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Topic: Computational group theory

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	Computational group theory - Wikipedia, the free encyclopedia
		Historically, other systems such as CAS (for character theory) and CAYLEY (a predecessor of MAGMA) were important.
		An excellent survey of the subject by Akos Seress of the Ohio State University, expanded from an article that appeared in the Notices of the American Mathematical Society is available on-line.
		There are also three books covering various parts of the subject: the Handbook of Computational Group Theory, by Holt, Eick and O'Brien ISBN 1584883723; Computation with Finitely-presented Groups by Sims ISBN 0521432138; and Algorithms for Permutation Groups by Seress ISBN 052166103X.
	en.wikipedia.org /wiki/Computational_group_theory (240 words)

	Category:Group theory - Wikipedia, the free encyclopedia
		In mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed in the above article.
		The branch of mathematics which studies groups is called group theory.
		Group theory is that branch of mathematics concerned with the study of groups.
	en.wikipedia.org /wiki/Category:Group_theory (105 words)

	UT Algorithms and Computational Theory Group
		Anna Gal (panni"at"cs.utexas.edu) --- Computational complexity; lower bounds; fault tolerant computing; randomness and computation; algorithms; combinatorics.
		Greg Plaxton (plaxton"at"cs.utexas.edu) --- Analysis of algorithms and theory of parallel computation.
		SIGACT sponsors the ACM Symposium on Theory of Computing (STOC) and is a co-sponsor of the ACM-SIAM Symposium on Discrete Algorithms (SODA) and the ACM Symposium on Parallel Algorithms and Architectures (SPAA).
	www.cs.utexas.edu /users/vlr/sac.html (278 words)

	20: Group Theory and Generalizations
		Group theory can be considered the study of symmetry: the collection of symmetries of some object preserving some of its structure forms a group; in some sense all groups arise this way.
		Groups acting on topological spaces are the basis of equivariant topology and homotopy theory in Algebraic Topology.
		Nielsen's theorem: subgroups of free groups are free.
	www.math.niu.edu /~rusin/known-math/index/20-XX.html (2774 words)

	References for Methods of Computational Group Theory
		Algorithms of Representation Theory by Gerhard Hiss (pp.
		Computational Aspects of Representation Theory of Finite Groups.
		Computational Aspects of Representation Theory of Finite Groups II.
	www.gap-system.org /~gap/Doc/references.html (535 words)

	Directory - Science: Math: Algebra: Group Theory
		Group Pub Forum Home Page · cached · These are the community pages for Group Theory, the mathematics of symmetry.
		Group Theory is a branch of algebra, but has strong connections with almost all parts of mathematics.
		Group Action Forum · cached · Association for the study of the theory of transformation groups and related topics.
	www.incywincy.com /default?p=388662 (523 words)

	The Computational Theory of Mind
		A second important issue in nineteenth and early twentieth century mathematics was one of delimiting the class of functions that are "computable" in the technical sense of being decidable or evaluable by the application of a rote procedure or algorithm.
		Turing's proposal was that the class of computable functions was equivalent to the class of functions that could be evaluated in a finite number of steps by a machine of the design he proposed.
		Regardless of one's outlook on the general prospects of causal theories of meaning, a sense of "meaning" that is cashed out in terms of causal covariance or causal etiology cannot be equivalent to either speaker meaning or conventional interpretability.
	plato.stanford.edu /entries/computational-mind (6686 words)

	Seminar "Group theory and topology"
		Finitely generated groups G become geometric objects when endowed with the word metric (depending on the generating set): the distance between a and b from G is the length of the shortest word representing a^{-1}b.
		The existence of a bounded-simple 2-generated group, containing a free non-cyclic subgroup, and the existence of an infinite simple bounded-generated 2-generated group are proven.
		group G and every homomorphism from F to a free group of rank 2 extends to G. Then F is a retract of G.
	math.vanderbilt.edu /~msapir/altopfall02.html (942 words)

	Computation with Finitely Presented Groups - Cambridge University Press
		It is the first text to present the fundamental algorithmic ideas which have been developed to compute with finitely presented groups, discussing techniques for computing with finitely presented groups which are infinite, or at least not obviously finite, and describing methods for working with elements, subgroups, and quotient groups of a finitely presented group.
		The author emphasizes the connections with fundamental algorithms from theoretical computer science, particularly the theory of automata and formal languages, computational number theory, and computational commutative algebra.
		The LLL lattice reduction algorithm and various algorithms for Hermite and Smith normal forms from computational number theory are used to study the abelian quotients of a finitely presented group.
	www.cup.cam.ac.uk /catalogue/catalogue.asp?ISBN=0521432138 (275 words)

	Mirago : Science: Math: Algebra: Group Theory
		Group Action Forum - Association for the study of the theory of transformation groups and related topics.
		Group Pub Forum Home Page - These are the community pages for Group Theory, the mathematics of symmetry.
		Permutation Groups Resources - Web-based resources for permutation groups and related areas in group theory and combinatorics.
	www.mirago.com /scripts/dir.aspx?cat=Top/Science/Math/Algebra/Group_Theory (589 words)

	Sylow theorem (Site not responding. Last check: 2007-10-26)
		In mathematics, especially group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G.
		The group G acts on itself or on the set of its p-subgroups in various ways, and each such action can be exploited to prove one of the Sylow theorems.
		In permutation groups, it has been proven by William Kantor that a Sylow p-subgroup can be found in polynomial time of the input (the degree of the group times the number of generators).
	www.toshare.info /en/Sylow.htm (1053 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		John Cannon (Sydney: Co-Chair) (Group theory) John has worked in several core areas of computational algebra for over twenty years but is best known for his contributions to computational group theory.
		His team was responsible for developing the Cayley system for group theory and related areas which has been in use for 15 years.
		Marty Isaacs (Madison: Group representations) Marty is an expert on group representations with particular emphasis on soluble groups.
	www.mscs.mu.edu /~mikes/CompAlg/committee.html (414 words)

	AT&T Worldnet Service - Directory
		Association for the study of the theory of transformation groups and related topics.
		The main point group symmetries of interest to defect physics by operation (reflection, rotations etc) and classification (trigonal, and cubic).
		Describes work to create a program that could be used to generate, identify, and analyze finite groups presented in the form of a Cayley Table as well as visualize the groups that are generated.
	www.att.net /cgi-bin/webdrill?catkey=gwd/Top/Science/Math/Algebra/Group_Theory (578 words)

	The Math Forum - Math Library - Group Theory (Site not responding. Last check: 2007-10-26)
		Group theory takes an abstract approach, dealing with many mathematical systems at once and requiring only that a mathematical system obey a few simple rules, seeking then to find properties common to all systems that obey these few rules.
		A short article designed to provide an introduction to finite groups - their internal properties: all those results about group theory for which a consideration of the order of elements is a central part of the question.
		A paper that presents an integrally generalized theory or conceptualization of life that includes elements of an axiomatic approach and is physically interpretable, formulated as the result of attempts to invent a holistic system of creative synthetic...more>>
	mathforum.org /library/topics/group_theory (2225 words)

	Preface of GAP 3.1
		Until well into the eighties the interest of pure mathematicians in computational group theory was stirred by, but in most cases also confined to the information that was produced by group theoretical software for their special research problems -- and hampered by the uneasy feeling that one was using fl boxes of uncontrollable reliability.
		The discussion of the plan of a system for computational group theory organized in a similar way started in the fall of 1985, programming only in the second half of 1986.
		We certainly enjoy seeing this happen, but we want to emphasize that our primary interest is the development of a group theory system and that we do not plan to try to extend it beyond our abilities into a general computer algebra system.
	www-groups.dcs.st-and.ac.uk /~gap/Doc/History/preface_3.1.html (1774 words)

	Warwick Mathematics Institute – Research Areas (Site not responding. Last check: 2007-10-26)
		A finite group can be conveniently represented in a computer either as a permutation group or as a matrix group, usually over a finite field.
		Improve algorithms for finding automorphism groups of a group and for testing two groups for isomorphism;
		The GAP and MAGMA computational algebra packages, which specialise in computational group theory, are both available at Warwick, and research in this area at Warwick is carried out in close collaboration with the administrators of these packages.
	www.maths.warwick.ac.uk /research/research_areas/comp_gp_thy.html (370 words)

	Open Questions: Algebra
		There are also indexes for group theory, (more here), ring theory, field theory, and linear algebra (more here).
		Computational group theory is one of the oldest and most developed branches of computational algebra.
		Although the classification theorem for finite simple groups was felt to be complete in 1983, and in spite of about 15,000 pages devoted to the proof, not all details had been published, and loose ends remained.
	www.openquestions.com /oq-ma007.htm (330 words)

	Seminar "Group theory and topology"
		Many of the CAT(0) groups one thinks of are known to be biautomatic (or at least automatic), but for reasons that really have nothing to do with the group being CAT(0).
		Unfortunately, the expanding constant of this family was unknown, because all proofs that a group has a property $T$ were not quantitative, and the expanding constants of this family of expanders was unknown.
		A diagram group is the fundamental group of the space of positive paths with fixed ends of a directed 2-complex.
	www.math.vanderbilt.edu /~msapir/altop.html (1495 words)

	Counting Complexity and Computational Group Theory (Site not responding. Last check: 2007-10-26)
		Computational complexity of many natural algorithmic problems are, in a sense, nicely understood.
		In the second part of the thesis we study the problem of computing a generator set of an unknown group, given a membership testing oracle for the group.
		These are the classes SYM of permutation groups, LIN(p) of linear spaces over the finite field of size p and the class 3-CNF of boolean functions represented in conjunctive normal form where each clause has at most three literals.
	www.eccc.uni-trier.de /eccc-local/ECCC-Theses/vinodchandran.html (499 words)

	MRC Research in Pure Mathematics
		A very active group works both on topics within combinatorics (especially finite geometry and design theory) and on links with algebra (permutation groups), logic (model theory), information and coding theory, and design of experiments.
		Topology and number theory are not only researched independently, but they are also used as research tools in group theory and dynamical systems.
		Research in Analysis is mostly in the areas of harmonic analysis and integral equations on groups; Jordan algebras and analysis on infinite-dimensional manifolds; operator algebras and functional analysis; and non-commutative geometry.
	www.maths.qmw.ac.uk /~mathres/res_pure.html (580 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		Much effort has been concentrated in Group Theory, as Group Theory seemed to be a suitable place to commence the development of algorithms and languages for Abstract Algebra.
		Our recent research was concentrated on the application of the computational group theory methods for investigating finitely presented groups and linear groups.
		Nowadays, they are part of GAP- a system for computational discrete algebra, elaborated with particular emphasis on Computational Group Theory.
	www.math.technion.ac.il /~techm/20041130163020041130nik (241 words)

	Colva M. Roney-Dougal (Site not responding. Last check: 2007-10-26)
		I am interested in most aspects of permutation group theory, and recently classified the primitive permutation groups of degree less than 2500.
		For the past few years I've been working off and on with the matrix group recognition problem, in particular on constructive recognition of the almost simple groups.
		For instance, I have recently developed a new algorithm for determining the conjugacy of subgroups of the general linear group.
	www.dcs.st-and.ac.uk /~colva (301 words)

	UT Algorithms and Computational Theory Group
		The algorithms and computational theory (ACT) group focuses on the theoretical foundations of computer science.
		The current research interests of faculty in the group include algorithm design, complexity theory, parallel and distributed computation, graph theory, randomized computation, computational learning theory, probabilistic methods and combinatorics.
		A major focus of the group is on the design and analysis of provably efficient algorithms for solving fundamental computational problems, where efficiency can be measured in terms of different resources such as time, space, number of processors, and number of random bits.
	www.cs.utexas.edu /~act (378 words)

	Resume
		``Computing the Discrete Fourier Transform Using Residue Number Systems in a Ring of Algebraic Integers'', with John Cozzens, IEEE Trans.
		``Computing with Matrix Groups using Permutation Representations'', with G. Cooperman and M. Tselman, Proc.
		ACM Symposium on Theory of Computer Science pp.
	www.ccs.neu.edu /home/laf/Resume.htm (1803 words)

	Alexandre V. Borovik: Research interests in Group Theory (Site not responding. Last check: 2007-10-26)
		In a joint work with AG Myasnikov and VN Remeslennikov, we made a surprising discovery that some classical algorithms of group theory may have very low complexity for most or "generic" inputs even if the corresponding problem on the group is algorithmically undecidable.
		Surprisingly there are very efficient probalilistic algorithms which allow, within certain classes of groups, to resolve this problem with any required degree of certainty.
		Altseimer and I have developed procedures for fl box recognition which originate in the model-theoretical algebra and happen to be noncommutative analogues of the Miller-Rabin primality test from the applied Number Theory.
	www.ma.umist.ac.uk /avb/grouptheory.html (371 words)

	Preface of GAP 2.4
		Up to this decade the interest of pure mathematicians in computational group theory was stirred by, but also mostly confined to the information that group theoretical software produced on their special research problems -- and hampered by the uneasy feeling that one was using fl boxes of uncontrollable reliability.
		However the last years have seen a rapid spread of interest in the understanding, design and even implementation of group theoretical algorithms, which are gradually getting accepted both as tools for a working group theoretician like standard methods of proof and as worthwhile objects of study like connections between notions expressed in theorems.
		So in contrast to the situation in older group theoretical software the GAP language is both the main implementation language and the user language of the system.
	www-groups.dcs.st-and.ac.uk /gap/Doc/History/preface_2.4.html (963 words)

	To Outside World
		A Collection of Meetings with relation to Group theory, related topics and in particular Computational Group Theory is maintained in St Andrews.
		In the Group Pub Forum you have a good chance to get answers to theoretical questions from group theory and related topics.
		ArXiv is an e-print service in the fields of physics, mathematics, non-linear science, computer science, and quantitative biology, operated by Cornell University.
	www-history.mcs.st-and.ac.uk /~gap/Contacts/outside.html (680 words)

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