In

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
* Modula ...

s over the

rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking
A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...

of $A$, 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.

rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking
A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...

: the cardinality of the maximal

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, an abelian group, also called a commutative group, is a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

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 commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

. With addition as an operation, the integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s and the real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

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
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

.
The concept of an abelian group underlies many fundamental algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s, such as fields
File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe''
FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...

, rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck
Ring may also refer to:
Sounds
* Ri ...

, vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, and algebras
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. 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, $A$, together with an operation $\backslash cdot$ that combines any two elements $a$ and $b$ of $A$ to form another element of $A,$ denoted $a\; \backslash cdot\; b$. The symbol $\backslash cdot$ is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, $(A,\; \backslash cdot)$, must satisfy five requirements known as the ''abelian group axioms'': ;Closure: For all $a$, $b$ in $A$, the result of the operation $a\; \backslash cdot\; b$ is also in $A$. ;Associativity: For all $a$, $b$, and $c$ in $A$, the equation $(a\; \backslash cdot\; b)\backslash cdot\; c\; =\; a\; \backslash cdot\; (b\; \backslash cdot\; c)$ holds. ;Identity element: There exists an element $e$ in $A$, such that for all elements $a$ in $A$, the equation $e\; \backslash cdot\; a\; =\; a\; \backslash cdot\; e\; =\; a$ holds. ;Inverse element: For each $a$ in $A$ there exists an element $b$ in $A$ such that $a\; \backslash cdot\; b\; =\; b\; \backslash cdot\; a\; =\; e$, where $e$ is the identity element. ;Commutativity: For all $a$, $b$ in $A$, $a\; \backslash cdot\; b\; =\; b\; \backslash cdot\; a$. 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 formodule
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
* Modula ...

s and rings. 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-ringIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s and partially ordered group
In abstract algebra, a partially ordered group is a group (mathematics), 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'', ...

s, where an operation is written additively even when non-abelian.
Multiplication table

To verify that afinite group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

is abelian, a table (matrix) – known as a Cayley table Named after the 19th century British
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** Britishness, the British identity a ...

– 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 binary operation, operation for an algebraic system.
The decimal multiplication table was traditionally taug ...

. If the group is $G\; =\; \backslash $ under the the entry of this table contains the product $g\_i\; \backslash cdot\; g\_j$.
The group is abelian if and only if this table is symmetric about the main diagonal. This is true since the group is abelian iff
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$g\_i\; \backslash cdot\; g\_j\; =\; g\_j\; \backslash cdot\; g\_i$ for all $i,\; j\; =\; 1,\; ...,\; n$, which is iff the $(i,\; j)$ entry of the table equals the $(j,\; i)$ entry for all $i,\; j\; =\; 1,\; ...,\; n$, i.e. the table is symmetric about the main diagonal.
Examples

* For theinteger
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s and the operation addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

$+$, denoted $(\backslash mathbb,\; +)$, the operation + combines any two integers to form a third integer, addition is associative, zero is the additive identity In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

, every integer $n$ has an additive inverse
In mathematics, the additive inverse of a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be ...

, $-n$, and the addition operation is commutative since $n\; +\; m\; =\; m\; +\; n$ for any two integers $m$ and $n$.
* Every cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

$G$ is abelian, because if $x$, $y$ are in $G$, then $xy\; =\; a^ma^n\; =\; a^\; =\; a^na^m\; =\; yx$. Thus the integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s, $\backslash mathbb$, form an abelian group under addition, as do the integers modulo $n$, $\backslash mathbb/n\; \backslash mathbb$.
* Every ring is an abelian group with respect to its addition operation. In a commutative ring
In ring theory
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...

the invertible elements, or units
Unit may refer to:
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* UNIT
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* Unit of action, a discrete piece of action (or beat) in ...

, form an abelian multiplicative group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. In particular, the real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.
* Every subgroup
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...

of an abelian group is normal, so each subgroup gives rise to a quotient group
A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...

. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple
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* "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 (mathematics), 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 ...

order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

.
* The concepts of abelian group and $\backslash mathbb$-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
* Modula ...

agree. More specifically, every $\backslash mathbb$-module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers $\backslash mathbb$ in a unique way.
In general, matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object.
Fo ...

, 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 $2\; \backslash times\; 2$ 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 a ...

named abelian groups after Norwegian
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*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 the ...

mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

, because Abel found that the commutativity of the group of a polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

implies that the roots of the polynomial can be calculated by using radicals.
Properties

If $n$ is anatural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

and $x$ is an element of an abelian group $G$ written additively, then $nx$ can be defined as $x\; +\; x\; +\; \backslash cdots\; +\; x$ ($n$ summands) and $(-n)x\; =\; -(nx)$. In this way, $G$ 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
* Modula ...

over the ring $\backslash mathbb$ of integers. In fact, the modules over $\backslash mathbb$ can be identified with the abelian groups.
Theorems about abelian groups (i.e. principal ideal domain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

$\backslash mathbb$) 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
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

s which is a specialization of the structure theorem for finitely generated modules over a principal ideal domainIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum
The direct sum is an operation from abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...

of a torsion group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by E ...

and a free abelian group
In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation that is associative, commutative, and invertible. A basis, also called ...

. The former may be written as a direct sum of finitely many groups of the form $\backslash mathbb/p^k\backslash mathbb$ for $p$ prime, and the latter is a direct sum of finitely many copies of $\backslash mathbb$.
If $f,\; g:\; G\; \backslash to\; H$ are two group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s between abelian groups, then their sum $f\; +\; g$, defined by $(f\; +\; g)(x)\; =\; f(x)\; +\; g(x)$, is again a homomorphism. (This is not true if $H$ is a non-abelian group.) The set $\backslash text(G,H)$ of all group homomorphisms from $G$ to $H$ is therefore an abelian group in its own right.
Somewhat akin to the dimension
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

of vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, every abelian group has a ''rank
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A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...

''. It is defined as the maximal cardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of a set of linearly independent
In the theory of vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...

(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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s have rank one, as well as every nonzero additive subgroup of the rationals. On the other hand, the multiplicative group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

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 (mathematics), 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 ...

s as a basis (this results from the fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...

).
The center
Center or centre may refer to:
Mathematics
*Center (geometry)
In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...

$Z(G)$ of a group $G$ is the set of elements that commute with every element of $G$. A group $G$ is abelian if and only if it is equal to its center $Z(G)$. The center of a group $G$ is always a characteristic
Characteristic (from the Greek word for a property, attribute or trait
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* Phenotypic trait in biology, which involve genes and characteristics of organisms
* Trait (computer programming), a model for structuring object-oriented ...

abelian subgroup of $G$. If the quotient group $G/Z(G)$ of a group by its center is cyclic then $G$ is abelian.
Finite abelian groups

Cyclic groups of integers modulo $n$, $\backslash mathbb/n\backslash mathbb$, 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. Theautomorphism group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

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
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and Ludwig Stickelberger
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*Swi ...

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 matrix (mat ...

.
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 $p$ there are (up to isomorphism) exactly two groups of order $p^2$, namely $\backslash mathbb\_$ and $\backslash mathbb\_p\backslash times\backslash mathbb\_p$.
Classification

The fundamental theorem of finite abelian groups states that every finite abelian group $G$ can be expressed as the direct sum of cyclic subgroups ofprime
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...

-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
A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...

; 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
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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
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of m ...

in 1801; see history
History (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approxima ...

for details.
The cyclic group $\backslash mathbb\_$ of order $mn$ is isomorphic to the direct sum of $\backslash mathbb\_m$ and $\backslash mathbb\_n$ if and only if $m$ and $n$ are coprime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

. It follows that any finite abelian group $G$ is isomorphic to a direct sum of the form
:$\backslash bigoplus\_^\backslash \; \backslash mathbb\_$
in either of the following canonical ways:
* the numbers $k\_1,\; k\_2,\; \backslash dots,\; k\_u$ are powers of (not necessarily distinct) primes,
* or $k\_1$ divides
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

$k\_2$, which divides $k\_3$, and so on up to $k\_u$.
For example, $\backslash mathbb\_$ can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: $\backslash mathbb\_\; \backslash cong\; \backslash \; \backslash oplus\; \backslash $. 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

.
For another example, every abelian group of order 8 is isomorphic to either $\backslash mathbb\_8$ (the integers 0 to 7 under addition modulo 8), $\backslash mathbb\_4\backslash oplus\; \backslash mathbb\_2$ (the odd integers 1 to 15 under multiplication modulo 16), or $\backslash mathbb\_2\backslash oplus\; \backslash mathbb\_2\; \backslash oplus\; \backslash mathbb\_2$.
See also list of small groups for finite abelian groups of order 30 or less.
Automorphisms

One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group $G$. To do this, one uses the fact that if $G$ splits as a direct sum $H\backslash oplus\; K$ of subgroups ofcoprime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

order, then
:$\backslash operatorname(H\backslash oplus\; K)\; \backslash cong\; \backslash operatorname(H)\backslash oplus\; \backslash operatorname(K).$
Given this, the fundamental theorem shows that to compute the automorphism group of $G$ it suffices to compute the automorphism groups of the Sylow $p$-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of $p$). Fix a prime $p$ and suppose the exponents $e\_i$ of the cyclic factors of the Sylow $p$-subgroup are arranged in increasing order:
:$e\_1\backslash leq\; e\_2\; \backslash leq\backslash cdots\backslash leq\; e\_n$
for some $n\; >\; 0$. One needs to find the automorphisms of
:$\backslash mathbf\_\; \backslash oplus\; \backslash cdots\; \backslash oplus\; \backslash mathbf\_.$
One special case is when $n\; =\; 1$, so that there is only one cyclic prime-power factor in the Sylow $p$-subgroup $P$. In this case the theory of automorphisms of a finite cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

can be used. Another special case is when $n$ is arbitrary but $e\_i\; =\; 1$ for $1\; \backslash le\; i\; \backslash le\; n$. Here, one is considering $P$ to be of the form
:$\backslash mathbf\_p\; \backslash oplus\; \backslash cdots\; \backslash oplus\; \backslash mathbf\_p,$
so elements of this subgroup can be viewed as comprising a vector space of dimension $n$ over the finite field of $p$ elements $\backslash mathbb\_p$. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so
:$\backslash operatorname(P)\backslash cong\backslash mathrm(n,\backslash mathbf\_p),$
where $\backslash mathrm$ is the appropriate general linear group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

. This is easily shown to have order
:$\backslash left,\; \backslash operatorname(P)\backslash =(p^n-1)\backslash cdots(p^n-p^).$
In the most general case, where the $e\_i$ and $n$ are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines
:$d\_k=\backslash max\backslash $
and
:$c\_k=\backslash min\backslash $
then one has in particular $k\; \backslash le\; d\_k$, $c\_k\; \backslash le\; k$, and
:$\backslash left,\; \backslash operatorname(P)\backslash \; =\; \backslash prod\_^n\; (p^-p^)\; \backslash prod\_^n\; (p^)^\; \backslash prod\_^n\; (p^)^.$
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'') $G=\backslash $ such that every element of the group is alinear combination
In mathematics, a linear combination 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'' and ''y'' would be ...

with integer coefficients of elements of .
Let be a free abelian group
In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation that is associative, commutative, and invertible. A basis, also called ...

with basis $B=\backslash .$
There is a unique group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

$p\backslash colon\; L\; \backslash to\; A,$ such that
:$p(b\_i)\; =\; x\_i\backslash quad\; \backslash text\; i=1,\backslash ldots,\; n.$
This homomorphism is surjective
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, 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 learnin ...

is finitely generated (since integers form a Noetherian ring
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

). 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...

of linear map defined by . Conversely every integer matrixIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

(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
Smith may refer to:
* Metalsmith, or simply smith, a craftsman fashioning tools or works of art out of various metals
People
* Smith (surname), a family name originating in England, Scotland and Ireland.
** List of people with surname Smith
* Sm ...

of is a matrix
:$S=UMV,$
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
The direct sum is an operation from abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...

:$\backslash Z^r\; \backslash oplus\; \backslash Z/d\_\backslash Z\; \backslash oplus\; \backslash cdots\; \backslash oplus\; \backslash Z/d\_\backslash Z,$
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
A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...

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 theinfinite cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

$\backslash mathbb$. Any finitely generated abelian group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

$A$ is isomorphic to the direct sum of $r$ copies of $\backslash mathbb$ and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

s of prime power
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

orders. Even though the decomposition is not unique, the number $r$, called the Divisible groupIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s, i.e. abelian groups $A$ in which the equation $nx\; =\; a$ admits a solution $x\; \backslash in\; A$ for any natural number $n$ and element $a$ of $A$, 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 $\backslash mathbb$ and Prüfer groups $\backslash mathbb\_p/Z\_p$ for various prime numbers $p$, and the cardinality of the set of summands of each type is uniquely determined. Moreover, if a divisible group $A$ is a subgroup of an abelian group $G$ then $A$ admits a direct complement: a subgroup $C$ of $G$ such that $G\; =\; A\; \backslash oplus\; C$. 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 (mathematics), module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q ...

s in the category of abelian groupsIn mathematics, the category theory, category Ab has the abelian groups as object (category theory), objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every Small category, small abelian category can ...

, 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 $\backslash mathbb/\backslash mathbb$ (periodic) and $\backslash mathbb$ (torsion-free).
Torsion groups

An abelian group is called periodic or torsion, if every element has finiteorder
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

. 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 $A$ is a periodic group, and it either has a bounded exponent, i.e., $nA\; =\; 0$ for some natural number $n$, or is countable and the $p$-heights of the elements of $A$ are finite for each $p$, then $A$ is isomorphic to a direct sum of finite cyclic groups.
The cardinality of the set of direct summands isomorphic to $\backslash mathbb/p^m\backslash mathbb$ in such a decomposition is an invariant of $A$. These theorems were later subsumed in the Kulikov criterion. In a different direction, Helmut Ulm found an extension of the second Prüfer theorem to countable abelian $p$-groups with elements of infinite height: those groups are completely classified by means of their Ulm invariants.
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 groups have been studied extensively: * Free abelian groups, i.e. arbitrary direct sums of $\backslash mathbb$ * Cotorsion group, Cotorsion and algebraically compact module, algebraically compact torsion-free groups such as the p-adic integer, $p$-adic integers * Slender groups An abelian group that is neither periodic nor torsion-free is called mixed. If $A$ is an abelian group and $T(A)$ is its torsion subgroup, then the factor group $A/T(A)$ is torsion-free. However, in general the torsion subgroup is not a direct summand of $A$, so $A$ is ''not'' isomorphic to $T(A)\; \backslash oplus\; A/T(A)$. Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group $\backslash mathbb$ of integers is torsion-free $\backslash mathbb$-module.Invariants and classification

One of the most basic invariants of an infinite abelian group $A$ is itslinearly independent
In the theory of vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...

subset of $A$. Abelian groups of rank 0 are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of $\backslash mathbb$ and can be completely described. More generally, a torsion-free abelian group of finite rank $r$ is a subgroup of $\backslash mathbb\_r$. On the other hand, the group of p-adic integer, $p$-adic integers $\backslash mathbb\_p$ is a torsion-free abelian group of infinite $\backslash mathbb$-rank and the groups $\backslash mathbb\_p^n$ with different $n$ 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 subgroup, pure and basic subgroup, basic subgroups. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. See the books by Irving Kaplansky, László Fuchs, Phillip Griffith, and David Arnold (mathematician), David Arnold, as well as the proceedings of the conferences on Abelian Group Theory published in ''Lecture Notes in Mathematics'' for more recent findings.
Additive groups of rings

The additive group of a ring 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 * Corner's results on countable torsion-free groups * Shelah's work to remove cardinality restrictions.Relation to other mathematical topics

Many large abelian groups possess a natural topology, which turns them into topological groups. The collection of all abelian groups, together with the Group homomorphism, homomorphisms between them, forms the category of abelian groups, category $\backslash textbf$, the prototype of an abelian category. proved that the first-order theory of abelian groups, unlike its non-abelian counterpart, is decidable. Mostalgebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s other than Boolean algebra (structure), Boolean algebras are Decidability (logic), undecidable.
There are still many areas of current research:
*Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the torsion-free abelian groups of rank 1, rank 1 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.
Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem: are all Whitehead groups of infinite order also free abelian group
In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation that is associative, commutative, and invertible. A basis, also called ...

s? In the 1970s, Saharon Shelah proved that the Whitehead problem is:
* list of statements undecidable in ZFC, 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 as an axiom;
* Positively answered if ZFC is augmented with the axiom of Constructible universe, constructibility (see statements true in L).
A note on the typography

Among mathematical adjectives derived from the proper name of amathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

, 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 acknowledgement 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