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	ARCC Workshop: Thompson's group at 40 years (Site not responding. Last check: 2007-11-05)
		F is the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval, where the slopes are powers of two and the places where the slope changes are dyadic rationals.
		F is a group of tree pair diagrams where elements are represented by pairs of rooted binary trees.
		F is the diagram group for one of the simplest presentations of the trivial semigroup.
	www.aimath.org /ARCC/workshops/thompsonsgroup.html (387 words)

	Thompson groups - Wikipedia, the free encyclopedia
		In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted F, T and V, which were first studied by the logician Richard Thompson in 1965.
		The Thompson groups, and F in particular, have a collection of unusual properties which have made them counterexamples to many general conjectures in group theory.
		The group F is "just non-abelian" in the sense that it is not abelian, but all its proper homomorphic images are abelian.
	en.wikipedia.org /wiki/Thompson_groups (322 words)

	Thompson_John biography
		Thompson's thesis, as is clear from its title, proved Frobenius's conjecture that a finite group with an automorphism which does not fix any group element is necessarily nilpotent.
		Thompson was an assistant at Harvard University in 1961-62, then, in 1962, he was appointed professor at the University of Chicago.
		To classify finite groups therefore reduces to two problems, namely the classification of finite simple groups and the solution of the extension problem, that is the problem of how to fit the building blocks together.
	www-history.mcs.st-andrews.ac.uk /Biographies/Thompson_John.html (1295 words)

	Group Theory for puzzles
		The tetrahedral group, the symmetry of a tetrahedron.
		The octahedral group, the symmetry of a cube or octahedron.
		The icosahedral group, the symmetry of a dodecahedron or icosahedron.
	www.geocities.com /jaapsch/puzzles/groups.htm (9058 words)

	Monoids and Groups. Group Theory and Symmetries - Numericana
		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).
		For any subset E of the group G, the subgroup generated by all the conjugates of the elements ofnbsp; E is called conjugate closure of E. It's a normal subgroup containing E. In fact, it's the smallest normal subgroup containing E (i.e, it's the intersection of all normal subgroups containing E).
		The derived subgroup of the Quaternion group is {+1,-1}.
	home.att.net /~numericana/answer/groups.htm (5267 words)

	Groups with Word Problem in NP, and Higman Embeddings
		Nevertheless not every finitely presented group with polynomial Dehn function has a simply connected asymptotic cone because if the cone is symply connected then the group has a linear isodiametric function, and Theorem 3 allows one to construct lots of groups with polynomial Dehn functions which cannot have linear isodiametric functions.
		So perhaps the class of groups with word problem in NP (which by Theorem 14 is the class of all subgroups of finitely presented groups with polynomial Dehn functions) can be considered as the class of ``tame" groups.
		The class of finitely presented groups with polynomial Dehn functions is, by Theorem 14, the ``universal" subclass of the class of all groups with word problem in NP.
	math.vanderbilt.edu /~msapir/Talk1/node6.html (1879 words)

	Math
		Let A be the automorphism group of a regular tree and let G be the subgroup generated by k random elements in A. We show that almost surely G is free of rank k and every nonidentity element of G has 0 or 2 fixpoints on the boundary of the tree.
		For locally compact topological groups with this property we show that almost all finite subsets of the group generate free subgroups.
		We characterise the abelianisation of a group, that has a presentation for which the set of relations is invariant under the full symmetric group acting on the set of generators.
	www.math.uchicago.edu /~abert/research.html (753 words)

	The Miracle Octad Generator
		The extreme case is the group corresponding to the identity, the geometry of which is too large to be of consequence.
		Perhaps the simplest example of a subgroup of the projective group in a plane is the set of all projective collineations which leave a line of the plane invariant.
		The group of the geometry of the affine n-space is the subgroup of the group of the projective n-space leaving the "hyperplane at infinity" invariant (as a set, not pointwise), as in Veblen's discussion above.
	finitegeometry.org /sc/24/MOG.html (858 words)

	Mac Lane Lecture
		Thompson, who was in the course, came to me to say that he wished to write a thesis on group theory.
		Adrian Albert was also interested in finite group theory, a subject active at Chicago from the very beginning in 1892, and found special funding to organize a special year on group theory at Chicago.
		Thompson and Walter Feit, a recent Ph.D. of Richard Brauer at Michigan, were two of the participants; during the year, they solved a famous problem by proving the Odd Order Theorem: A finite simple group that is not cyclic cannot be of odd order, which was a famous conjecture of Burnside.
	www.math.ufl.edu /fac/thompson/maclane.html (4591 words)

	Thompson_John (print-only)
		The reason was that suddenly progress began to be made on one of the main problems of finite group theory, namely the classification of finite simple groups.
		Another major early step by Thompson towards the classification of finite simple groups was his classification of those finite simple groups in which every soluble subgroup has a soluble normaliser.
		Thompson was awarded a Fields Medal for his work at the International Congress of Mathematicians in Nice in 1970.
	www-groups.dcs.st-and.ac.uk /~history/Printonly/Thompson_John.html (1290 words)

	Algebra Seminar
		It is then natural for us to define the notion of a permutably detectable group as a group G where for any direct product of finitely many copies of G, the only permutable subgroups isomorphic to G are the direct factors.
		Conjecture 4: For a given prime p, the Sylow p-subgroup of a group G is always normal in the group.
		Abstract: The commutator [M,N], where M,N are normal subgroups of a group, is the normal subgroup generated by all commutators [m,n] with m in M and n in N. This is a binary operation on the lattice of normal subgroups of a group.
	www.math.binghamton.edu /dept/AlgebraSem/s05.html (959 words)

	Simple Groups of Finite Morley rank
		The connected component of a Sylow 2-subgroup of a group of finite Morley rank is a central product UT with U 2-unipotent and T a 2-torus; the intersection of U and T is finite*.
		More generally, there are no simple groups of finite Morley rank of mixed type, and any simple group of finite Morley rank is algebraic.
		In the odd type case, any simple K-group of finite* Morley rank is either algebraic, or is minimal connected simple with Prüfer 2-rank at most two.
	www.rci.rutgers.edu /~cherlin/FMR (555 words)

	[No title]
		Thompson, J. Rational functions associated to presentations of finite groups.
		Thompson, J. A finiteness theorem for subgroups of ${\rm PSL}(2,\, R)$ which are commensurable with ${\rm PSL}(2,\, Z)$.
		Thompson, John G. Nonsolvable finite groups all of whose local subgroups are solvable.
	www.math.ufl.edu /fac/facmr/Thompson.html (902 words)

	[No title]
		Finite simple groups and localization Jose L. Rodriguez, Jer^ome Scherer and Jacques Thevenaz * Abstract The purpose of this paper is to explore the concept of localization, wh* ich comes from homotopy theory, in the context of finite* simple groups.
		Thus we restrict ourselves to the stu* dy of finite* groups and wonder if it would be possible to understand the finite localization* s of a given finite* simple group.
		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)

	Sean Cleary's Mathematics Research Interests
		Thompson's group F is not almost convex, (with Jennifer Taback) (Journal of Algebra, Vol 270}, No. 1, December 2003, pp.
		Parafree groups are groups which are residually nilpotent and have quotients with the terms in their lower central series which are isomorphic to the corresponding quotients for a free group.
		The distance from the origin in the word metric for generalizations F(p) of Thompson's group F is quasi-isometric to the number of carets in the reduced rooted tree diagrams representing the elements of F(p).
	www.sci.ccny.cuny.edu /~cleary/research.html (2769 words)

	NEIL G. THOMPSON, Ph.D. \| Staff
		Thompson has directed or contributed to 32 major research projects and numerous field studies and testing projects examining various aspects of corrosion science, corrosion monitoring and cathodic protection.
		Thompson, N. and Sosnin, H. A., "Corrosion of 50-50 Tin-Lead Solder in Household Plumbing," Welding Journal, April 1985, pp.
		Thompson, N. G., Lawson, K. M., and Beavers, J. A., "Application of Electrochemical Impedance Spectroscopy for the Detection of Ongoing Corrosion of Rebar in Concrete Structures," proceedings of the Transportation Research Board Annual Meeting, National Research Council (1989).
	www.cctechnologies.com /staff/thompson/index.htm (2978 words)

	Publications of Mark Sapir
		This group is torsion of exponent n>>1 by cyclic.
		Here is a paper (joint with Olshanskii) where we present easy quasi-isometric embeddings of relatively free groups in finitely based varieties (in particular, the free Burnside groups) and Baumslag-Solitar groups into finitely presented groups with small Dehn functions.
		In particular, we proved that a finitely generated group G has word problem in NP if and only if G is a subgroup of a finitely presented group H with polynomial Dehn function (moreover this subgroup has bounded length distortion).
	atlas.math.vanderbilt.edu /~msapir/publications.html (1130 words)

	[No title]
		A finitely generated group G is called hyperbolic if for one (and hence all) finite symmetric set S of generators the metric space (G, dS) is a hyperbo* *lic metric space.
		Denote by Out(Fn) the group of outer automorphisms of Fn, i.e.
		On fundamental groups of manifolds of nonnegative curvature.
	hopf.math.purdue.edu /Lueck/lueck_classifyingspaces1203.txt (13567 words)

	[No title]
		A polycyclic group has breadth n if it has a normal series whose factors are abelian groups with n generators.
		A group G is of polycyclic breadth n (PBn-group) if it has a normal series whose factors are abelian groups with ≤n generators.
		Beidleman was correct, and I have realized that polycyclic breadth of a group is a generalization of supersolvability, especially in the case of polycyclic breadth 2.
	www.ndsu.nodak.edu /ndsu/foguel/Research.html (365 words)

	Computer: Interview with Ken Thompson, May 1999
		People used them in smaller groups, and it was the beginning of the demise of the monster comp center, where the bureaucracy hidden behind the guise of a multimillion dollar machine would dictate the way computing ran.
		Thompson: First, I have to say that the language itself is almost exclusively the work of Sean Dorward, and in my talking about it I don't want to imply I had much to do with it.
		Thompson: Well, at first we thought it was simple: You just write a finite state machine for this phone system.
	www.cs.princeton.edu /courses/archive/spring03/cs333/thompson.html (4577 words)

	Publications of Mark Sapir
		This group is torsion of exponent n>>1 by cyclic.
		In particular, we proved that a finitely generated group G has word problem in NP if and only if G is a subgroup of a finitely presented group H with polynomial Dehn function (moreover this subgroup has bounded length distortion).
		"A variety with undecidable set of subalgebras of finite simple algebras" we construct a finitely based variety of algebras with two binary operations where the embeddability into finite simple algebras is undecidable (in semigroups, rings and groups this problem is always decidable).
	www.math.vanderbilt.edu /~msapir/publications.html (1129 words)

	Group Theory Seminar
		Each group in the class is a branch group with positive Hausdorff dimension in the topology induced by level stabilizers.
		Abstract Given a connected semisimple Lie group G, the Kazhdan-Margulis lemma says that there exists a positive lower bound for the covolume of cocomplact lattices in G. This is no longer true when G is the automorphism group of a locally finite tree or buildings.
		Because of this, it is very hard to determine an explicit finite set of instances of the 4-Engel identity that defines the group E(2,4), the two-generator, relatively free group in the variety of 4-Engel groups.
	www.math.rutgers.edu /~seminars/GroupTheory.html (970 words)

	BACKGROUND
		Note that the growth rate of the free metabelian group of rank 2 should be strictly less than 3 (which is the growth rate of the free group of rank 2) since a free metabelian group is not free.
		In this case, one may encounter a situation where neither finitely presented groups in the given class nor finitely presented groups outside of the the given class are recursively enumerable.
		The automorphism group of a free metabelian group of finite rank is known to be finitely generated unless the rank equals 3 -- see [S.Bachmuth, H.Mochizuki, Aut(F) \to Aut(F/F") is surjective for free group F of rank \geq 4, Trans.
	www.cs.gc.cuny.edu /~cryptlab/gworld/problems/Back2.html (4773 words)

	Geometric Functional Analysis Seminar Abstract (Site not responding. Last check: 2007-11-05)
		Guba and Sapir have shown, for example, that Thompson's group F is the diagram group over P= based at x.
		In this talk, an explicit construction of a contractible cubical free G-complex is given for any diagram group G. Theorem 1: When P is a finite presentation, this complex is a proper CAT(0) space and the action of G is by isometries.
		Theorem 2: If P is a finite presentation of a finite semigroup, all diagram groups over P are of type F-infinity.
	www.math.psu.edu /gfa/PastSeminars/SP00Abstracts/farley.html (129 words)

	mahdavi (Site not responding. Last check: 2007-11-05)
		Using finite state automata to study a group is a new idea that was introduced by W. Thurston, based on the work of J. Cannon [2].
		The concept of finite state automaton [4] has emerged as a significant tool in many branches of human knowledge and understanding including: linguistics, computer science, philosophy, biology, logic, etc, and in particular, recently, group theory.
		We say a group is automatic if for a set of semigroup generators A of a group G there is a regular subset R of A*(words on A) which can be mapped onto G under an appropriate evaluation map.
	www2.potsdam.edu /mahdavk/Research.htm (704 words)

	retro (Site not responding. Last check: 2007-11-05)
		UNIX was never a ``project;'' it was not designed to meet any specific need except that felt by its major author, Ken Thompson, and soon after its origin by the author of this paper, for a pleasant environment in which to write and use programs.
		Instead they must be split into groups, say ``/usr1'' and ``/usr2''; this is somewhat inconvenient, especially when space on one device runs out so that some users must be moved.
		In fact, most installations do not use groups at all (all users are in the same group), and even those that do would be happy to have more possible user IDs and fewer group IDs.
	cm.bell-labs.com /cm/cs/who/dmr/retro.html (8567 words)

	Cockcroft Properties of Thompson's Group (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Abstract: In a study of the word problem for groups, R. Thompson considered a certain group F of self-homeomorphisms of the Cantor set and showed, among other things, that F is finitely presented.
		Using results of K. Brown and R. Geoghegan, M. Dyer showed that F is the fundamental group of a finite two-complex Z 2 having Euler characteristic one and which is Cockcroft, in the sense that each map of the two-sphere into Z 2 is homologically trivial.
		1 The ubiquity of Thompson's group F in groups of piecewise li..
	citeseer.ist.psu.edu /474305.html (308 words)

	Burnside problem
		It is clear that any finite group is periodic.
		A finitely generated linear group which is finite dimensional and has finite exponent is finite i.e.
		Now (moving ahead), the classification of finite simple groups in the 1980's shows that ii.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Burnside_problem.html (1011 words)

	Mathematics Colloquium #1 (Site not responding. Last check: 2007-11-05)
		In the 1960's, Richard Thompson invented a triple of groups (F in T in V), which have since appeared throughout many different branches of mathematics.
		For example, they have provided a technique for constructing an elementary example of a finitely presented group which has unsolvable word problem, the universal obstruction to a problem in homotopy theory, the structure group for the associative law, and the first example of a group which is torsion-free, infinite dimensional, and of type infinity.
		The group F is conjectured to be an example of a finitely presented, nonamenable group which has no free subgroup on two generators.
	www.unomaha.edu /wwwmath/Colloquium/coll1.html (136 words)

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