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Topic: Special linear group

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General linear group

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Far North (district), New Zealand

	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.
		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 special linear group, SL(n, F), is the group of all matrices with determinant 1.
	en.wikipedia.org /wiki/General_linear_group (1100 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.
		The special linear group SL_n(F) is the subgroup of GL_n(F) whose elements have determinant equal to 1.
		A group that is not trivial is nontrivial.
	www.math.harvard.edu /~knill/sofia/data/group.txt (4457 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)

	PlanetMath: projective special linear group (Site not responding. Last check: 2007-10-20)
		"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/ProjectiveSpecialLinearGroup.html (125 words)

	PUBLICATIONS
		Bireflectionality of orthogonal and symplectic groups of characteristic 2.
		Linear and Multilinear Algebra 39 3 (1995) 209-230.
		Linear and Multilinear Algebra 35 1 (1993) 11-36.
	www.math.toronto.edu /ellers/publications.html (665 words)

	Cornell Math - MATH 767, Spring 2000 (Site not responding. Last check: 2007-10-20)
		Linear groups are subgroups of GL_n(F) for some field F and some n.
		Linear algebraic groups are those subgroups that are defined by polynomial equations in the matrix entries and hence are endowed with the structure of algebraic variety.
		A typical example is the special linear group SL_n(F), which is defined by the single polynomial equation det(A) = 1.
	www.math.cornell.edu /~www/Courses/GradCourses/SP00/767.html (208 words)

	Lie group - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-20)
		In mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps.
		Lie groups are important in mathematical analysis, physics and geometry because they serve to describe the symmetry of analytical structures.
		If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra g over F there is a unique (up to isomorphism) simply connected Lie group G with g as Lie algebra.
	xahlee.org /_p/wiki/Lie_group.html (1378 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/SpecialLinearGroup.html (108 words)

	Standard Matrix Groups
		Construct the orthogonal group Omega(n, K) (which is the kernel of the spinor norm map on SO(n, K)) corresponding to the n-dimensional vector space V over the field K = GF(q), where n is odd, n >= 3 and q is a prime power.
		The Suzuki groups are specified slightly differently, as the degree of the group is always four.
		Given a matrix group G and a permutation group H, construct action of the wreath product on the tensor power of G by H, which is the (image of) the wreath product in its action on the tensor power (of the space that G acts on).
	www.math.lsu.edu /magma/text296.htm (1648 words)

	The linear groups (Site not responding. Last check: 2007-10-20)
		A linear map will in general map a rectangle to a parallelogram and so even if it manages to preserve the lengths of the sides of the rectangle, it will in general stretch the diagonals and so will not preserve all lengths.
		It is a standard result about linear transformations from n dimensional spaces to n dimensional spaces that dim(null space) + dim(range) = n and so the dimension of the range of T is n and it is hence an onto map.
		The group O(n) is the union of SO(n) and the coset K.
	www-history.mcs.st-and.ac.uk /~john/geometry/Lectures/L2.html (718 words)

	Divisible Homology Classes in the Special Linear Group of a Number Field (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		Abstract: Introduction Let F be a number field and SL(F) denote the infinite special linear group over F.
		The integral homology groups of SL(F) are in general not finitely generated, but it was shown by the first author in Section 2 of [A1] that, for all integers i 0, H i (SL(F); Z) is the direct sum of a free abelian group of finite rank and a torsion group.
		1.5: Linear Group Homology Properties of the Inclusion of a Ring..
	citeseer.ist.psu.edu /257308.html (291 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.
		the insolvable affine primitive permutation groups of degree
		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.
	www.math.colostate.edu /manuals/gap/CHAP045.htm (3533 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		General Linear Group $GL(n)$: The set of regular linear transformations on $R^n$.
		Special Linear Group $SL(n)$: A subgroup of $GL(n)$, it is the set of linear transformations on $R^n$ with determinant 1.
		Special Orthogonal Group $SO(n)$: corresponds to rotations in an n-dimensional Euclidean space.
	draal.physics.wisc.edu /Notes/GroupTheory/src (295 words)

	Properties of a Matrix Group (Site not responding. Last check: 2007-10-20)
		True if the matrix group G of degree n over the field K is the general linear group GL(n, K).
		Given a group G and elements g and h belonging to G, return the value true if g and h are conjugate in G. The function returns a second value in the event that the elements are conjugate: an element k which conjugates g into h.
		True if the subgroup H of the group G is a maximal subgroup of G. This function is evaluated by constructing the permutation representation of G on the cosets of H and testing this representation for primitivity.
	www.math.colostate.edu /manuals/magma/htmlhelp/text324.html (393 words)

	PSL(2,7) (Site not responding. Last check: 2007-10-20)
		The projective special linear group G = PSL(2,7) is a finite group in mathematics that has important applications in algebra, geometry, and number theory.
		It is the automorphism group of the Klein quartic, and it is the second-smallest nonabelian simple group, next to the alternating group A
		However, PSL(2,7) is also isomorphic to SL(3,2) (= GL(3,2)), the special (general) linear group of 3×3 matrices over the field with 2 elements.
	www.worldhistory.com /wiki/P/PSL(2,7).htm (673 words)

	NCGOA Seminar, Spring 2004
		Thompson's group is the subgroup of a locally rigid group whose associated groupoid is the Cuntz groupoid.
		Abstract: In the 1940's, Hochschild introduced cohomology groups for algebras, and these were adapted by Kadison and Ringrose in the 1970's to the functional analytic setting of von Neumann algebras, the weakly closed self-adjoint subalgebras of bounded operators on a Hilbert space.
		In particular, the unitary local cocycles form a group that may be viewed as the automorphism group of the flow.
	math.vanderbilt.edu /~bisch/NCGOA_seminar_spring04.html (1461 words)

	Dihedral and General Linear Groups (Site not responding. Last check: 2007-10-20)
		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)

	[No title] (Site not responding. Last check: 2007-10-20)
		The general linear group of degree n over a ring R is the group of all invertible nxn matrices with entries in a R. Examples of rings that we know already are: integers, rational, reals, complex numbers, finite field with prime p elements, ring of numbers modulo n.
		For example, the special linear group of matrices with determinant 1 is obtained as follows: / S := SL(2,3); S; Generators(S); / 4.
		Permutation groups 4.1 Basics of Permutation groups Permutation groups are groups of functions from a set X to itself which are all one to one and onto.
	www.math.bgu.ac.il /~bessera/magma/lecture2 (732 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)

	A Constructive Recognition Algorithm for the Special Linear Group - Celler, Leedham-Green (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		Abstract: In the rst part of this note we present an algorithm to recognise constructively the special linear group.
		1 Introduction It seems possible, using Aschbacher's celebrated analysis of subgroups of classical groups [5], to develop algorithms that will answer basic questions about the group G generated by a subset X of GL(d; q), for modest values of d and q, as is already possible for permutation groups.
		Celler and C. Leedham-Green, A constructive recognition algorithm for the special linear group (to appear in Proc.
	citeseer.ist.psu.edu /196390.html (483 words)

	Modern Algebra Lecture Notes, 11/13/02 (Site not responding. Last check: 2007-10-20)
		In the case n=2, the special orthogonal group SO is the group of rotations of the plane that send the origin to itself.
		Theorem: The converse of Lagrange's Theorem is false.
		That is, there exist positive integers n, d such that d divides n, and yet there exists a group G of order n that has no subgroup H of order d.
	www.assumption.edu /Alfano/MAT351-FA02/Notes/111302.html (380 words)

	c6s3p4df1ex (Site not responding. Last check: 2007-10-20)
		The multiplicative groups of Q, R and C.
		(R) is a group, called the general linear group.
		(R) of matrices of determinant 1 also is a group, called the special linear group.
	www.win.tue.nl /~ida/alge/c6s3p4df1ex.html (176 words)

	Notes Group Theory (Site not responding. Last check: 2007-10-20)
		It can also be viewed as the factor group of
		is non-compact, meaning elements in the group matrices are not bounded and can be arbitrarily large.
		this corresponds to the Lorentz group in 4 dimensions (
	draal.physics.wisc.edu /Notes/GroupTheory?subject=congruencesubgroup... (79 words)

	Amazon.ca: Books: Generalization Steinberg Groups (Site not responding. Last check: 2007-10-20)
		The Steinberg relations are the commutator relations which hold between elementary matrices in a special linear group.
		The groups obtained here, called linkage groups, have an enormous number of interesting images, finite and infinite.
		Part of the work involves theoretical group combinatorics and the other part involves computer calculations to study the linkage structure of various interesting groups.
	www.amazon.ca /exec/obidos/ASIN/9810220286 (276 words)

	DISCRETA GAP interface
		The first string is a label for the group in ASCII, the second is the corresponding label in tex.
		This composes a group on 36 points of order 4896, isomorphic to PSL(2,17) acting diagonally on 2 x 18 points.
		Retrieves the generators for the group from the KM-file.
	www.mathe2.uni-bayreuth.de /discreta/GAP/gap_group.html (519 words)

	GAP Manual: 7 Groups
		They are represented as permutation groups, matrix groups, ag groups or even more complicated constructs as for instance automorphism groups, direct products or semi-direct products where the group elements are represented by records.
		Standard group elements may be compared with objects of other types while generic group elements may disallow such a comparison.
		must be a parent group, i.e., it must not be a subgroup of a parent group, and this parent group will be the parent of the constructed subgroup.
	www.mcs.kent.edu /system/documentation/gap/CHAP007.htm (8612 words)

	Table of contents for Library of Congress control number 98021344 (Site not responding. Last check: 2007-10-20)
		A constructive recognition algorithm for the special linear group F. Celler and C. Leedham-Green 3.
		On the characterization of finite groups by characters W. Kimmerle 10.
		Intersections of Sylow subgroups in finite groups V. Mazurov and V. Zenkov 15.
	www.loc.gov /catdir/toc/cam027/98021344.html (280 words)

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