In
mathematics, an abelian group, also called a commutative 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 ...
in which the result of applying the group
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
to two group elements does not depend on the order in which they are written. That is, the group operation is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
. With addition as an operation, 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 and the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
.
The concept of an abelian group underlies many fundamental
algebraic structures, such as
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
,
rings
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 ...
,
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, and
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
. The theory of abelian groups is generally simpler than that of their
non-abelian counterparts, and finite abelian groups are very well understood and
fully classified.
Definition
An abelian group is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, together with an
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
that combines any two
elements and
of
to form another element of
denoted
. The symbol
is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation,
, must satisfy four requirements known as the ''abelian group axioms'' (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation is ''defined'' for any ordered pair of elements of , that the result is ''
well-defined
In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
'', and that the result ''
belongs to'' ):
;Associativity: For all
,
, and
in
, the equation
holds.
;Identity element: There exists an element
in
, such that for all elements
in
, the equation
holds.
;Inverse element: For each
in
there exists an element
in
such that
, where
is the identity element.
;Commutativity: For all
,
in
,
.
A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".
Facts
Notation
There are two main notational conventions for abelian groups – additive and multiplicative.
Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for
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 ...
s and
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. The additive notation may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered, some notable exceptions being
near-ring In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.
Definition
A set ''N'' together with two binary operations ...
s and
partially ordered group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' ...
s, where an operation is written additively even when non-abelian.
Multiplication table
To verify that a
finite group is abelian, a table (matrix) – known as a
Cayley table Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplic ...
– can be constructed in a similar fashion to a
multiplication table
In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.
The decimal multiplication table was traditionally taught as an essenti ...
. If the group is
under the the entry of this table contains the product
.
The group is abelian
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
this table is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
about the main diagonal. This is true since the group is abelian
iff
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicon ...
for all
, which is iff the
entry of the table equals the
entry for all
, i.e. the table is symmetric about the main diagonal.
Examples
* For 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 and the operation
addition , denoted
, the operation + combines any two integers to form a third integer, addition is associative, zero is the
additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from elemen ...
, every integer
has an
additive inverse,
, and the addition operation is commutative since
for any two integers
and
.
* Every
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
is abelian, because if
,
are in
, then
. Thus 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,
, form an abelian group under addition, as do the
integers modulo ,
.
* Every
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 ...
is an abelian group with respect to its addition operation. In a
commutative ring the invertible elements, or
units
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* Unit (album), ...
, form an abelian
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
. In particular, the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.
* Every
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 an abelian group is
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
, so each subgroup gives rise to a
quotient group. Subgroups, quotients, and
direct sums of abelian groups are again abelian. The finite
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
abelian groups are exactly the cyclic groups of
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
order.
* The concepts of abelian group and
-
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 ...
agree. More specifically, every
-module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers
in a unique way.
In general,
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. However, some groups of matrices are abelian groups under matrix multiplication – one example is the group of
rotation matrices.
Historical remarks
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...
named abelian groups after
Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
* Norwegians, both a nation and an ethnic group native to Norway
* Demographics of Norway
*The Norwegian language, including ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
, because Abel found that the commutativity of the group of a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
implies that the roots of the polynomial can be
calculated by using radicals.
Properties
If
is a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
and
is an element of an abelian group
written additively, then
can be defined as
(
summands) and
. In this way,
becomes 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 ...
of integers. In fact, the modules over
can be identified with the abelian groups.
Theorems about abelian groups (i.e.
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 ...
s over the
principal ideal domain ) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of
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 ...
s which is a specialization of the
structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a
direct sum of a
torsion group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements.
For examp ...
and a
free abelian group. The former may be written as a direct sum of finitely many groups of the form
for
prime, and the latter is a direct sum of finitely many copies of
.
If
are two
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
s between abelian groups, then their sum
, defined by
, is again a homomorphism. (This is not true if
is a non-abelian group.) The set
of all group homomorphisms from
to
is therefore an abelian group in its own right.
Somewhat akin to the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, every abelian group has a ''
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
* ...
''. It is defined as the maximal
cardinality of a set of
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
(over the integers) elements of the group. Finite abelian groups and torsion groups have rank zero, and every abelian group of rank zero is a torsion group. The integers and the
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 have rank one, as well as every nonzero
additive subgroup of the rationals. On the other hand, the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of the nonzero rationals has an infinite rank, as it is a free abelian group with the set of the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s as a basis (this results from the
fundamental theorem of arithmetic
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
).
The
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of a group
is the set of elements that commute with every element of
. A group
is abelian if and only if it is equal to its center
. The center of a group
is always a
characteristic abelian subgroup of
. If the quotient group
of a group by its center is cyclic then
is abelian.
Finite abelian groups
Cyclic groups of
integers modulo ,
, were among the first examples of groups. It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of
Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous ...
and
Ludwig Stickelberger
Ludwig Stickelberger (18 May 1850 – 11 April 1936) was a Swiss mathematician who made important contributions to linear algebra (theory of elementary divisors) and algebraic number theory (Stickelberger relation in the theory of cyclotomi ...
and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices ...
.
Any group of prime order is isomorphic to a cyclic group and therefore abelian. Any group whose order is a square of a prime number is also abelian. In fact, for every prime number
there are (up to isomorphism) exactly two groups of order
, namely
and
.
Classification
The fundamental theorem of finite abelian groups states that every finite abelian group
can be expressed as the direct sum of cyclic subgroups of
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups. This is generalized by 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 ...
, with finite groups being the special case when ''G'' has zero
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
* ...
; this in turn admits numerous further generalizations.
The classification was proven by
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...
in 1870, though it was not stated in modern group-theoretic terms until later, and was preceded by a similar classification of quadratic forms by
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
in 1801; see
history
History (derived ) is the systematic study and the documentation of the human activity. The time period of event before the invention of writing systems is considered prehistory. "History" is an umbrella term comprising past events as well ...
for details.
The cyclic group
of order
is isomorphic to the direct sum of
and
if and only if
and
are
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
. It follows that any finite abelian group
is isomorphic to a direct sum of the form
:
in either of the following canonical ways:
* the numbers
are powers of (not necessarily distinct) primes,
* or
divides
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
, which divides
, and so on up to
.
For example,
can be expressed as the direct sum of two cyclic subgroups of order 3 and 5:
. The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are
isomorphic.
For another example, every abelian group of order 8 is isomorphic to either
(the integers 0 to 7 under addition modulo 8),
(the odd integers 1 to 15 under multiplication modulo 16), or
.
See also
list of small groups
The following list in mathematics contains the finite groups of small order up to group isomorphism.
Counts
For ''n'' = 1, 2, … the number of nonisomorphic groups of order ''n'' is
: 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5 ...
for finite abelian groups of order 30 or less.
Automorphisms
One can apply the
fundamental theorem to count (and sometimes determine) the
automorphisms
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
of a given finite abelian group
. To do this, one uses the fact that if
splits as a direct sum
of subgroups of
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
order, then
:
Given this, the fundamental theorem shows that to compute the automorphism group of
it suffices to compute the automorphism groups of the
Sylow -subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of
). Fix a prime
and suppose the exponents
of the cyclic factors of the Sylow
-subgroup are arranged in increasing order:
:
for some
. One needs to find the automorphisms of
:
One special case is when
, so that there is only one cyclic prime-power factor in the Sylow
-subgroup
. In this case the theory of automorphisms of a finite
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
can be used. Another special case is when
is arbitrary but
for
. Here, one is considering
to be of the form
:
so elements of this subgroup can be viewed as comprising a vector space of dimension
over the finite field of
elements
. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so
:
where
is the appropriate
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
. This is easily shown to have order
:
In the most general case, where the
and
are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines
:
and
:
then one has in particular
,
, and
:
One can check that this yields the orders in the previous examples as special cases (see Hillar, C., & Rhea, D.).
Finitely generated abelian groups
An abelian group is finitely generated if it contains a finite set of elements (called ''generators'')
such that every element of the group is a
linear combination with integer coefficients of elements of .
Let be a
free abelian group with basis
There is a unique
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
such that
:
This homomorphism is
surjective, and its
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
is finitely generated (since integers form a
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
). Consider the matrix with integer entries, such that the entries of its th column are the coefficients of the th generator of the kernel. Then, the abelian group is isomorphic to the
cokernel
The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of .
Cokernels are dual to the kernels of category theory, hence the nam ...
of linear map defined by . Conversely every
integer matrix In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Integ ...
defines a finitely generated abelian group.
It follows that the study of finitely generated abelian groups is totally equivalent with the study of integer matrices. In particular, changing the generating set of is equivalent with multiplying on the left by a
unimodular matrix
In mathematics, a unimodular matrix ''M'' is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix ''N'' that is its inverse (these are equiv ...
(that is, an invertible integer matrix whose inverse is also an integer matrix). Changing the generating set of the kernel of is equivalent with multiplying on the right by a unimodular matrix.
The
Smith normal form In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can b ...
of is a matrix
:
where and are unimodular, and is a matrix such that all non-diagonal entries are zero, the non-zero diagonal entries are the first ones, and is a divisor of for . The existence and the shape of the Smith normal proves that the finitely generated abelian group is the
direct sum
:
where is the number of zero rows at the bottom of (and also the
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
* ...
of the group). This is 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 ...
.
The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums.
Infinite abelian groups
The simplest infinite abelian group is the
infinite cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binar ...
. Any
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 ...
is isomorphic to the direct sum of
copies of
and a finite abelian group, which in turn is decomposable into a direct sum of finitely many
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
s of
prime power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number.
For example: , and are prime powers, while
, and are not.
The sequence of prime powers begins:
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
orders. Even though the decomposition is not unique, the number
, called the
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
* ...
of
, and the prime powers giving the orders of finite cyclic summands are uniquely determined.
By contrast, classification of general infinitely generated abelian groups is far from complete.
Divisible group In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an ''n''th multiple for each positive in ...
s, i.e. abelian groups
in which the equation
admits a solution
for any natural number
and element
of
, constitute one important class of infinite abelian groups that can be completely characterized. Every divisible group is isomorphic to a direct sum, with summands isomorphic to
and
Prüfer group
In mathematics, specifically in group theory, the Prüfer ''p''-group or the ''p''-quasicyclic group or ''p''∞-group, Z(''p''∞), for a prime number ''p'' is the unique ''p''-group in which every element has ''p'' different ''p''-th roots.
...
s
for various prime numbers
, and the cardinality of the set of summands of each type is uniquely determined. Moreover, if a divisible group
is a subgroup of an abelian group
then
admits a direct complement: a subgroup
of
such that
. Thus divisible groups are
injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule o ...
s in the
category of abelian groups, and conversely, every injective abelian group is divisible (
Baer's criterion). An abelian group without non-zero divisible subgroups is called reduced.
Two important special classes of infinite abelian groups with diametrically opposite properties are ''torsion groups'' and ''torsion-free groups'', exemplified by the groups
(periodic) and
(torsion-free).
Torsion groups
An abelian group is called
periodic or
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
, if every element has finite
order. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second
Prüfer theorems state that if
is a periodic group, and it either has a bounded exponent, i.e.,
for some natural number
, or is countable and the
-heights of the elements of
are finite for each
, then
is isomorphic to a direct sum of finite cyclic groups.
The cardinality of the set of direct summands isomorphic to
in such a decomposition is an invariant of
. These theorems were later subsumed in the Kulikov criterion. In a different direction,
Helmut Ulm
Helmut Ulm (born 21 June 1908 in Gelsenkirchen; died 13 June 1975) was a German mathematician who established the classification of countable periodic abelian groups by means of their Ulm invariants.
Career
Helmut Ulm's father was an elementary ...
found an extension of the second Prüfer theorem to countable abelian
-groups with elements of infinite height: those groups are completely classified by means of their
Ulm invariant In mathematics, the height of an element ''g'' of an abelian group ''A'' is an invariant that captures its divisibility properties: it is the largest natural number ''N'' such that the equation ''Nx'' = ''g'' has a solution ''x'' ∈ ''A'' ...
s.
Torsion-free and mixed groups
An abelian group is called torsion-free if every non-zero element has infinite order. Several classes of
torsion-free abelian group
In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only e ...
s have been studied extensively:
*
Free abelian groups, i.e. arbitrary direct sums of
*
Cotorsion and
algebraically compact
In mathematics, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means. The solutions to these s ...
torsion-free groups such as the
-adic integers
*
Slender group
In mathematics, a slender group is a torsion-free abelian group that is "small" in a sense that is made precise in the definition below.
Definition
Let ZN denote the Baer–Specker group, that is, the group of all integer sequences, with term ...
s
An abelian group that is neither periodic nor torsion-free is called mixed. If
is an abelian group and
is its
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
, then the factor group
is torsion-free. However, in general the torsion subgroup is not a direct summand of
, so
is ''not'' isomorphic to
. Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group
of integers is torsion-free
-module.
Invariants and classification
One of the most basic invariants of an infinite abelian group
is its
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
* ...
: the cardinality of the maximal
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
subset of
. Abelian groups of rank 0 are precisely the periodic groups, while
torsion-free abelian groups of rank 1
In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only elem ...
are necessarily subgroups of
and can be completely described. More generally, a torsion-free abelian group of finite rank
is a subgroup of
. On the other hand, the group of
-adic integers is a torsion-free abelian group of infinite
-rank and the groups
with different
are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups.
The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before 1950 and form a foundation of the classification of more general infinite abelian groups. Important technical tools used in classification of infinite abelian groups are
pure
Pure may refer to:
Computing
* A pure function
* A pure virtual function
* PureSystems, a family of computer systems introduced by IBM in 2012
* Pure Software, a company founded in 1991 by Reed Hastings to support the Purify tool
* Pure-FTPd, F ...
and
basic subgroups. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. See the books by
Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andr ...
,
László Fuchs László Fuchs (born June 24, 1924) is a Hungarian-born American mathematician, the Evelyn and John G. Phillips Distinguished Professor Emeritus in Mathematics at Tulane University. ,
Phillip Griffith, and
David Arnold
David Arnold (born 23 January 1962) is a British film composer whose credits include scoring five James Bond films, as well as ''Stargate'' (1994), '' Independence Day'' (1996), ''Godzilla'' (1998) and the television series ''Little Britain'' ...
, as well as the proceedings of the conferences on Abelian Group Theory published in ''
Lecture Notes in Mathematics
''Lecture Notes in Mathematics'' is a book series in the field of mathematics, including articles related to both research and teaching. It was established in 1964 and was edited by A. Dold, Heidelberg and B. Eckmann, Zürich. Its publisher is Sp ...
'' for more recent findings.
Additive groups of rings
The additive group of a
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 ...
is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication). Some important topics in this area of study are:
*
Tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
* A.L.S. Corner's results on countable torsion-free groups
* Shelah's work to remove cardinality restrictions
*
Burnside ring
Relation to other mathematical topics
Many large abelian groups possess a natural
topology
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 ...
, which turns them into
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s.
The collection of all abelian groups, together with the
homomorphisms between them, forms the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
, the prototype of an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
.
proved that the first-order theory of abelian groups, unlike its non-abelian counterpart, is decidable. Most
algebraic structures other than
Boolean algebras are
undecidable.
There are still many areas of current research:
*Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the
rank 1
Rank 1 is a Dutch trance group, formed in the Netherlands in 1997. Widely regarded as one of the originators of the Dutch trance sound, the group have produced a number of dancefloor hits since their conception. Although the two members of the g ...
case are well understood;
*There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups;
*While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the case of countable mixed groups is much less mature.
*Many mild extensions of the first-order theory of abelian groups are known to be undecidable.
*Finite abelian groups remain a topic of research in
computational group theory
In mathematics, computational group theory is the study of
groups by means of computers. It is concerned
with designing and analysing algorithms and
data structures to compute information about groups. The subject
has attracted interest because f ...
.
Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
commonly assumed to underlie all of mathematics. Take the
Whitehead problem
In group theory, a branch of abstract algebra, the Whitehead problem is the following question:
Saharon Shelah proved that Whitehead's problem is independent of ZFC, the standard axioms of set theory.
Refinement
Assume that ''A'' is an abel ...
: are all Whitehead groups of infinite order also
free abelian groups? In the 1970s,
Saharon Shelah proved that the Whitehead problem is:
*
Undecidable in ZFC (
Zermelo–Fraenkel axioms), the conventional
axiomatic set theory from which nearly all of present-day mathematics can be derived. The Whitehead problem is also the first question in ordinary mathematics proved undecidable in ZFC;
* Undecidable even if
ZFC is augmented by taking the
generalized continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
as an axiom;
* Positively answered if ZFC is augmented with the axiom of
constructibility (see
statements true in L
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann universe and the constructib ...
).
A note on typography
Among mathematical
adjective
In linguistics, an adjective (abbreviated ) is a word that generally modifies a noun or noun phrase or describes its referent. Its semantic role is to change information given by the noun.
Traditionally, adjectives were considered one of the ma ...
s derived from the
proper name
A proper noun is a noun that identifies a single entity and is used to refer to that entity (''Africa'', ''Jupiter'', ''Sarah'', ''Microsoft)'' as distinguished from a common noun, which is a noun that refers to a class of entities (''continent, ...
of a
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
, the word "abelian" is rare in that it is often spelled with a lowercase a, rather than an uppercase A, the lack of capitalization being a tacit acknowledgment not only of the degree to which Abel's name has been institutionalized but also of how ubiquitous in modern mathematics are the concepts introduced by him.
See also
*
*
*, the smallest non-abelian group
*
*
*
Notes
References
*
*
*
*
*
*
*
* Unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978.
*
*
External links
*
{{DEFAULTSORT:Abelian Group
Abelian group theory
Properties of groups
Niels Henrik Abel