
In
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, a finitely generated group is a
group ''G'' that has some
finite 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
inverses of such elements.
By definition, every
finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be
countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
but countable groups need not be finitely generated. The additive group of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
s Q is an example of a countable group that is not finitely generated.
Examples
* Every
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
canonical projection.
* A group that is generated by a single element is called
cyclic. Every infinite cyclic group is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the additive group of the
integers
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
Z.
** A
locally cyclic group is a group in which every finitely generated
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
is cyclic.
* The
free group on a finite set is finitely generated by the elements of that set (
§Examples).
*
A fortiori, every
finitely presented group (
§Examples) is finitely generated.
Finitely generated abelian groups

Every
abelian group can be seen as a
module over the
ring of integers Z, and in a
finitely generated abelian group with generators ''x''
1, ..., ''x''
''n'', every group element ''x'' can be written as a
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of these generators,
:''x'' = ''α''
1â‹…''x''
1 + ''α''
2â‹…''x''
2 + ... + ''α''
''n''â‹…''x''
''n''
with integers ''α''
1, ..., ''α''
''n''.
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 a
free abelian group of finite
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
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
Index (: indexes or indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on the Halo Array in the ...
in a finitely generated group is always finitely generated, and the
Schreier index formula Schreier is a surname of German language, German origin. Notable people with the surname include:
*Christian Schreier (born 1959), German footballer
*Dan Moses Schreier, American sound designer and composer
*Jake Schreier (born 1981), American dir ...
gives a bound on the number of generators required.
In 1954,
Albert G. Howson showed that the
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of two finitely generated subgroups of a free group is again finitely generated. Furthermore, if
and
are the numbers of generators of the two finitely generated subgroups then their intersection is generated by at most
generators.
This upper bound was then significantly improved by
Hanna Neumann to
; see
Hanna Neumann conjecture.
The
lattice of subgroups of a group satisfies the
ascending chain condition
In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings. These conditions p ...
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
all subgroups of the group are finitely generated. A group such that all its subgroups are finitely generated is called
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
.
A group such that every finitely generated subgroup is finite is called
locally finite. Every locally finite group is
periodic, i.e., every element has finite
order.
Conversely, every periodic abelian group is locally finite.
Applications
Finitely generated groups arise in diverse mathematical and scientific contexts. A frequent way they do so is by the
Å varc-Milnor lemma, or more generally thanks to an
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
topological and
geometric properties of
spaces on which these groups act.
Differential geometry and topology
*
Fundamental groups of compact
manifolds are finitely generated. Their geometry coarsely reflects the possible geometries of the manifold: for instance, non-positively curved compact manifolds have
CAT(0) fundamental groups, whereas uniformly positively-curved manifolds have finite fundamental group (see
Myers' theorem).
*
Mostow's rigidity theorem: for compact
hyperbolic manifolds of dimension at least 3, an isomorphism between their fundamental groups extends to a
Riemannian isometry.
*
Mapping class groups of surfaces are also important finitely generated groups in low-dimensional topology.
Algebraic geometry and number theory
*
Lattices in Lie groups,
in p-adic groups...
*
Superrigidity,
Margulis' arithmeticity theorem
Combinatorics, algorithmics and cryptography
* Infinite families of
expander graphs can be constructed thanks to finitely generated groups with
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 walks on
Cayley graphs of finitely generated groups provide approachable examples of
random walks on graphs
*
Percolation
In physics, chemistry, and materials science, percolation () refers to the movement and filtration, filtering of fluids through porous materials. It is described by Darcy's law. Broader applications have since been developed that cover connecti ...
on Cayley graphs
Physics and chemistry
*
Crystallographic groups
* Mapping class groups appear in
topological quantum field theories
Biology
*
Knot groups are used to study
molecular knots
Related notions
The
word problem for a finitely generated group is the
decision problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...
of whether two
word
A word is a basic element of language that carries semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no consensus among linguist ...
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
In the mathematical subject of group theory, the rank of a group ''G'', denoted rank(''G''), can refer to the smallest cardinality of a generating set for ''G'', that is
: \operatorname(G)=\min\.
If ''G'' is a finitely generated group, then the ...
is often defined to be the smallest
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
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
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
Notes
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
Group theory
Properties of groups