In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers 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.^[1]

The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. 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.

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Definition

An abelian group is a set ${\displaystyle A}$, together with an operation ${\displaystyle \cdot }$ that combines any two elements ${\displaystyle a}$ and ${\displaystyle b}$ of ${\displaystyle A}$ to form another element of ${\displaystyle A,}$ denoted ${\displaystyle a\cdot b}$. The symbol ${\displaystyle \cdot }$ is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, ${\displaystyle (A,\cdot )}$, 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 A, that the result is well-defined, and that the result belongs to A):

Associativity

For all ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ in ${\displaystyle A}$, the equation ${\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}$ holds.

Identity element

There exists an element ${\displaystyle e}$ in ${\displaystyle A}$, such that for all elements ${\displaystyle a}$ in ${\displaystyle A}$, the equation ${\displaystyle e\cdot a=a\cdot e=a}$ holds.

Inverse element

For each ${\displaystyle a}$ in ${\displaystyle A}$ there exists an element ${\displaystyle b}$ in ${\displaystyle A}$ such that ${\displaystyle a\cdot b=b\cdot a=e}$, where ${\displaystyle e}$ is the identity element.

Commutativity

For all ${\displaystyle a}$, ${\displaystyle b}$ in ${\displaystyle A}$, ${\displaystyle a\cdot b=b\cdot a}$.

A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".^[2]: 11

Facts

Notation

See also: Additive group and Multiplicative group

There are two main notational conventions for abelian groups – additive and multiplicative.

Convention	Operation	Identity	Powers	Inverse
Addition	${\displaystyle x+y}$	0	${\displaystyle nx}$	${\displaystyle -x}$
Multiplication	${\displaystyle x\cdot y}$ or ${\displaystyle xy}$	1	${\displaystyle x^{n}}$	${\displaystyle x^{-1}}$

Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules 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-rings and partially ordered groups, where an operation is written additively even when non-abelian.^{[3]: 28–29 [4]: 9–14}

Examples

• For the integers and the operation addition ${\displaystyle +}$, denoted ${\displaystyle (\mathbb {Z} ,+)}$, the operation + combines any two integers to form a third integer, addition is associative, zero is the additive identity, every integer ${\displaystyle n}$ has an additive inverse, ${\displaystyle -n}$, and the addition operation is commutative since ${\displaystyle n+m=m+n}$ for any two integers ${\displaystyle m}$ and ${\displaystyle n}$.
• Every cyclic group ${\displaystyle G}$ is abelian, because if ${\displaystyle x}$, ${\displaystyle y}$ are in ${\displaystyle G}$, then ${\displaystyle xy=a^{m}a^{n}=a^{m+n}=a^{n}a^{m}=yx}$. Thus the integers, ${\displaystyle \mathbb {Z} }$, form an abelian group under addition, as do the integers modulo ${\displaystyle n}$, ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$.
• Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.
• Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime order.^[6]: 32
• The concepts of abelian group and ${\displaystyle \mathbb {Z} }$-module agree. More specifically, every ${\displaystyle \mathbb {Z} }$-module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers ${\displaystyle \mathbb {Z} }$ in a unique way.

In general, matrices, 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 ${\displaystyle 2\times 2}$ rotation matrices.

Historical remarks

Camille Jordan named abelian groups after Norwegian mathematician Niels Henrik Abel, as Abel had found that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated by using radicals.^{[7]: 144–145 [8]: 157–158}

Properties

If ${\displaystyle n}$ is a natural number and ${\displaystyle x}$ is an element of an abelian group ${\displaystyle G}$ written additively, then ${\displaystyle nx}$ can be defined as ${\displaystyle x+x+\cdots +x}$ (${\displaystyle n}$ summands) and ${\displaystyle (-n)x=-(nx)}$. In this way, ${\displaystyle G}$ becomes a module over the ring ${\displaystyle \mathbb {Z} }$ of integers. In fact, the modules over ${\displaystyle \mathbb {Z} }$ can be identified with the abelian groups.^{[9]: 94–97}

Theorems about abelian groups (i.e. modules over the principal ideal domain ${\displaystyle \mathbb {Z} }$) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups 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 and a free abelian group. The former may be written as a direct sum of finitely many groups of the form ${\displaystyle \mathbb {Z} /p^{k}\mathbb {Z} }$ for ${\displaystyle p}$ prime, and the latter is a direct sum of finitely many copies of ${\displaystyle \mathbb {Z} }$.

If ${\displaystyle f,g:G\to H}$ are two group homomorphisms between abelian groups, then their sum ${\displaystyle f+g}$, defined by ${\displaystyle (f+g)(x)=f(x)+g(x)}$, is again a homomorphism. (This is not true if ${\displaystyle H}$ is a non-abelian group.) The set ${\displaystyle {\text{Hom}}(G,H)}$ of all group homomorphisms from ${\displaystyle G}$ to ${\displaystyle H}$ is therefore an abelian group in its own right.

Somewhat akin to the dimension of vector spaces, every abelian group has a rank. It is defined as the maximal cardinality of a set of linearly independent (over the integers) elements of the group.^{[10]: 49–50} 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 numbers have rank one, as well as every nonzero additive subgroup of the rationals. On the other hand, the multiplicative group of the nonzero rationals has an infinite rank, as it is a free abelian group with the set of the prime numbers as a basis (this results from the fundamental theorem of arithmetic).

The center ${\displaystyle Z(G)}$ of a group ${\displaystyle G}$ is the set of elements that commute with every element of ${\displaystyle G}$. A group ${\displaystyle G}$ is abelian if and only if it is equal to its center ${\displaystyle Z(G)}$. The center of a group ${\displaystyle G}$ is always a characteristic abelian subgroup of ${\displaystyle G}$. If the quotient group ${\displaystyle G/Z(G)}$ of a group by its center is cyclic then ${\displaystyle G}$ is abelian.^[11]

Finite abelian groups

Cyclic groups of integers modulo ${\displaystyle n}$, ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$, 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 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 and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra.

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.^[12] In fact, for every prime number ${\displaystyle p}$ there are (up to isomorphism) exactly two groups of order ${\displaystyle p^{2}}$, namely ${\displaystyle \mathbb {Z} _{p^{2}}}$ and ${\displaystyle \mathbb {Z} _{p}\times \mathbb {Z} _{p}}$.

Finitely generated abelian groups

An abelian group A is finitely generated if it contains a finite set of elements (called generators) ${\displaystyle G=\{x_{1},\ldots ,x_{n}\}}$ such that every element of the group is a linear combination with integer coefficients of elements of G.

Let L be a free abelian group with basis ${\displaystyle B=\{b_{1},\ldots ,b_{n}\}.}$ There is a unique group homomorphism ${\displaystyle p\colon L\to A,}$ such that

${\displaystyle p(b_{i})=x_{i}\quad {\text{for }}i=1,\ldots ,n.}$

This homomorphism is surjective, and its kernel is finitely generated (since integers form a Noetherian ring). Consider the matrix M with integer entries, such that the entries of its jth column are the coefficients of the jth generator of the kernel. Then, the abelian group is isomorphic to the cokernel of linear map defined by M. Conversely every integer matrix 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 A is equivalent with multiplying M on the left by a unimodular matrix (that is, an invertible integer matrix whose inverse is also an integer matrix). Changing the generating set of the kernel of M is equivalent with multiplying M on the right by a unimodular matrix.

The Smith normal form of M is a matrix

${\displaystyle S=UMV,}$

where U and V are unimodular, and S is a matrix such that all non-diagonal entries are zero, the non-zero diagonal entries ${\displaystyle d_{1,1},\ldots ,d_{k,k}}$ are the first ones, and ${\displaystyle d_{j,j}}$ is a divisor of ${\displaystyle d_{i,i}}$ for i > j. The existence and the shape of the Smith normal proves that the finitely generated abelian group A is the direct sum

${\displaystyle \mathbb {Z} ^{r}\oplus \mathbb {Z} /d_{1,1}\mathbb {Z} \oplus \cdots \oplus \mathbb {Z} /d_{k,k}\mathbb {Z} ,}$

where r is the number of zero rows at the bottom of r (and also the rank of the group). This is the fundamental theorem of finitely generated abelian groups.

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.^{[14]: 26–27}

Infinite abelian groups

The simplest infinite abelian group is the infinite cyclic group ${\displaystyle \mathbb {Z} }$. Any finitely generated abelian group ${\displaystyle A}$ is isomorphic to the direct sum of ${\displaystyle r}$ copies of ${\displaystyle \mathbb {Z} }$ and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of prime power orders. Even though the decomposition is not unique, the number ${\displaystyle r}$, called the rank of ${\displaystyle A}$, 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. Divisible groups, i.e. abelian groups ${\displaystyle A}$ in which the equation ${\displaystyle nx=a}$ admits a solution ${\displaystyle x\in A}$ for any natural number ${\displaystyle n}$ and element ${\displaystyle a}$ of ${\displaystyle A}$, 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 ${\displaystyle \mathbb {Q} }$ and Prüfer groups ${\displaystyle \mathbb {Q} _{p}/Z_{p}}$ for various prime numbers ${\displaystyle p}$, and the cardinality of the set of summands of each type is uniquely determined.^[15] Moreover, if a divisible group ${\displaystyle A}$ is a subgroup of an abelian group ${\displaystyle G}$ then ${\displaystyle A}$ admits a direct complement: a subgroup ${\displaystyle C}$ of ${\displaystyle G}$ such that ${\displaystyle G=A\oplus C}$. Thus divisible groups are injective modules 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 ${\displaystyle \mathbb {Q} /\mathbb {Z} }$ (periodic) and ${\displaystyle \mathbb {Q} }$ (torsion-free).

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 homomorphisms between them, forms the category ${\displaystyle {\textbf {Ab}}}$, the prototype of an abelian category.

Wanda Szmielew (1955) 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 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 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 as an axiom;
• Positively answered if ZFC is augmented with the axiom of constructibility (see statements true in L).

A note on typography

Among mathematical adjectives derived from the proper name of a mathematician, 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.^[21]

See also

• Commutator subgroup – Smallest normal subgroup by which the quotient is commutative
• Abelianization – Quotienting a group by its commutator subgroup
• Dihedral group of order 6 – Non-commutative group with 6 elements, the smallest non-abelian group
• Elementary abelian group – Commutative group in which all nonzero elements have the same order
• Grothendieck group – Abelian group extending a commutative monoid
• Pontryagin duality – Duality for locally compact abelian groups

Notes

References

• Cox, David (2004). Galois Theory. Wiley-Interscience. ISBN 9781118031339. MR 2119052.
• Fuchs, László (1970). Infinite Abelian Groups. Pure and Applied Mathematics. Vol. 36-I. Academic Press. MR 0255673.
• Fuchs, László (1973). Infinite Abelian Groups. Pure and Applied Mathematics. Vol. 36-II. Academic Press. MR 0349869.
• Griffith, Phillip A. (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7.
• Herstein, I. N. (1975). Topics in Algebra (2nd ed.). John Wiley & Sons. ISBN 0-471-02371-X.
• Hillar, Christopher; Rhea, Darren (2007). "Automorphisms of finite abelian groups" (PDF). American Mathematical Monthly. 114 (10): 917–923. arXiv:math/0605185. Bibcode:2006math......5185H. doi:10.1080/00029890.2007.11920485. JSTOR 27642365. S2CID 1038507.
• Jacobson, Nathan (2009). Basic Algebra I (2nd ed.). Dover Publications. ISBN 978-0-486-47189-1.
• Rose, John S. (2012). A Course on Group Theory. Dover Publications. ISBN 978-0-486-68194-8. Unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978.
• Szmielew, Wanda (1955). "Elementary Properties of Abelian Groups" (PDF). Fundamenta Mathematicae. 41 (2): 203–271. doi:10.4064/fm-41-2-203-271. MR 0072131. Zbl 0248.02049.
• Robinson, Abraham; Zakon, Elias (1960). "Elementary Properties of Ordered Abelian Groups" (PDF). Transactions of the American Mathematical Society. 96 (2): 222–236. doi:10.2307/1993461. JSTOR 1993461. Archived (PDF) from the original on 2022-10-09.

External links

• "Abelian group". Encyclopedia of Mathematics. EMS Press. 2001 [1994].