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Topic: General linear group

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	PlanetMath: general linear group
		; this is isomorphic to the group of two-by-two matrices with integer entries having determinant
		The general linear group is an example of a group scheme; viewing it in this way ties together the properties of
		This is version 5 of general linear group, born on 2002-02-22, modified 2007-01-30.
	planetmath.org /encyclopedia/GeneralLinearGroup.html (257 words)

	Monoids and Groups. Group Theory and Symmetries - Numericana
		The derived subgroup of a group is generated by its commutators.
		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 derived subgroup of the Quaternion group is {+1,-1}.
	home.att.net /~numericana/answer/groups.htm (5271 words)

	NationMaster - Encyclopedia: General linear group
		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.
		The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n by n invertible matrices with entries in K. www.ebroadcast.com.au /lookup/encyclopedia/li/Linear_map.html
		More generally, the general linear group of degree n over any field F (such as the complex numbers), or a ring R (such as the ring of integers), is the set of n×n invertible matrices with entries from F (or R), again with matrix multiplication as the group operation.
	www.nationmaster.com /encyclopedia/General_linear_group/Special_linear_group (1138 words)

	General linear group: Definition and Links by Encyclopedian.com
		In abstract algebra, the general linear group of degree n over a field F (written as GL(n,F)) is the group of n-by-n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication.
		GL(n, F) and subgroups of GL(n, F) are important in the development 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.
		The special linear group, SL(n, F), is the group of all matrices with determinant 1 (that this forms a group follows from the rule of multiplication of determinants).
	www.encyclopedian.com /ge/General-linear-group.html (641 words)

	[No title]
		Group theory is a powerful method for analyzing abstract and physical systems in which symmetry --the intrinsic property of an object to remain invariant under certain classes of transformations-- is present because the mathematical study of symmetry is systematized and formalized in group theory.
		general linear group +------------------------------------------------------------ The n x n general linear group GL_n(F) is the set of n x n matrices with entries in the field F and nonzero determinant, under the law of composition of matrix multiplication.
		A group that is not trivial is nontrivial.
	abel.math.harvard.edu /~knill/sofia/data/group.txt (4457 words)

	Projective linear group - Biocrawler (Site not responding. Last check: )
		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) of Möbius transformations (sometimes called the Möbius group), which acts on the projective line.
	www.biocrawler.com /encyclopedia/Projective_linear_group (228 words)

	Science Fair Projects - General linear group
		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.
		The special linear group, SL(n, F), is the group of all matrices with determinant 1.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/General_linear_group (1104 words)

	[No title]
		As a result, we may find a positive correlation in the first group (the more intelligent, the better the performance), but a zero or slightly negative correlation in the second group (the more intelligent the student, the less likely he or she is to acquire math skills from the particular textbook).
		General Linear Models: This extremely comprehensive chapter discusses a complete implementation of the general linear model, and describes the sigma-restricted as well as the overparameterized approach.
		General Regression Models: This chapter discusses the between subject designs and multivariate designs which are appropriate for stepwise regression as well as discussing how to perform stepwise and best-subset model building (for continuous as well as categorical predictors).
	www.statsoft.com /textbook/stanman.html (5994 words)

	Unitary group - ExampleProblems.com
		In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices with complex entries, with the group operation that of matrix multiplication.
		The unitary group is a subgroup of the general linear group GL(n, C).
		The unitary group U(n) is endowed with the relative topology as a subset of M
	www.exampleproblems.com /wiki/index.php?title=Unitary_group&redirect=no (432 words)

	PlanetMath: general linear group (Site not responding. Last check: )
		; this is isomorphic to the group of two-by-two matrices with integer entries having determinant
		The general linear group is an example of a group scheme; viewing it in this way ties together the properties of
		This is version 5 of general linear group, born on 2002-02-22, modified 2007-01-30.
	www.planetmath.org /encyclopedia/GeneralLinearGroup.html (264 words)

	Contents
		The ``standard treatment'', which emphasizes the classification of groups, is put aside in favor of a treatment which emphsizes examples and computations.
		The Cayley graph of a group, homomorphisms, and quotient groups are introduced.
		The automorphism group of a linear error-correcting code and more campanology and the theory of the Rubik's cube are discussed as applications.
	web.usna.navy.mil /~wdj/book/node1.html (878 words)

	Glossary of terms for Fermat's Last Theorem
		A homomorphism from an abstract group to a general linear group.
		Group representations have numerous theoretical and applied uses, since matrix groups have well-known properties and are easy to compute with.
		This is a generalization of the concept of nearness obtained from a numerical measure of "distance" between two points.
	cgd.best.vwh.net /home/flt/flt10.htm (2633 words)

	The Dispatch - Serving the Lexington, NC - News (Site not responding. Last check: )
		A (real) Lie group is a mathematical group which is also a finite-dimensional real smooth manifold, and in which the group operations of multiplication and inversion are smooth maps.
		The group of upper triangular n by n matrices is a solvable Lie group of dimension n(n + 1)/2.
		The group of smooth maps from a manifold to a finite dimensional group is called a gauge group, and is used in quantum field theory and Donaldson theory.
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Lie_group (3298 words)

	Lie group Summary
		The Lorentz group and the Poincare group of isometries of spacetime are Lie groups of dimensions 6 and 10 that are used in special relativity.
		The group U(1)×SU(2)×SU(3) is a Lie group of dimension 1+3+8=12 that is the gauge group of the standard model, whose dimension corresponds to the 1 photon + 3 vector bosons + 8 gluons of the standard model.
		The group of smooth maps from a manifold to a finite dimensional group is called a gauge group, and is used in quantum field theory and Donaldson theory.
	www.bookrags.com /Lie_group (4005 words)

	Glossary of terms for Fermat's Last Theorem (Site not responding. Last check: )
		A homomorphism from an abstract group to a general linear group.
		Group representations have numerous theoretical and applied uses, since matrix groups have well-known properties and are easy to compute with.
		This is a generalization of the concept of nearness obtained from a numerical measure of "distance" between two points.
	gyral.blackshell.com /flt/flt10.htm (2633 words)

	PlanetMath: special linear group (Site not responding. Last check: )
		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.
	www.planetmath.org /encyclopedia/SpecialLinearGroup.html (108 words)

	Maths - Group Theory - Martin Baker
		In general, for groups, there is no requirement for commutativity, so a * b is not necessarily equal to b * a.
		The group SU(2) is equivalent to (is isomorphic to) the set of unit length quaternions with multiplication.
		SU(n) is a subgroup of the unary group U(n) which is a subgroup of the general linear group GL(n).
	www.euclideanspace.com /maths/algebra/groups/types/index.htm (1177 words)

	Finite group Summary
		A general notation for finite arithmetic sequences is a, a+d, a+2d, a+3d,..., a+(n-1)d.
		Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the local theory, and the theory of solvable groups and nilpotent groups.
		Finite groups are directly relevant to symmetry, when that is restricted to a finite number of transformations.
	www.bookrags.com /Finite_group (1026 words)

	Springer Online Reference Works
		Although groups took up a position in the theory of differential equations somewhat different from that in the theory of algebraic equations, this led to the creation of the theory of Lie groups, and also to the theory of algebraic groups, which has deep connections with many branches of mathematics.
		is a linear Lie group, that is, a subgroup of the general linear group
		In the study of the structure of semi-simple Lie groups an important role is played by their maximal compact subgroups, studied by Cartan in close connection with the theory of symmetric spaces (see [10]).
	eom.springer.de /l/l058590.htm (2286 words)

	How to Make the Mathieu Group M(24) (Site not responding. Last check: )
		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)

	ABSTRACT ALGEBRA ON LINE: Groups
		A group G is said to be a finite group if the set G has a finite number of elements.
		Let G be a group, and let H be a subset of G. Then H is called a subgroup of G if H is itself a group, under the operation induced by G. Proposition.
		Any subgroup of the symmetric group Sym(S) on a set S is called a permutation group or group of permutations.
	www.math.niu.edu /~beachy/aaol/groups.html (1115 words)

	general linear group@Everything2.com
		The general linear group is a useful entry point into the mathematical realm of group theory (roughly, the abstract study of symmetries), since it ties that subject into the more familiar (for most undergraduates) linear algebra.
		Translating abstract notions of groups into more recognisable properties of matrices (or, in turn, linear maps) can make it easier to grasp them at first, and provides some indication of the power of such abstraction once it is subsequently applied to other groups (such as those of permutations).
		The notion of the general linear group, besides providing a compact way of refering to the invertible matrices, is a starting point for several relations and groups of interest in linear algebra/group theory.
	www.everything2.com /index.pl?node_id=1638638 (604 words)

	[General] PHP still the king, ruby follows deep down!
		I agree the hype's created but users switching to rails are doing it mostly following the hype and I know quite a many who returned to the good old environment of PHP after switching to rails.
		Unless you have a real strong reason to switch to another language, you should stick with the one you're familiar and comfortable with.
		I like php too and I am sure it will remain the language of choice for many but I think ruby has a bright future ahead of it and its worth learning.
	spicefuse.com /general-php-still-king-ruby-follows-deep-down-t-38.html (810 words)

	Group White Sheet FAQ
		The important consequence of faithful representations is that the original group is embedded as a subgroup of the symmetric group.
		A primitivity block is a subset of the set on which the group acts in which either the group moves the elements within the subset one to another, or it swaps all the elements of the block with the elements of another block.
		Every group action has at least two block systems of imprimitivity which correspond to the two trivial partitions: first, the partition of every element into on equivalence class, and second the partition where every element is in its own class.
	www.uoregon.edu /~jwilson7/math/WhiteSheetFAQ.html (795 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.
	euler.slu.edu /~blyth/gap/htm/ref/CHAP045.htm (3533 words)

	Dihedral and General Linear Groups (Site not responding. Last check: )
		All other group elements are involutions, except for the identity element (leave the coin alone).
		The general linear group of order n is the set of nonsingular n×n matrices under multiplication.
		The special linear group is the kernel of the homomorphism implemented by the determinant, namely the n×n matrices with determinant 1.
	www.mathreference.com /grp,dih.html (310 words)

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