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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

''G'' that has some

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...

generating set In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...

''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 In mathematics, a set is countable if either it is 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 from it into the natural numbers ...

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 rat ...

s Q is an example of a countable group that is not finitely generated.

Examples

* Every

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

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 In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...

. * A

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is 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 s ...

. Every infinite cyclic group is isomorphic to the

additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structures ...

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 is a group in which every finitely generated subgroup is cyclic. * The

free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...

on a finite set is finitely generated by the elements of that set ( §Examples). *

A fortiori ''Argumentum a fortiori'' (literally "argument from the stronger eason) (, ) is a form of argumentation that draws upon existing confidence in a proposition to argue in favor of a second proposition that is held to be implicit in, and even more cer ...

, 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 * Mo ...

over the

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

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, n ...

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

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, n ...

states that a finitely generated Abelian group is the direct sum of a free Abelian group of finite

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...

and a finite Abelian group, each of which are unique up to isomorphism.

Subgroups

of a finitely generated group need not be finitely generated. The

commutator subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...

of the

F_2

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

are finitely generated. A subgroup of finite index in a finitely generated group is always finitely generated, and the

Schreier index formula Schreier is a surname of German origin. Notable people with the surname include: *Christian Schreier (born 1959), German footballer *Dan Moses Schreier, American sound designer and composer *Jake Schreier, American director *Józef Schreier, Polis ...

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

m

and

n

are the numbers of generators of the two finitely generated subgroups then their intersection is generated by at most

2mn - m - n + 1

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 ...

2(m-1)(n-1) + 1

, see

Hanna Neumann conjecture In the mathematical subject of group theory, the Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957.Hanna Neumann. ''On t ...

. The

lattice of subgroups In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their uni ...

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 con ...

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 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 lengt ...

. 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 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 Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...

studies the connections between algebraic properties of finitely generated groups and

topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

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 wheth ...

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 . Info ...

. 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 th ...

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.

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