL(2,q)$ as group of automorphisms and with $(t,k) \in \{(4,5), (4,6), (5,6)\}$ is included.
	www.maths.tcd.ie /EMIS/journals/CMUC/cmuc9604/abstrall.htm (1333 words)

	week79
		The symmetry group of the tetrahedron is the group A
		The symmetry group of the icosahedron is A
		Again, it's good to consider the example of sets and groups: if c is a set, RLc is the underlying set of the free group on c.
	math.ucr.edu /home/baez/week79.html (2417 words)

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