Editing Finitely generated group

[[File:Dih4 cycle graph.svg|thumb|The [[dihedral group of order 8]] requires two generators, as represented by this [[Cycle graph (algebra)|cycle diagram]].]]
In [[algebra]], a '''finitely generated group''' is a [[group (mathematics)|group]] ''G'' that has some [[Finite set|finite]] [[generating set of a group|generating set]] ''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 [[Inverse element|inverses]] 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]] is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be [[countable set|countable]] but countable groups need not be finitely generated. The additive group of [[rational number]]s '''Q''' is an example of a countable group that is not finitely generated.

== Examples ==
* Every [[quotient group|quotient]] of a finitely generated group ''G'' is finitely generated; the quotient group is generated by the images of the generators of ''G'' under the [[Quotient group#Properties|canonical projection]].
* A group that is generated by a single element is called [[cyclic group|cyclic]]. Every infinite cyclic group is [[group isomorphism|isomorphic]] to the additive group of the [[Integer#Algebraic properties|integers]] '''Z'''.
** A [[locally cyclic group]] is a group in which every finitely generated [[subgroup]] is cyclic.
* The [[free group]] on a finite set is finitely generated by the elements of that set ([[Generating set of a group#Examples|§Examples]]).
* [[Argumentum a fortiori|A fortiori]], every [[Presentation of a group#Definition|finitely presented group]] ([[Presentation of a group#Examples|§Examples]]) is finitely generated.

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

Every [[abelian group]] can be seen as a [[module (mathematics)|module]] over the [[ring (mathematics)|ring]] of integers '''Z''', and in a [[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]] 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]] states that a finitely generated abelian group is the [[direct sum of groups|direct sum]] of a [[free abelian group]] of finite [[rank of an abelian group|rank]] and a finite abelian group, each of which are unique [[up to]] isomorphism.

==Subgroups==
A subgroup of a finitely generated group need not be finitely generated. The [[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 of a subgroup|index]] in a finitely generated group is always finitely generated, and the [[Schreier index formula]] gives a bound on the number of generators required.{{sfnp|Rose|2012|p=55}}

In 1954, [[Albert G. Howson]] showed that the [[intersection (set theory)|intersection]] 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]] |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]] to <math>2(m-1)(n-1) + 1</math>; see [[Hanna Neumann conjecture]].

The [[lattice of subgroups]] of a group satisfies the [[ascending chain condition]] [[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 group|Noetherian]].

A group such that every finitely generated subgroup is finite is called [[locally finite group|locally finite]]. Every locally finite group is [[periodic group|periodic]], i.e., every element has finite [[order (group theory)|order]]. [[Converse (logic)|Conversely]], 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|Švarc-Milnor lemma]], or more generally thanks to an [[Group action|action]] through which a group inherits some finiteness property of a space. [[Geometric group theory]] studies the connections between algebraic properties of finitely generated groups and [[topology|topological]] and [[geometry|geometric]] properties of [[space (mathematics)|spaces]] on which these groups act.

=== Differential geometry and topology ===

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

=== Algebraic geometry and number theory ===

* [[Lattice (discrete subgroup)#Lattices in semisimple Lie groups|Lattices in Lie groups]], [[Lattice (discrete subgroup)#Lattices in p-adic Lie groups|in p-adic groups]]...
* [[Superrigidity]], [[Arithmetic group#Margulis arithmeticity theorem|Margulis' arithmeticity theorem]]

=== Combinatorics, algorithmics and cryptography ===

* Infinite families of [[Expander graph|expander graphs]] can be constructed thanks to finitely generated groups with [[Kazhdan's property (T)|property T]]
* Algorithmic problems in [[combinatorial group theory]]
* [[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 walk|Random walks]] on [[Cayley graph|Cayley graphs]] of finitely generated groups provide approachable examples of [[Random walk#On graphs|random walks on graphs]]
* [[Percolation theory|Percolation]] on Cayley graphs

=== Physics and chemistry ===

* [[Crystallographic group|Crystallographic groups]]
* Mapping class groups appear in [[topological quantum field theories]]

=== Biology ===

* [[Knot group|Knot groups]] are used to study [[Molecular knot|molecular knots]]

==Related notions==
The [[Word problem for groups|word problem]] for a finitely generated group is the [[decision problem]] of whether two [[word (group theory)|word]]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]].

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

==See also==
* [[Finitely generated module]]
* [[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]]