# Finitely generated group

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Finitely_generated_group
> Markdown URL: https://mediated.wiki/source/Finitely_generated_group.md
> Source: https://en.wikipedia.org/wiki/Finitely_generated_group
> Source revision: 1342967029
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Group type in algebra}}
[[File:Dih4 cycle graph.svg|thumb|The [dihedral group of order 8](/source/dihedral_group_of_order_8) requires two generators, as represented by this [cycle diagram](/source/Cycle_graph_(algebra)).]]
In [algebra](/source/algebra), a '''finitely generated group''' is a [group](/source/group_(mathematics)) ''G'' that has some [finite](/source/Finite_set) [generating set](/source/generating_set_of_a_group) ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of  ''S'' and of [inverses](/source/Inverse_element) of such elements.<ref>{{cite journal|doi=10.1090/S0002-9939-1967-0215904-3|title=A note on finitely generated groups|journal=Proceedings of the American Mathematical Society|volume=18|issue=4|pages=756–758|year=1967|last1=Gregorac|first1=Robert J.|doi-access=free}}</ref> 

By definition, every [finite group](/source/finite_group) is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be [countable](/source/countable_set) but countable groups need not be finitely generated. The additive group of [rational number](/source/rational_number)s '''Q''' is an example of a countable group that is not finitely generated.

== Examples ==
* Every [quotient](/source/quotient_group) of a finitely generated group ''G'' is finitely generated; the quotient group is generated by the images of the generators of ''G'' under the [canonical projection](/source/Quotient_group).
* A group that is generated by a single element is called [cyclic](/source/cyclic_group). Every infinite cyclic group is [isomorphic](/source/group_isomorphism) to the additive group of the [integers](/source/Integer) '''Z'''.
** A [locally cyclic group](/source/locally_cyclic_group) is a group in which every finitely generated [subgroup](/source/subgroup) is cyclic.
* The [free group](/source/free_group) on a finite set is finitely generated by the elements of that set ([§Examples](/source/Generating_set_of_a_group)).
* [A fortiori](/source/Argumentum_a_fortiori), every [finitely presented group](/source/Presentation_of_a_group) ([§Examples](/source/Presentation_of_a_group)) is finitely generated.

==Finitely generated abelian groups==
[[File:Cyclic group.svg|right|thumb|200px|The six 6th [complex](/source/complex_number) [roots of unity](/source/roots_of_unity) form a [cyclic group](/source/cyclic_group) under multiplication.]]
{{main|Finitely generated abelian group}}

Every [abelian group](/source/abelian_group) can be seen as a [module](/source/module_(mathematics)) over the [ring](/source/ring_(mathematics)) of integers '''Z''', and in a [finitely generated abelian group](/source/finitely_generated_abelian_group) with generators ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>, every group element ''x'' can be written as a [linear combination](/source/linear_combination) of these generators,
:''x'' = ''α''<sub>1</sub>⋅''x''<sub>1</sub> + ''α''<sub>2</sub>⋅''x''<sub>2</sub> + ... + ''α''<sub>''n''</sub>⋅''x''<sub>''n''</sub>
with integers ''α''<sub>1</sub>, ..., ''α''<sub>''n''</sub>.

Subgroups of a finitely generated abelian group are themselves finitely generated.

The [fundamental theorem of finitely generated abelian groups](/source/fundamental_theorem_of_finitely_generated_abelian_groups) states that a finitely generated abelian group is the [direct sum](/source/direct_sum_of_groups) of a [free abelian group](/source/free_abelian_group) of finite [rank](/source/rank_of_an_abelian_group) and a finite abelian group, each of which are unique [up to](/source/up_to) isomorphism.

==Subgroups==
A subgroup of a finitely generated group need not be finitely generated. The [commutator subgroup](/source/commutator_subgroup) of the free group <math>F_2</math> on two generators is an example of a subgroup of a finitely generated group that is not finitely generated.

On the other hand, all subgroups of a finitely generated abelian group are finitely generated.

A subgroup of finite [index](/source/Index_of_a_subgroup) in a finitely generated group is always finitely generated, and the [Schreier index formula](/source/Schreier_index_formula) gives a bound on the number of generators required.{{sfnp|Rose|2012|p=55}}

In 1954, [Albert G. Howson](/source/Albert_G._Howson) showed that the [intersection](/source/intersection_(set_theory)) of two finitely generated subgroups of a free group is again finitely generated. Furthermore, if <math>m</math> and <math>n</math> are the numbers of generators of the two finitely generated subgroups then their intersection is generated by at most <math>2mn - m - n + 1</math> generators.<ref>{{cite journal |last=Howson |first=Albert G. |date=1954 |title=On the intersection of finitely generated free groups |journal=[Journal of the London Mathematical Society](/source/Journal_of_the_London_Mathematical_Society) |volume=29 |issue=4 |pages=428–434 |doi=10.1112/jlms/s1-29.4.428|mr=0065557}}</ref> This upper bound was then significantly improved by [Hanna Neumann](/source/Hanna_Neumann) to <math>2(m-1)(n-1) + 1</math>; see [Hanna Neumann conjecture](/source/Hanna_Neumann_conjecture).

The [lattice of subgroups](/source/lattice_of_subgroups) of a group satisfies the [ascending chain condition](/source/ascending_chain_condition) [if and only if](/source/if_and_only_if) all subgroups of the group are finitely generated. A group such that all its subgroups are finitely generated is called [Noetherian](/source/Noetherian_group).

A group such that every finitely generated subgroup is finite is called [locally finite](/source/locally_finite_group). Every locally finite group is [periodic](/source/periodic_group), i.e., every element has finite [order](/source/order_(group_theory)). [Conversely](/source/Converse_(logic)), every periodic abelian group is locally finite.{{sfnp|Rose|2012|p=75}}

== Applications ==
{{Expand section|date=September 2017}}
Finitely generated groups arise in diverse mathematical and scientific contexts. A frequent way they do so is by the [Švarc-Milnor lemma](/source/%C5%A0varc%E2%80%93Milnor_lemma), or more generally thanks to an [action](/source/Group_action) through which a group inherits some finiteness property of a space. [Geometric group theory](/source/Geometric_group_theory) studies the connections between algebraic properties of finitely generated groups and [topological](/source/topology) and [geometric](/source/geometry) properties of [spaces](/source/space_(mathematics)) on which these groups act.

=== Differential geometry and topology ===

* [Fundamental groups](/source/Fundamental_group) of compact [manifolds](/source/Manifold) are finitely generated. Their geometry coarsely reflects the possible geometries of the manifold: for instance, non-positively curved compact manifolds have [CAT(0)](/source/CAT(0)_group) fundamental groups, whereas uniformly positively-curved manifolds have finite fundamental group (see [Myers' theorem](/source/Myers's_theorem)).
* [Mostow's rigidity theorem](/source/Mostow_rigidity_theorem): for compact [hyperbolic manifolds](/source/Hyperbolic_manifold) of dimension at least 3, an isomorphism between their fundamental groups extends to a [Riemannian isometry](/source/Isometry_(Riemannian_geometry)).
* [Mapping class groups of surfaces](/source/Mapping_class_group_of_a_surface) are also important finitely generated groups in low-dimensional topology.

=== Algebraic geometry and number theory ===

* [Lattices in Lie groups](/source/Lattice_(discrete_subgroup)), [in p-adic groups](/source/Lattice_(discrete_subgroup))...
* [Superrigidity](/source/Superrigidity), [Margulis' arithmeticity theorem](/source/Arithmetic_group)

=== Combinatorics, algorithmics and cryptography ===

* Infinite families of [expander graphs](/source/Expander_graph) can be constructed thanks to finitely generated groups with [property T](/source/Kazhdan's_property_(T))
* Algorithmic problems in [combinatorial group theory](/source/combinatorial_group_theory)
* [Group-based cryptography](/source/Group-based_cryptography) attempts to make use of hard algorithmic problems related to group presentations in order to construct quantum-resilient cryptographic protocols

=== Analysis ===

=== Probability theory ===

* [Random walks](/source/Random_walk) on [Cayley graphs](/source/Cayley_graph) of finitely generated groups provide approachable examples of [random walks on graphs](/source/Random_walk)
* [Percolation](/source/Percolation_theory) on Cayley graphs

=== Physics and chemistry ===

* [Crystallographic groups](/source/Crystallographic_group)
* [Molecular symmetry groups](/source/Molecular_symmetry)
* Mapping class groups appear in [topological quantum field theories](/source/topological_quantum_field_theories)

=== Biology ===

* [Knot groups](/source/Knot_group) are used to study [molecular knots](/source/Molecular_knot)

==Related notions==
The [word problem](/source/Word_problem_for_groups) for a finitely generated group is the [decision problem](/source/decision_problem) of whether two [word](/source/word_(group_theory))s in the generators of the group represent the same element. The word problem for a given finitely generated group is solvable if and only if the group can be embedded in every [algebraically closed group](/source/algebraically_closed_group).

The [rank of a group](/source/rank_of_a_group) is often defined to be the smallest [cardinality](/source/cardinality) of a generating set for the group. By definition, the rank of a finitely generated group is finite.

==See also==
* [Finitely generated module](/source/Finitely_generated_module)
* [Presentation of a group](/source/Presentation_of_a_group)

==Notes==
{{reflist}}

==References==
* {{cite book |last=Rose |first=John S. |date=2012 |title=A Course on Group Theory |publisher=Dover Publications |isbn=978-0-486-68194-8 |orig-year=unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978 }}

Category:Group theory
Category:Properties of groups

---
Adapted from the Wikipedia article [Finitely generated group](https://en.wikipedia.org/wiki/Finitely_generated_group) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Finitely_generated_group?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.