
	Where results make sense

Topic: Finitely generated group

Related Topics

Prime ideal

Integer

Irrational number

Rational number

	PlanetMath: finitely generated group
		However, a finitely generated group may have subgroups that are not finitely generated.
		The finitely generated groups all of whose subgroups are also finitely generated are precisely the groups satisfying the maximal condition.
		This is version 20 of finitely generated group, born on 2002-02-03, modified 2006-04-11.
	planetmath.org /encyclopedia/FinitelyGenerated.html (192 words)

	Encyclopedia :: encyclopedia : Abelian group (Site not responding. Last check: 2007-09-21)
		The multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules.
		A typical example is the classification of finitely generated abelian groups.
		This is a special application of the fundamental theorem of finitely generated abelian groups in the case when G has torsion-free rank equal to 0.
	www.hallencyclopedia.com /Abelian_group (819 words)

	Kleinian group - Wikipedia, the free encyclopedia
		In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i.e.
		The group generated by inversion in each circle is a Kleinian group.
		The group of symmetries of the tessellation is a Kleinian group.
	en.wikipedia.org /wiki/Kleinian_group (430 words)

	PlanetMath: Zeta function of a group
		Note that these numbers are finite since a finitely generated group has only finitely many subgroups of a given index.
		However, a simpler proof for the normal zeta function is provided by the fact that a finite nilpotent group decomposes into a direct product of its Sylow subgroups.
		This is version 4 of Zeta function of a group, born on 2005-05-14, modified 2005-05-18.
	planetmath.org /encyclopedia/ZetaFunctionOfAGroup.html (307 words)

	Algebraic topology (Site not responding. Last check: 2007-09-21)
		The fundamental group of a (finite) simplicial complex does have a finite presentation.
		Finitely generated abelian group s can be completely classified and are particularly easy to work with.
		The free rank of the ''n -th homology group of a simplicial complex is equal to the n -th Betti number, so one can use the homology groups of a simplicial complex to calculate its Euler-Poincaré characteristic.
	www.nebulasearch.com /encyclopedia/article/Algebraic_topology.html (457 words)

	Automatic Groups
		When the group is a finitely generated group given in terms of generators and relations, the question is known as the word problem.
		For example, assuming that all elements of a finite group are generators, for any x in the group, the identity word and the word (x)(x^(-1)) represent the same group element.
		In order to define word-hyperbolic groups, we need to discuss Cayley graphs; the Cayley graph is a graph with a point for each element of the group and with a directed line from one point to a second point whenever the second element is the first element times one of the generators.
	www.geom.uiuc.edu /docs/forum/automaticgroups/automaticgroups.html (1192 words)

	Finitely generated abelian groups
		We first remark that any subgroup of a finitely generated free abelian group is finitely generated.
		Then we see that finitely generated abelian groups can be presented as quotients of finite rank free abelian groups, and such a presentation can be reinterpreted in terms of matrices over the integers.
		We obtain a proof of the theorem by reinterpreting in terms of groups.
	modular.fas.harvard.edu /papers/ant/html/node9.html (725 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.
		Moreover, every finitely generated recursively presented group G can be embedded into a finitely presented group H in such a way that the degree of unsolvability of the conjugacy problem in H coinsides with the degree of undecidability of the conjugacy problem in G.
		In the case when the initial group might have torsion, the solvability of the order problem is essential because it is easy to costruct a finitely generated group with solvable power problem but unsolvable order problem, which is not embeddable into any group with solvable conjugacy problem (hint: conjugated elements must have the same order).
	math.vanderbilt.edu /~msapir/Talk1/node6.html (1879 words)

	2. Residual Finiteness Results - Residually Finite Groups
		Theorem: (Mal'cev [Mal40]) A finitely generated residually finite group is Hopfian.
		Theorem: The automorphism group of a finitely generated, residually finite group is residually finite.
		A and B are finitely generated torsion-free nilpotent and C is closed in A and B. A and B are finitely generated torsion-free nilpotent and C is cyclic (recall that Higman's example [Higm51] shows that one cannot weaken the conditions to A and B residually finite)
	www.math.umbc.edu /~campbell/CombGpThy/RF_Thesis/2_RF_Results.html (3347 words)

	Magnus (Site not responding. Last check: 2007-09-21)
		Since each such group is given by finitely many generators and finitely many defining relators in the category of abelian groups, i.e., as a quotient of a free abelian group of finite rank by a subgroup generated by finitely many elements, such a decomposition can be obtained by Gaussian Elimination.
		Each of the generators occuring in the given word is then re-expressed in terms of the new generators that have been obtained in the primary decomposition, leading to a representation of the given word in terms of the primary decomposition, i.e., its primary (canonical) form.
		Automorphisms of the free abelian group of rank n are in one-to-one correspondence with matrices from the group GL_n(Z).
	zebra.sci.ccny.cuny.edu /web/caiss/MAGNUS/website/pages/checkinabeliangroup.html (1968 words)

	Indecomposable module - Wikipedia, the free encyclopedia
		Every finitely-generated abelian group is a direct sum of (finitely many) indecomposable abelian groups.
		There are, however, other indecomposable abelian groups which are not finitely generated; the rational numbers Q form the simplest example.
		A module of finite length is indecomposable if and only if its endomorphism ring is local.
	en.wikipedia.org /wiki/Indecomposable_module (416 words)

	Seminar "Group theory and topology"
		For every finitely generated group G on approach proposed by A.Olshanskii and the author is applied.
		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.
	math.vanderbilt.edu /~msapir/altopfall02.html (942 words)

	Summary of my results so far (Site not responding. Last check: 2007-09-21)
		However, given a finitely generated nilpotent group G and any subgroup H of G, the dominion of H in G in the category of all nilpotent groups is equal to H.
		The dominion of a subgroup H of a nil-2 group G in the variety of all nil-2 groups is equal to the dominion in the variety generated by G.
		In fact, there is a finitely generated group G, and a subgroup H of G, which is abelian of finite rank, such that the dominion of H in G is abelian of infinite rank.
	www.matem.unam.mx /~magidin/research/results.html (722 words)

	"Automorphisms of Graph Groups" (Site not responding. Last check: 2007-09-21)
		A graph group is a finitely generated group whose only relations are that certain of the generators are allowed to commute with one another.
		Such a group may be used to model a finite set of reversible processes, with those processes that may be run in parallel corresponding to generators which commute.
		A graph group may be described by the graph G whose vertices are the generators with commuting generators joined by an edge, and the graph is denoted by F(G).
	users.wpi.edu /~hservat/auto.html (131 words)

	Group theory Research (Site not responding. Last check: 2007-09-21)
		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.
		Finiteness properties of asynchronously automatic groups.dvi, appeared in ``Geometric Group Theory", Proceedings of a Special Research Quarter at The Ohio state University, Spring 1992, edited by Ruth Charney, Michael Davis, and Michael Shapiro, pp.
	www.math.utah.edu /~sg/old-eprints.htm (630 words)

	[ref] 42 Finitely Presented Groups
		A finitely presented group (in short: FpGroup) is a group generated by a finite set of abstract generators subject to a finite set of relations that these generators satisfy.
		Finitely presented groups are obtained by factoring a free group by a set of relators.
		Finitely presented groups are groups and so all operations for groups should be applicable to them (though not necessarily efficient methods are available.) Most methods for finitely presented groups rely on coset enumeration.
	www.math.colostate.edu /manuals/gap/CHAP042.htm (2642 words)

	Group theory Research (Site not responding. Last check: 2007-09-21)
		It concerns statements of the graph theoretic Conjecture H and its group theoretic consequences.
		This is the first of 3 projected papers on filling length in group theory, as the analogue of "space" in computer science, its calculation using duality of planar graphs, and its relation with isoperimetric inequalities for central extensions.
		A finitely presented group which occurs as the kernel of a homomorphism of a hyperbolic group to a free group satisfies a polynomial isoperimetric inequality.
	www.math.utah.edu:8080 /~sg/eprints.htm (401 words)

	Montreal Geometric Group Theory Seminar
		Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces.
		According to the definition, any acd group is a torsion-free abelian group of finite rank which has a completely decomposable subgroup of finite index.
		However, acd group properties tightly connected with their endomorphism ring characteristics are very different from those of completely decomposable groups.
	www.math.mcgill.ca /wise/ggt/seminar.html (1390 words)

	[No title] (Site not responding. Last check: 2007-09-21)
		The central theme is to consider a finitely generated group as a geometric object through its Cayley graph.
		A group is an algebraic object having a multiplication defined on it.
		In the case of infinite groups, they are often studied by associating with them finite objects of some type.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9200433.txt (137 words)

	Read This: Combinatorial Group Theory
		However, by the end of the century, finitely generated abelian groups had been classified, and von Dyck had introduced free groups and presentations of groups by generators and relations.
		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)

	What is Geometric Group Theory? (Site not responding. Last check: 2007-09-21)
		A simple definition of Geometric group theory is that it is the study of groups as geometric objects.
		Geometric group theory draws upon techniques from, and solves problems in the theory of 3-manifolds, hyperbolic geometry, combinatorial group theory, Lie groups...
		In this case we are using a generator a which is a 120 degree rotation, and a generator b which is a reflection.
	www.math.mcgill.ca /wise/ggt/cayley.html (307 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)

	Colloquia and Seminars - UNL - Department of Mathematics (Site not responding. Last check: 2007-09-21)
		We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure.
		This is deduced from the main result about finite groups: let $w$ be a locally finite group word and $d\in\mathbb{N}$.
		An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the lower central series is closed.
	www.math.unl.edu /pi/colloquia/seminarabstract-gst-20051129.txt (111 words)

	BACKGROUND (Site not responding. Last check: 2007-09-21)
		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.
		\ge 2, the matrices (1, a ; 0, 1) and (1, 0 ; a, 1) generate a free group.
		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.sci.ccny.cuny.edu /~shpil/gworld/problems/Back2.html (4848 words)

	The Math Forum - Math Library - Group Theory (Site not responding. Last check: 2007-09-21)
		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.
		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.
	www.mathforum.org /library/topics/group_theory (2240 words)

	Burnside problem
		It is clear that any finite group is periodic.
		Is a finitely generated periodic group of bounded exponent necessarily finite?
		A finitely generated linear group which is finite dimensional and has finite exponent is finite i.e.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/Burnside_problem.html (1011 words)

	Math Seminars (Site not responding. Last check: 2007-09-21)
		The spectrum of a graph is the spectrum of the discrete Laplace operator associated to this graph.
		The spectral theory of graphs and groups is extremely interesting subject related to many other fascinating topics (Ramanujan graphs and expenders, Ihara zeta function, Poisson and Martin boundary, amenability and random walks, reduced C*-algebras and idempotents).
		These groups will be generated by finite automata and will be of branch type.
	www.math.psu.edu /dynsys/abstracts-2004/grigorchuk2.html (323 words)

	Topology Festival Abstracts (Site not responding. Last check: 2007-09-21)
		We prove that every finitely generated nilpotent group of class c admits a polynomial isoperimetric function of degree c+1 and a linear upper bound on its filling length function.
		Ever since Gromov showed that the group of symplectomorphisms of the product of two 2-spheres of equal size has the homotopy type of an extension of SO(3) x SO(3) by Z/2Z, people have been interested in understanding the special properties of groups of symplectomorphisms.
		The classification theory of hyperbolic 3-manifolds (with finitely generated fundamental group) hinges on Thurston's conjecture from the late 70's, that such a manifold is uniquely determined by its topological type and a finite number of invariants that describe the asymptotic structure of its ends.
	www.math.cornell.edu /~festival/2002/abstracts.html (392 words)

Try your search on: Qwika (all wikis)