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	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)

	Geometric group theory - Wikipedia, the free encyclopedia
		Geometric group theory and combinatorial group theory are two closely related branches of mathematics, which study infinite discrete groups.
		Geometric group theory uses topological and geometric methods to study groups; the main philosophy is to deduce information about a group by analyzing how it acts on topological spaces.
		Combinatorial group theory studies discrete groups as quotients of free groups, typically described using presentations.
	en.wikipedia.org /wiki/Geometric_group_theory (275 words)

	Combinatorial game theory -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-18)
		The founders of the general theory were (additional info and facts about Elwyn Berlekamp) Elwyn Berlekamp, (additional info and facts about John Conway) John Conway and Richard Guy, in collaborative work during the 1960s that took some time fully to be published.
		The (A group of things of the same kind that belong together and are so used) set of (additional info and facts about nimber) nimbers is defined as the smallest subcollection containing 0 and containing for every G in the subcollection.
		Nimbers are the combinatorial game theoretic analogue of the (The number designating place in an ordered sequence) ordinal numbers.
	www.absoluteastronomy.com /encyclopedia/c/co/combinatorial_game_theory.htm (865 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 (120 words)

	Research and publications
		Many combinatorial properties of free Lie algebras are very similar to those of free groups, but to work with free Lie algebras is easier, so I often turn to free Lie algebras to try one or another conjecture originally made for free groups.
		During 1993-2000, my research in group theory was primarily focused on free groups and their automorphisms; in particular, on various properties of orbits under the action of the group of automorphisms of a free group.
		On the palindromic and primitive widths of a free group, J. Algebra 285 (2005), 574-585.
	www.sci.ccny.cuny.edu /~shpil/res.html (998 words)

	Amazon.com: Combinatorial Group Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete): Books: R. Lyndon,P. E. Schapp
		An Introduction to the Theory of Groups by Joseph J. Rotman
		Advanced textbook outlining combinatorial group theory, with emphasis on connections with topology, logic, and arguments of a primitive geometric nature.
		Combinatorial and Geometric Group Theory, Edinburgh 1993: Proceedings of ICMS Workshop Held at Heriot-Watt University 23 March to 8 April 1993 (London Mathematical Society Lecture Note Series) by Andrew J. Duncan on 7 pages
	www.amazon.com /Combinatorial-Theory-Ergebnisse-Mathematik-Grenzgebiete/dp/0387076425 (442 words)

	Combinatorial Group Theory (Site not responding. Last check: 2007-10-18)
		A presentation of a group G consists of a set of generators of G, together with a collection of relations amongst these generators such that any other relation amongst the generators is derivable (in a precise sense) from the given relations.
		Combinatorial group theory is the study of groups given by presentations.
		R.C. Lyndon and P.E. Schupp, Combinatorial group theory, Springer 1977.
	www.maths.gla.ac.uk /homepages/sjp/combgrpthy.html (497 words)

	Read This: Combinatorial Group Theory
		Thus, when Combinatorial Group Theory appeared in 1966, it described a relatively small corner of group theory, though one in which new significant theorems were being proved.
		Thus in the first chapter the group defined by a presentation is constructed as a group of equivalence classes of words.
		A free group is then defined as one given by a presentation whose set of relators is empty, and the definition by the universal property is relegated to an exercise.
	www.maa.org /reviews/MagnusKarrassSolitar.html (1163 words)

	Group theory Research
		Dehn functions and L_1-norms of finite presentions,.pdf, appeared in "Algorithms and Classification in Combinatorial Group Theory", edited by G. Baumslag and C. Miller III, Springer-Verlag, MSRI series vol.
		Isoperimetric and isodiametric functions of finite presentations.ps,.pdf,.dvi, appeared in Geometric Group Theory, Volume 1, edited by G. Niblo and M Roller, London Math.
		Banff lectures on hyperbolic and automatic groups.ps,.pdf,.dvi, appeared in the Proceedings of the CRM Summer School in Combinatorial Group Theory, Banff 1996, editor O. Kharlampovich, published by the AMS in the CRM Proceedings and Lecture Notes Series, vol.
	www.math.utah.edu /~sg/old-eprints.htm (630 words)

	ARCC Workshop: Dichotomy Amenable/Nonamenable in Combinatorial Group Theory
		This workshop, sponsored by AIM and the NSF, will be devoted to various incarnations of the notion of amenability for a finitely generated group.
		The main goal of the workshop is to gain better understanding of the meaning of being amenable or nonamenable for a discrete, finitely generated group.
		Algebraic, geometric and probabilistic structure of amenable and nonamenable groups (is R. Thompson's group $F$ amenable?
	www.aimath.org /ARCC/workshops/nonamenable.html (321 words)

	Read This! --- Index to book reviews on MAA Online
		Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, by W. Magnus, A. Karrass, and D. Solitar.
		Representation Theory of Finite Reductive Groups, by Marc Cabanes and Michel Enguehard.
		The Theory of Finite Groups: An Introduction, by Hans Kurzweil and Bernd Stellmacher.
	www.maa.org /reviews/titles-index.html (6696 words)

	Computational Group Theory
		Computational Group Theory (CGT) is the study of algorithms, theoretical as well as for concrete calculations, for computing with groups.
		Such algorithms find use both in group theory itself and in areas (such as combinatorics, topology or physics) that use groups to describe symmetries.
		While the language is interpreted, the implementation of fundamental routines in the system's kernel (in C) gives the convenience of an interpreted language (important for complicated mathematical algorithms) while still preserving much of the speed of a compiled language (“90% or the runtime is spent in 10% of the code'”).
	www.math.colostate.edu /~hulpke/CA/CGT.html (739 words)

	Group Theory - ABC Directory (Site not responding. Last check: 2007-10-18)
		Group theory section of the mathematics e-print arXiv.
		Group Theory is a branch of algebra, but has strong connections with almost all parts of mathematics.
		The main point group symmetries of interest to defect physics by operation (reflection, rotations etc) and classification (trigonal, and cubic).
	www.abc.net /directory/Science/Math/Algebra/Group_Theory (405 words)

	Constructing Public Key Cryptosystems Via Combinatorial Group Theory
		Combinatorial Group Theory (CGT) is the study of groups by means of generators and defining relators.
		The central methodology employed is that of Combinatorial Group Theory -- the algorithmic study of groups specified by presentations, that is by means of generators and defining relators.
		A sytematic study of braid groups was undertaken by E. Artin in 1925.
	www-cs.engr.ccny.cuny.edu /~csmma/pkc-cgt.html (3407 words)

	MIT OpenCourseWare \| Mathematics \| 18.315 Combinatorial Theory: Hyperplane Arrangements, Fall 2004 \| Readings
		On the Foundations of Combinatorial Theory: Combinatorial Geometries.
		Terao, H. "Free Arrangements of Hyperplanes and Unitary Reflection Groups." Proc.
		Your use of the MIT OpenCourseWare site and course materials is subject to the conditions and terms of use in our Legal Notices section.
	ocw.mit.edu /OcwWeb/Mathematics/18-315Fall-2004/Readings/index.htm (191 words)

	Group Theory (Site not responding. Last check: 2007-10-18)
		It is ideal for the undergraduate or the chemist who needs only a cursory understanding of group theory.
		It is divided into five parts: group theory, ring theory, modules and vector spaces, field theory and galois theory, and finite group...
		Mathematics: Loops in Group Theory and Lie Theory (De...
	www.growinglifestyle.co.uk /uk/j69647 (185 words)

	Theory Group
		The theory group brings together researchers in many areas of mathematics and physics to develop novel approaches to problems in computer science and information technology.
		Among the areas of expertise we have are statistical physics, probability theory, combinatorics, geometry and topology, theoretical computer science, and algorithms.
		In addition to the more novel efforts of the group, we also do a substantial amount of work on more traditional combinatorics, including graph theory, extremal combinatorics, random graphs, and enumeration.
	research.microsoft.com /theory (462 words)

	Montreal Geometric Group Theory Seminar
		I am going to show how one can develop a theory of algebraic extensions of subgroups of free groups.
		This is analogues to the classical theory of algebraic extensions of fields.
		I will discuss some non-trivial applications of this theory to open problems in combinatorial group theory.
	www.math.mcgill.ca /wise/ggt/seminar0203/Nov602.html (47 words)

	Powell's Books - Combinatorial Methods: Free Groups, Polynomials, Free Algebras by Alexander Mikhalev (Site not responding. Last check: 2007-10-18)
		The main purpose of this book is to show how ideas from combinatorial group theory have spread to two other areas of mathematics: the theory of Lie algebras and affine algebraic geometry.
		Some of these ideas, in turn, came to combinatorial group theory from low-dimensional topology at the beginning of the 20th Century.
		Here the authors show how the aforementioned techniques of combinatorial group theory found their way into affine algebraic geometry, a fascinating area of mathematics that studies polynomials and polynomial mappings.
	www.powells.com /cgi-bin/biblio?inkey=4-0387405623-0 (202 words)

	Combinatorial Group Theory And Applications To Geometry: Foundations Of The Classical Theory; Author: Collins, D. J.; ...
		Combinatorial Group Theory And Applications To Geometry: Foundations Of The Classical Theory; Author: Collins, D. J.; Editor: Kostrikin, A. I.; Paperback
		Beginning with a comprehensive description of the part of group theory that is rooted in topology, the book brings the reader to advanced recent work on groups relating to topological manifolds.
		This will be a valuable guide to researchers in the field and contains numerous examples, sketches of proofs, and open problems.A comprehensive description of the roots of topology, consisting of two monographs on geometric aspects of group theory.
	www.netstoreusa.com /mabooks/354/3540637044.shtml (277 words)

	Combinatorial Group Theory and Public Key Cryptography (ResearchIndex) (Site not responding. Last check: 2007-10-18)
		Combinatorial Group Theory and Public Key Cryptography (2004)
		In this paper we address the following questions: (1) whether choosing a di#erent group, or a class of groups, can remedy the situation; (2) whether some other "hard" problem from combinatorial group...
		3 decision problems in group theory and random walks (context) - Kapovich, Myasnikov et al.
	citeseer.ist.psu.edu /714268.html (400 words)

	Faculty at the Mathematics Ph.D. Program at CUNY
		Kulkarni, Ravi: differential geometry, Riemann surfaces, discontinuous groups
		Osin, Denis: Geometric group theory; Coarse geometry; Algebraic topology.
		Terilla, John: algebraic topology, differential geometry, topological quantum field theory, and deformation theory jterilla@qc.cuny.edu [Queens College]
	math.gc.cuny.edu /faculty/faculty.html (330 words)

	5B1459 Combinatorial Group Theory
		The fundamental group of a bouquet of circles is free.
		The fundamental group of a connected graph is free.
		Nielsen-Schreier Theorem: a subgroup of a free group is free.
	www.math.kth.se /~tatiana/5B1459/index.html (247 words)

	Geometric and Combinatorial Group Theory
		The course begins with a general discussion of the role of group theory in mathematics and from a discussion of how groups come to us in nature we move to the idea of (finite) generation, group presentations, and free groups; the fundamental decision problems of group theory (particularly the word problem) are introduced early.
		Group actions, with emphasis on topological and geometric representations of groups: covering spaces and the fundamental group; pi_1 as homotopy classes of loops; pushouts and the Seifert-van Kampen Theorem; every finitely presented group is the fundamental group of a compact 2-complex and a closed 4-manifold.
		Constructions: every countable group is a subgroup of a 2-generator group; infinite simple groups, etc.
	www.ma.imperial.ac.uk /~mbrids/gcgt (284 words)

	Scrooble.com for : Combinatorial Group Theory
		The common misspelling of the word combinatorial group theory are shown below.
		Google and the two satellite-TV providers have endorsed combinatorial bidding, which allows a bidder to win a package of individual licenses without needing...
		This combinatorial, hierarchical approach to memory formation provides a way for the brain to generate an almost unlimited number of unique network-level...
	www.scrooble.com /mathematics/combinatorial-group-theory.htm (100 words)

	Abstract: Combinatorial Group Theory, Dr. Carl Droms (Site not responding. Last check: 2007-10-18)
		Abstract: This talk will be a general introduction to the subject of Combinatorial Group Theory.
		My research is in Combinatorial Group Theory and Low-Dimensional Topology.
		lies at the boundary of Combinatorial Group Theory and Topological Graph Theory.
	www.math.jmu.edu /~brownet/colloquia/11092006.html (140 words)

	Groups & Group Theory (Site not responding. Last check: 2007-10-18)
		Matrix Groups: An Introduction to Lie Group Theory...
		Brings singularities and bifurcation theory to life - Singularities and Groups in Bifurcation Theory has been written in a frankly partisian spirit.
		The authors believe that singularity theory offers an extremeley useful approach to bifurcation...
	www.growinglifestyle.co.uk /uk/j202836 (111 words)

	Amazon.com: Classical Topology and Combinatorial Group Theory: Books: John Stillwell (Site not responding. Last check: 2007-10-18)
		A Combinatorial Introduction to Topology by Michael Henle
		Focusing on historical background and visual interpretation of results, it emphasizes spaces with few dimensions, where visualization is possible, and interaction with combinatorial group theory via the fundamental group.
		Most of the results and proofs are known, but some have been simplified or placed in a new perspective.
	www.amazon.com /Classical-Topology-Combinatorial-Group-Theory/dp/0387979700 (584 words)

	Bibliography
		A Generalization of Gröbner Basis Algorithms to Polycyclic Group Rings.
		A Generalization of Gröbner Bases Algorithms to Nilpotent Group Rings.
		Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations.
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/14/paper_html/node8.html (359 words)

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