In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, 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 is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be
countable 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 (e.g. ). The set of all ra ...
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
subgroup of a finitely generated group need not be finitely generated.
* A group that is generated by a single element is called
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
. Every infinite cyclic group is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...
to the
additive group of the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s Z.
** A
locally cyclic group
In mathematics, a locally cyclic group is a group (''G'', *) in which every finitely generated subgroup is cyclic group, cyclic.
Some facts
* Every cyclic group is locally cyclic, and every locally cyclic group is abelian group, abelian.
* Every f ...
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 (
§Examples).
*
A fortiori, every
finitely presented 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 ...
(
§Examples) is finitely generated.
Finitely generated Abelian groups
Every
Abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
can be seen as a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
over the
ring of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s Z, and in a
finitely generated Abelian group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, ...
with generators ''x''
1, ..., ''x''
''n'', every group element ''x'' can be written as a
linear combination of these generators,
:''x'' = ''α''
1⋅''x''
1 + ''α''
2⋅''x''
2 + ... + ''α''
''n''⋅''x''
''n''
with integers ''α''
1, ..., ''α''
''n''.
Subgroups of a finitely generated
Abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
are themselves finitely generated.
The
fundamental theorem of finitely generated abelian groups
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, ...
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
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
are finitely generated.
A subgroup of finite
index in a finitely generated group is always finitely generated, and the
Schreier index formula gives a bound on the number of generators required.
In 1954,
Albert G. Howson showed that the intersection 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
Johanna (Hanna) Neumann (née von Caemmerer; 12 February 1914 – 14 November 1971) was a German-born mathematician who worked on group theory.
Biography
Neumann was born on 12 February 1914 in Lankwitz, Steglitz-Zehlendorf (today a distr ...
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 ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These c ...
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.
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
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
. Conversely, every periodic
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
is locally finite.
Applications
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.
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 of the input values. An example of a decision problem is deciding by means of an algorithm whe ...
whether two
word
A word is a basic element of language that carries an objective or practical 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 conse ...
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 In group theory, a group A\ is algebraically closed if any finite set of equations and inequations that are applicable to A\ have a solution in A\ without needing a group extension. This notion will be made precise later in the article in .
Inf ...
.
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
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