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Group type in algebra
countable group that is not finitely generated. Every quotient of a finitely generated group G is finitely generated; the quotient group is generated by the
Finitely_generated_group
Commutative group where every element is the sum of elements from one finite subset
a finite (hence finitely generated) abelian group. Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian
Finitely generated abelian group
Finitely_generated_abelian_group
Abstract algebra concept
generated (the images of the generators in the quotient give a finite generating set), a subgroup of a finitely generated group need not be finitely generated
Generating_set_of_a_group
Mathematical term in group theory
growth rate of a finitely generated group is at most exponential and it was also understood early on that finitely generated nilpotent groups have polynomial
Grigorchuk_group
Area in mathematics devoted to the study of finitely generated groups
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties
Geometric_group_theory
Specification of a mathematical group by generators and relations
a finite presentation. A group is finitely generated (respectively finitely related, finitely presented) if it has a presentation that is finitely generated
Presentation_of_a_group
In algebra, module with a finite generating set
concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional
Finitely_generated_module
Theorem in group theory
the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G {\displaystyle G} has more
Stallings theorem about ends of groups
Stallings_theorem_about_ends_of_groups
If G is a finitely generated group with exponent n, is G necessarily finite?
Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William
Burnside_problem
Smallest cardinality of a generating set for a group
are finitely generated subgroups of a free group F(X) such that the intersection L = H ∩ K {\displaystyle L=H\cap K} is nontrivial, then L is finitely generated
Rank_of_a_group
Concept in category theory
generated free group. Finitely generated group Finitely generated monoid Finitely generated abelian group Finitely generated module Finitely generated ideal Finitely
Finitely_generated_object
Type of algebra
mathematics, a finitely generated algebra (also called an algebra of finite type) over a (commutative) ring R {\displaystyle R} , or a finitely generated R {\displaystyle
Finitely_generated_algebra
infinitely generated such as ( Q , + ) {\displaystyle (\mathbb {Q} ,+)} or any Lie group; finitely presented such as any free group (on a finite set of generators)
Infinite_group
Problem in finite group theory
of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G {\displaystyle G} is the algorithmic problem
Word_problem_for_groups
Statement in abstract algebra
for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
Commutative group (mathematics)
This is generalized by the fundamental theorem of finitely generated abelian groups, with finite groups being the special case when G has zero rank; this
Abelian_group
Theorem in geometric group theory
finitely generated abelian groups: any group which is quasi-isometric to a finitely generated abelian group contains a free abelian group of finite index
Gromov's theorem on groups of polynomial growth
Gromov's_theorem_on_groups_of_polynomial_growth
Mathematical concept
is finitely generated, and any Cayley graph for G {\displaystyle G} is quasi-isometric to Y {\displaystyle Y} . Thus a group is (finitely generated and)
Hyperbolic_group
Group with subnormal series where all factors are abelian
finitely generated. The alternating group A4 is an example of a finite solvable group that is not supersolvable. If we restrict ourselves to finitely
Solvable_group
Type of mathematical group
{\displaystyle |m|\geq 2} is finitely generated, residually finite, and Hopfian , but B ( 2 , 3 ) {\displaystyle B(2,3)} is finitely generated yet not Hopfian, and
Residually_finite_group
Type of group used in topology and geometric group theory
In mathematics, a CAT(0) group is a finitely generated group with a group action on a CAT(0) space that is geometrically proper, cocompact, and isometric
CAT(0)_group
Mathematical group based upon a finite number of elements
generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra. A group of Lie type is a group closely
Finite_group
group is also boundedly generated. A finitely generated torsion group must be finite if it is boundedly generated; equivalently, an infinite finitely
Boundedly_generated_group
automatic group is a finitely generated group equipped with several finite-state automata. These automata represent the Cayley graph of the group. That is
Automatic_group
Mathematical property
the group. It is mostly of interest for the study of infinite groups. Special cases of groups with finiteness properties are finitely generated and finitely
Finiteness properties of groups
Finiteness_properties_of_groups
Theorem in algebra
David Muller and Paul Schupp in 1983. Let G be a finitely generated group with a finite marked generating set X, that is a set X together with the map π
Muller–Schupp_theorem
Mathematical concept
precisely the finitely generated virtually nilpotent groups. This terminology is also used when P is just another group. That is, if G and H are groups then G
Virtually
Branch of mathematics that studies the properties of groups
are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the
Group_theory
Every subgroup of a cyclic group is cyclic, and if finite, its order divides its parent's
More generally, a finitely generated group is cyclic if and only if its lattice of subgroups is distributive and an arbitrary group is locally cyclic
Subgroups_of_cyclic_groups
Type of mathematical group
linear group is locally finite. In particular, if it is finitely generated then it is finite. Selberg's lemma: any finitely generated linear group contains
Linear_group
exactly the finitely generated subgroups of finitely presented groups. Since every countable group is a subgroup of a finitely generated group, the theorem
Higman's_embedding_theorem
Ukrainian mathematician
of finitely generated groups of intermediate growth. Grigorchuk's group has a number of other remarkable mathematical properties. It is a finitely generated
Rostislav_Grigorchuk
Group in which each element has finite order
all dihedral groups. None of these examples has a finite generating set. Explicit examples of finitely generated infinite periodic groups were constructed
Torsion_group
Topics referred to by the same term
finitely presented may refer to: finitely presented group finitely presented monoid finitely presented module finitely presented algebra finitely presented
Finitely_presented
Limit groups over free groups
group is a finitely generated group for which a presentation arises in this way for some 1 ≤ k ≤ n {\textstyle 1\leq k\leq n} . A finitely generated group
Limit_group
finite cyclic group is a torsion abelian group. A finitely generated abelian group is torsion iff it is finite, see Finitely generated abelian group#Classification
Torsion_abelian_group
Group that admits a formal description in terms of reflections
edge by an edge labelled 6. Also note that every finitely generated Coxeter group is an automatic group. Dynkin diagrams have the additional restriction
Coxeter_group
Group whose Cayley graph is an initially subamenable graph
products. A finitely generated group is sofic if it is the limit of a sequence of sofic groups. The limit of a sequence of amenable groups (that is, an
Sofic_group
Mathematics concept
\varphi } can be generated by the conjugates of finitely many elements of F {\displaystyle F} , then G {\displaystyle G} is finitely presented. If S {\displaystyle
Free_group
a finitely generated co-Hopfian group need not be co-Hopfian, and being co-Hopfian is not a quasi-isometry invariant for finitely generated groups. Baumslag–Solitar
Co-Hopfian_group
Equivalence relation of groups
H2. For example: A group is finite if and only if it is commensurable with the trivial group. Any two finitely generated free groups on at least 2 generators
Commensurability (group theory)
Commensurability_(group_theory)
Abelian group with no non-trivial torsion elements
with finite order. While finitely generated abelian groups are completely classified, not much is known about infinitely generated abelian groups, even
Torsion-free_abelian_group
Set of mathematical functions concerning algebraic group isomorphism
including computational group theory, k-theory, and knot theory. Let F n {\textstyle F_{n}} be a finitely generated free group of rank n {\textstyle n}
Nielsen_transformation
Set with associative invertible operation
Any finite abelian group is isomorphic to a product of finite cyclic groups; this statement is part of the fundamental theorem of finitely generated abelian
Group_(mathematics)
Locally compact topological group with an invariant averaging operation
but not finitely presented. However, in 2002 Sapir and Olshanskii found finitely presented counterexamples: non-amenable finitely presented groups that have
Amenable_group
Concept in mathematics
mapping class group of a surface is a finitely generated group. The smallest number of Dehn twists that can generate the mapping class group of a closed
Mapping class group of a surface
Mapping_class_group_of_a_surface
be written as a product of length n. Suppose G is a finitely generated group; and T is a finite symmetric set of generators (symmetric means that if
Growth_rate_(group_theory)
classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or
List_of_finite_simple_groups
Topological group that is in a certain sense assembled from a system of finite groups
group is finitely generated (as a topological group) if and only if there exists d ∈ N {\displaystyle d\in \mathbb {N} } such that every group in the system
Profinite_group
contained in one of the factors. In particular, a finitely generated group with property FA has finite abelianization. Property FA is equivalent for countable
Serre's_property_FA
Outer automorphism group of a free group on n generators
has a finitely generated free abelian subgroup of finite index (Bestvina, Feighn, Handel, 2004); For i > 0 {\displaystyle i>0} , all but finitely many
Out(Fn)
Type of group
group for which every finitely generated subgroup is finite. Since the cyclic subgroups of a locally finite group are finitely generated hence finite
Locally_finite_group
Theorem classifying finite simple groups
classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every finite simple group is either cyclic
Classification of finite simple groups
Classification_of_finite_simple_groups
Mathematical group that can be generated as the set of powers of a single element
every finite group. An infinite group is virtually cyclic if and only if it is finitely generated and has exactly two ends; an example of such a group is
Cyclic_group
British mathematician
conjectured that every finitely generated group is accessible. The conjecture motivated much progress in the understanding of splittings of groups. In 1985 Martin
C._T._C._Wall
Group without normal subgroups other than the trivial group and itself
{\displaystyle n\geq 2} . It is much more difficult to construct finitely generated infinite simple groups. The first existence result is non-explicit; it is due
Simple_group
Cardinality of a mathematical group, or of the subgroup generated by an element
the generated group. Lagrange's theorem states that for any subgroup H of a finite group G, the order of the subgroup divides the order of the group; that
Order_(group_theory)
hyperbolic groups, allowing one to produce many quotients of such groups with prescribed properties. Every finitely generated acylindrically hyperbolic group has
Acylindrically hyperbolic group
Acylindrically_hyperbolic_group
Function between two metric spaces that only respects their large-scale geometry
relation on the class of metric spaces. Given a finite generating set S of a finitely generated group G, we can form the corresponding Cayley graph of
Quasi-isometry
British mathematician
finitely generated groups are accessible. Roughly, this means that every finitely generated group can be constructed from finite and one-ended groups via a
Martin_Dunwoody
Subgroup of a root system's isometry group
which is generated by reflections through the hyperplanes orthogonal to at least one of the roots, and as such is a finite reflection group. In fact it
Weyl_group
that is, a group with a finite number of elements. finitely generated group A group G is finitely generated if there is a finite generating set, that is
Glossary_of_group_theory
Representations of finite groups, particularly on vector spaces
G}sHs^{-1}.} Since R ( G ) {\displaystyle {\mathcal {R}}(G)} is finitely generated as a group, the first point can be rephrased as follows: For each character
Representation theory of finite groups
Representation_theory_of_finite_groups
Discrete group of Möbius transformations
finitely many points removed, and the covering is ramified at finitely many points. A Kleinian group is called finitely generated if it has a finite number
Kleinian_group
Mathematical theory
Kleinian groups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem
Ahlfors_finiteness_theorem
Direct summand of a free module (mathematics)
However, it is true that for finitely presented modules M over a commutative ring R (in particular if M is a finitely generated R-module and R is Noetherian)
Projective_module
Type of solvable group in mathematics
cyclic group by a cyclic group. Examples of polycyclic groups include finitely generated abelian groups, finitely generated nilpotent groups, and finite solvable
Polycyclic_group
moduli theory, given an algebraic, reductive, Lie group G {\displaystyle G} and a finitely generated group π {\displaystyle \pi } , the G {\displaystyle G}
Character_variety
Theorem in group theory
lemma implies that every subgroup of finite index of a finitely generated group is again finitely generated. The group Z 3 = Z / 3 Z {\displaystyle \mathbb
Schreier's_lemma
a finitely generated group are defined to be the ends of the corresponding Cayley graph; this definition is insensitive to the choice of generating set
End_(topology)
Theorem in group theory
following lemma. Let F be a finitely generated free group, with n generators. Let G1 and G2 be two finitely presented groups. Suppose there exists a surjective
Grushko_theorem
On constructing an aspherical CW complex whose fundamental group is a given group
algebraic topology, the Eilenberg–Ganea theorem states for every finitely generated group G with certain conditions on its cohomological dimension (namely
Eilenberg–Ganea_theorem
Discrete group type in group theory
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space
Reflection_group
Concept in metric geometry
infinite groups in the context of geometric group theory, as a quasi-isometry invariant of finitely generated groups. As shown by Guoliang Yu, finitely generated
Asymptotic_dimension
Algebra of formal sums
subgroup of a free abelian group is free abelian (below) can be used to show that every finitely generated abelian group is finitely presented. For, if G {\displaystyle
Free_abelian_group
American mathematician (1924–2008)
geometric group theory Muller is known for the Muller–Schupp theorem, joint with Paul Schupp, characterizing finitely generated virtually free groups as finitely
David_E._Muller
finitely generated groups. Finitely generated infinite groups have one, two, or infinitely many ends, and the Stallings theorem about ends of groups provides
End_(graph_theory)
normal subgroup is a finitely generated infinite simple group. Higman (1974) later found some finitely presented infinite groups Gn,r that are simple
Higman_group
Part of the mathematical subject of group theory
subgroup of finite index. If A is finite and all the vertex groups Av are finitely generated then G is finitely generated. If A is finite and all the
Bass–Serre_theory
Mathematical space with a notion of distance
distance. In geometric group theory this construction is applied to the Cayley graph of a (typically infinite) finitely-generated group, yielding the word
Metric_space
most 2, finitely generated discrete groups are geometrically finite, but Greenberg (1966) showed that there are examples of finitely generated discrete
Geometric_finiteness
Mathematical abelian group
product of two copies of the cyclic group of order 2 by the Fundamental Theorem of Finitely Generated Abelian Groups. It was named Vierergruppe (German:
Klein_four-group
Graph defined from a mathematical group
geometric group theory. For a finitely generated group, this is independent of choice of finite set of generators, hence an intrinsic property of the group. This
Cayley_graph
imply algebraic, geometric and algorithmic properties of the group. Finitely presented groups satisfying sufficiently strong small cancellation conditions
Small_cancellation_theory
Type of group in abstract algebra
the normal subgroup S of permutations that fix all but finitely many elements, which is generated by transpositions. Those elements of S that are products
Symmetric_group
Decision problem
algorithm to run and how (finitely) much memory is available. In fact the problem of deciding whether a finitely presented group is trivial is undecidable
Group_isomorphism_problem
Type of group in mathematics
of groups in two such expressions, which maps each group to one that is isomorphic. Primary cyclic groups are characterised among finitely generated abelian
Primary_cyclic_group
Algebraic structure with an associative operation and an identity element
to generate M if the smallest submonoid of M containing S is M. If there is a finite set that generates M, then M is said to be a finitely generated monoid
Monoid
Mathematical theorem in group theory
of finitely generated linear groups. The theorem, proven by Tits, is stated as follows. Theorem— Let G {\displaystyle G} be a finitely generated linear
Tits_alternative
Group theory function
groups. In particular, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation of this group is
Dehn_function
Topic in algebraic number theory
multiplicative group containing the units of R. Dirichlet's unit theorem holds for S-units: the group of S-units is finitely generated, with rank (maximal
S-unit
subgroups are finitely generated nilpotent (Dahmani, Touikan) Any hyperbolic group, such as a free group of finite rank or the fundamental group of a hyperbolic
Relatively_hyperbolic_group
Duality for locally compact abelian groups
topological groups, these methods introduced systematic use of limiting processes that later became important in the study of non-finitely generated groups and
Pontryagin_duality
Group in which the order of every element is a power of p
conjectures described the set of all finite p-groups of fixed coclass as perturbations of finitely many pro-p groups. The coclass conjectures were proven
P-group
transformations generate the full automorphism group of a finitely generated free group. Nielsen, and later Bernhard Neumann used these ideas to give finite presentations
Automorphism group of a free group
Automorphism_group_of_a_free_group
Commutative group in which all nonzero elements have the same order
abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary
Elementary_abelian_group
Geometric group theory the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Group of even permutations of a finite set
alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree
Alternating_group
Mathematical property
saying that the intersection of any two finitely generated subgroups of this group is again finitely generated. The property is named after Albert G. Howson
Howson_property
geometry is any proper, geodesic metric space. An action of a finitely-generated group G on a geometry X is geometric if it satisfies the following conditions:
Geometric_group_action
FINITELY GENERATED-GROUP
FINITELY GENERATED-GROUP
Girl/Female
Biblical
Penetrated.
Boy/Male
Hindu, Indian
Self Generated; Lord Shiva
Girl/Female
Tamil
Anindita | அநிஂதிதா
Beautiful, Virtuous, Venerated
Anindita | அநிஂதிதா
Boy/Male
American, Australian, Christian, Danish, French, Gaelic, Latin, Scottish
Son of; Taken from Mackenzie; Greatest; Finely Made; Comely
Boy/Male
Indian, Telugu
Generated
Boy/Male
Indian
Honored, Venerated
Boy/Male
Muslim
Honored, Venerated
Girl/Female
American, Australian, Chinese, Scottish
Son of the Fair One; Fair Skinned; Comely; Finely Made
Girl/Female
Indian
Beautiful, Virtuous, Venerated
Girl/Female
Indian
Beautiful, Virtuous, Venerated
Biblical
penetrated
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Sindhi, Tamil, Telugu
Virtuous; Venerated
Girl/Female
African, American, British, Celtic, Chinese, Christian, English, Irish
Handsome; Beautiful; Knowing; Good Looking; Born of Fire; Finely Made
Boy/Male
Indian, Sanskrit
Devoted; Venerated
Boy/Male
Arabic, Muslim, Sindhi
Venerated; Honoured
Girl/Female
Indian
Who is to be Venerated and Respected
Girl/Female
Hindu
Generates harmony in dance and music
Girl/Female
Tamil
Aninditha | அநிஂதிதா
Beautiful, Virtuous, Venerated
Aninditha | அநிஂதிதா
Girl/Female
Tamil
Generates harmony in dance and music
Male
Gaelic
Variant form of Gaelic Cainneach, COINNEACH means "comely; finely made."
FINITELY GENERATED-GROUP
FINITELY GENERATED-GROUP
Boy/Male
Tamil
Arrow like
Female
Egyptian
, another form of Ratta or Ritho.
Girl/Female
English American Spanish
The gemstone emerald.
Girl/Female
Greek Italian
Reaper.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Lord Vishnu
Surname or Lastname
English
English : patronymic from a reduced form of the personal name Steven.
Boy/Male
Gujarati, Hindu, Indian, Malayalam, Marathi, Sanskrit, Telugu
Love; Mind
Girl/Female
Hindu, Indian
Early Winter
Male
Hungarian
 Hungarian form of Roman Latin Constantine, KONSTANTIN means "steadfast." Compare with other forms of Konstantin.
Biblical
the stone of help
FINITELY GENERATED-GROUP
FINITELY GENERATED-GROUP
FINITELY GENERATED-GROUP
FINITELY GENERATED-GROUP
FINITELY GENERATED-GROUP
n.
One who, or that which, generates, begets, causes, or produces.
n.
Capability of being generated.
a.
Having a limit; limited in quantity, degree, or capacity; bounded; -- opposed to infinite; as, finite number; finite existence; a finite being; a finite mind; finite duration.
a.
Relating to autogenesis; self-generated.
adv.
In a finite manner or degree.
adv.
Without bounds or limits; beyond or below assignable limits; as, an infinitely large or infinitely small quantity.
n.
That which generates.
imp. & p. p.
of Venerate
v. t.
To generate; to produce.
a.
Capable of being generated or produced.
p. pr. & vb. n.
of Generate
v. t.
To beget; to procreate; to propagate; to produce (a being similar to the parent); to engender; as, every animal generates its own species.
adv.
Infinitely.
imp. & p. p.
of Generate
a.
Self-generated; produced independently.
a.
Generated by water.
a.
First formed or generated; original; primigenial.
a.
Not begot; not yet generated; also, having never been generated; self-existent; eternal.
v. t.
To regard with reverential respect; to honor with mingled respect and awe; to reverence; to revere; as, we venerate parents and elders.