Abelian group

For the group described by the archaic use of the related term "Abelian linear group", seeSymplectic group.

Inmathematics,anabelian group,also called acommutative group,is agroupin which the result of applying thegroup operationto two group elements does not depend on the order in which they are written. That is, the group operation iscommutative.With addition as an operation, theintegersand thereal numbersform 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 mathematicianNiels Henrik Abel.

The concept of an abelian group underlies many fundamentalalgebraic structures,such asfields,rings,vector spaces,andalgebras.The theory of abelian groups is generally simpler than that of theirnon-abeliancounterparts, and finite abelian groups are very well understood andfully classified.

Definition

An abelian group is aset$A$,together with anoperation$\cdot$that combines any twoelements$a$and$b$of$A$to form another element of$A,$denoted$a\cdot b$.The symbol$\cdot$is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation,$(A,\cdot )$,must satisfy four requirements known as theabelian group axioms(some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation isdefinedfor any ordered pair of elements ofA,that the result iswell-defined,and that the resultbelongs toA):

Associativity

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

Identity element

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

Inverse element

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

Commutativity

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

A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".

Facts

Notation

See also:Additive groupandMultiplicative group

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

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

Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation formodulesandrings.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 beingnear-ringsandpartially ordered groups,where an operation is written additively even when non-abelian.

Multiplication table

To verify that afinite groupis abelian, a table (matrix) – known as aCayley table– can be constructed in a similar fashion to amultiplication table.If the group is$G=\{g_{1}=e,g_{2},\dots,g_{n}\}$under theoperation$\cdot$,the$(i,j)$-thentry of this table contains the product$g_{i}\cdot g_{j}$.

The group is abelianif and only ifthis table issymmetricabout the main diagonal. This is true since the group is abelianiff$g_{i}\cdot g_{j}=g_{j}\cdot g_{i}$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 theintegersand the operationaddition$+$,denoted$(\mathbb {Z},+)$,the operation + combines any two integers to form a third integer, addition is associative, zero is theadditive identity,every integer$n$has anadditive inverse,$-n$,and the addition operation is commutative since$n+m=m+n$for any two integers$m$and$n$.
• Everycyclic group$G$is abelian, because if$x$,$y$are in$G$,then$xy=a^{m}a^{n}=a^{m+n}=a^{n}a^{m}=yx$.Thus theintegers,$\mathbb {Z}$,form an abelian group under addition, as do theintegers modulo$n$,$\mathbb {Z} /n\mathbb {Z}$.
• Everyringis an abelian group with respect to its addition operation. In acommutative ringthe invertible elements, orunits,form an abelianmultiplicative group.In particular, thereal numbersare an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.
• Everysubgroupof an abelian group isnormal,so each subgroup gives rise to aquotient group.Subgroups, quotients, anddirect sumsof abelian groups are again abelian. The finitesimpleabelian groups are exactly the cyclic groups ofprimeorder.
• The concepts of abelian group and$\mathbb {Z}$-moduleagree. More specifically, every$\mathbb {Z}$-module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers$\mathbb {Z}$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$2\times 2$rotation matrices.

Historical remarks

Camille Jordannamed abelian groups afterNorwegianmathematicianNiels Henrik Abel,as Abel had found that the commutativity of the group of apolynomialimplies that the roots of the polynomial can becalculated by using radicals.

Properties

If$n$is anatural numberand$x$is an element of an abelian group$G$written additively, then$nx$can be defined as$x+x+\cdots +x$($n$summands) and$(-n)x=-(nx)$.In this way,$G$becomes amoduleover thering$\mathbb {Z}$of integers. In fact, the modules over$\mathbb {Z}$can be identified with the abelian groups.

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

If$f,g:G\to H$are twogroup homomorphismsbetween 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${\text{Hom}}(G,H)$of all group homomorphisms from$G$to$H$is therefore an abelian group in its own right.

Somewhat akin to thedimensionofvector spaces,every abelian group has arank.It is defined as the maximalcardinalityof a set oflinearly independent(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 therational numbershave rank one, as well as every nonzeroadditive subgroupof the rationals. On the other hand, themultiplicative groupof the nonzero rationals has an infinite rank, as it is a free abelian group with the set of theprime numbersas a basis (this results from thefundamental theorem of arithmetic).

Thecenter$Z(G)$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 acharacteristicabelian 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 ofintegers modulo$n$,$\mathbb {Z} /n\mathbb {Z}$,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. Theautomorphism groupof a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper ofGeorg FrobeniusandLudwig Stickelbergerand later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter oflinear 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.In fact, for every prime number$p$there are (up to isomorphism) exactly two groups of order$p^{2}$,namely$\mathbb {Z} _{p^{2}}$and$\mathbb {Z} _{p}\times \mathbb {Z} _{p}$.

Classification

Thefundamental theorem of finite abelian groupsstates that every finite abelian group$G$can be expressed as the direct sum of cyclic subgroups ofprime-power order; it is also known as thebasis theorem for finite abelian groups.Moreover, automorphism groups of cyclic groups are examples of abelian groups.This is generalized by thefundamental theorem of finitely generated abelian groups,with finite groups being the special case whenGhas zerorank;this in turn admits numerous further generalizations.

The classification was proven byLeopold Kroneckerin 1870, though it was not stated in modern group-theoretic terms until later, and was preceded by a similar classification of quadratic forms byCarl Friedrich Gaussin 1801; seehistoryfor details.

The cyclic group$\mathbb {Z} _{mn}$of order$mn$is isomorphic to the direct sum of$\mathbb {Z} _{m}$and$\mathbb {Z} _{n}$if and only if$m$and$n$arecoprime.It follows that any finite abelian group$G$is isomorphic to a direct sum of the form

$\bigoplus _{i=1}^{u}\ \mathbb {Z} _{k_{i}}$

in either of the following canonical ways:

• the numbers$k_{1},k_{2},\dots,k_{u}$are powers of (not necessarily distinct) primes,
• or$k_{1}$divides$k_{2}$,which divides$k_{3}$,and so on up to$k_{u}$.

For example,$\mathbb {Z} _{15}$can be expressed as the direct sum of two cyclic subgroups of order 3 and 5:$\mathbb {Z} _{15}\cong \{0,5,10\}\oplus \{0,3,6,9,12\}$.The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 areisomorphic.

For another example, every abelian group of order 8 is isomorphic to either$\mathbb {Z} _{8}$(the integers 0 to 7 under addition modulo 8),$\mathbb {Z} _{4}\oplus \mathbb {Z} _{2}$(the odd integers 1 to 15 under multiplication modulo 16), or$\mathbb {Z} _{2}\oplus \mathbb {Z} _{2}\oplus \mathbb {Z} _{2}$.

See alsolist of small groupsfor finite abelian groups of order 30 or less.

Automorphisms

One can apply thefundamental theoremto count (and sometimes determine) theautomorphismsof a given finite abelian group$G$.To do this, one uses the fact that if$G$splits as a direct sum$H\oplus K$of subgroups ofcoprimeorder, then

$\operatorname {Aut} (H\oplus K)\cong \operatorname {Aut} (H)\oplus \operatorname {Aut} (K).$

Given this, the fundamental theorem shows that to compute the automorphism group of$G$it suffices to compute the automorphism groups of theSylow$p$-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:

$e_{1}\leq e_{2}\leq \cdots \leq e_{n}$

for some$n>0$.One needs to find the automorphisms of

$\mathbf {Z} _{p^{e_{1}}}\oplus \cdots \oplus \mathbf {Z} _{p^{e_{n}}}.$

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 finitecyclic groupcan be used. Another special case is when$n$is arbitrary but$e_{i}=1$for$1\leq i\leq n$.Here, one is considering$P$to be of the form

$\mathbf {Z} _{p}\oplus \cdots \oplus \mathbf {Z} _{p},$

so elements of this subgroup can be viewed as comprising a vector space of dimension$n$over the finite field of$p$elements$\mathbb {F} _{p}$.The automorphisms of this subgroup are therefore given by the invertible linear transformations, so

$\operatorname {Aut} (P)\cong \mathrm {GL} (n,\mathbf {F} _{p}),$

where$\mathrm {GL}$is the appropriategeneral linear group.This is easily shown to have order

$\left|\operatorname {Aut} (P)\right|=(p^{n}-1)\cdots (p^{n}-p^{n-1}).$

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

$d_{k}=\max\{r\mid e_{r}=e_{k}\}$

and

$c_{k}=\min\{r\mid e_{r}=e_{k}\}$

then one has in particular$k\leq d_{k}$,$c_{k}\leq k$,and

$\left|\operatorname {Aut} (P)\right|=\prod _{k=1}^{n}(p^{d_{k}}-p^{k-1})\prod _{j=1}^{n}(p^{e_{j}})^{n-d_{j}}\prod _{i=1}^{n}(p^{e_{i}-1})^{n-c_{i}+1}.$

One can check that this yields the orders in the previous examples as special cases (see Hillar & Rhea).

Finitely generated abelian groups

Main article:Finitely generated abelian group

An abelian groupAis finitely generated if it contains a finite set of elements (calledgenerators)$G=\{x_{1},\ldots,x_{n}\}$such that every element of the group is alinear combinationwith integer coefficients of elements ofG.

LetLbe afree abelian groupwith basis$B=\{b_{1},\ldots,b_{n}\}.$ There is a uniquegroup homomorphism $p\colon L\to A,$such that

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

This homomorphism issurjective,and itskernelis finitely generated (since integers form aNoetherian ring). Consider the matrixMwith integer entries, such that the entries of itsjth column are the coefficients of thejth generator of the kernel. Then, the abelian group is isomorphic to thecokernelof linear map defined byM.Conversely everyinteger matrixdefines 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 ofAis equivalent with multiplyingMon the left by aunimodular matrix(that is, an invertible integer matrix whose inverse is also an integer matrix). Changing the generating set of the kernel ofMis equivalent with multiplyingMon the right by a unimodular matrix.

TheSmith normal formofMis a matrix

$S=UMV,$

whereUandVare unimodular, andSis a matrix such that all non-diagonal entries are zero, the non-zero diagonal entries⁠$d_{1,1},\ldots,d_{k,k}$⁠are the first ones, and⁠$d_{j,j}$⁠is a divisor of⁠$d_{i,i}$⁠fori>j.The existence and the shape of the Smith normal form proves that the finitely generated abelian groupAis thedirect sum

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

whereris the number of zero rows at the bottom ofS(and also therankof the group). This is thefundamental 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.

Infinite abelian groups

The simplest infinite abelian group is theinfinite cyclic group$\mathbb {Z}$.Anyfinitely generated abelian group$A$is isomorphic to the direct sum of$r$copies of$\mathbb {Z}$and a finite abelian group, which in turn is decomposable into a direct sum of finitely manycyclic groupsofprime powerorders. Even though the decomposition is not unique, the number$r$,called therankof$A$,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$A$in which the equation$nx=a$admits a solution$x\in A$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$\mathbb {Q}$andPrüfer groups$\mathbb {Q} _{p}/Z_{p}$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$G=A\oplus C$.Thus divisible groups areinjective modulesin thecategory of abelian groups,and conversely, every injective abelian group is divisible (Baer's criterion). An abelian group without non-zero divisible subgroups is calledreduced.

Two important special classes of infinite abelian groups with diametrically opposite properties aretorsion groupsandtorsion-free groups,exemplified by the groups$\mathbb {Q} /\mathbb {Z}$(periodic) and$\mathbb {Q}$(torsion-free).

Torsion groups

An abelian group is calledperiodicortorsion,if every element has finiteorder.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 secondPrüfer theoremsstate that if$A$is a periodic group, and it either has abounded exponent,i.e.,$nA=0$for some natural number$n$,or is countable and the$p$-heightsof 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$\mathbb {Z} /p^{m}\mathbb {Z}$in such a decomposition is an invariant of$A$.These theorems were later subsumed in theKulikov criterion.In a different direction,Helmut Ulmfound 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 theirUlm invariants.

Torsion-free and mixed groups

An abelian group is calledtorsion-freeif every non-zero element has infinite order. Several classes oftorsion-free abelian groupshave been studied extensively:

• Free abelian groups,i.e. arbitrary direct sums of$\mathbb {Z}$
• Cotorsionandalgebraically compacttorsion-free groups such as the$p$-adic integers
• Slender groups

An abelian group that is neither periodic nor torsion-free is calledmixed.If$A$is an abelian group and$T(A)$is itstorsion 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$A$isnotisomorphic to$T(A)\oplus A/T(A)$.Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group$\mathbb {Z}$of integers is torsion-free$\mathbb {Z}$-module.

Invariants and classification

One of the most basic invariants of an infinite abelian group$A$is itsrank:the cardinality of the maximallinearly independentsubset of$A$.Abelian groups of rank 0 are precisely the periodic groups, whiletorsion-free abelian groups of rank 1are necessarily subgroups of$\mathbb {Q}$and can be completely described. More generally, a torsion-free abelian group of finite rank$r$is a subgroup of$\mathbb {Q} _{r}$.On the other hand, the group of$p$-adic integers$\mathbb {Z} _{p}$is a torsion-free abelian group of infinite$\mathbb {Z}$-rank and the groups$\mathbb {Z} _{p}^{n}$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 arepureandbasicsubgroups. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. See the books byIrving Kaplansky,László Fuchs,Phillip Griffith,andDavid Arnold,as well as the proceedings of the conferences on Abelian Group Theory published inLecture Notes in Mathematicsfor more recent findings.

Additive groups of rings

The additive group of aringis 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
• 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 naturaltopology,which turns them intotopological groups.

The collection of all abelian groups, together with thehomomorphismsbetween them, forms thecategory${\textbf {Ab}}$,the prototype of anabelian category.

Wanda Szmielew(1955) proved that the first-order theory of abelian groups, unlike its non-abelian counterpart, is decidable. Mostalgebraic structuresother thanBoolean algebrasareundecidable.

There are still many areas of current research:

• Amongst torsion-free abelian groups of finite rank, only the finitely generated case and therank 1case 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 incomputational group theory.

Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about theset theorycommonly assumed to underlie all of mathematics. Take theWhitehead problem:are all Whitehead groups of infinite order alsofree abelian groups?In the 1970s,Saharon Shelahproved that the Whitehead problem is:

• Undecidable in ZFC(Zermelo–Fraenkel axioms), the conventionalaxiomatic set theoryfrom 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 ifZFCis augmented by taking thegeneralized continuum hypothesisas an axiom;
• Positively answered if ZFC is augmented with the axiom ofconstructibility(seestatements true in L).

A note on typography

Among mathematicaladjectivesderived from theproper nameof amathematician,the word "abelian" is rare in that it is often spelled with a lowercasea,rather than an uppercaseA,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

• 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

References

• Cox, David(2004).Galois Theory.Wiley-Interscience.ISBN9781118031339.MR2119052.
• Fuchs, László(1970).Infinite Abelian Groups.Pure and Applied Mathematics. Vol. 36-I.Academic Press.MR0255673.
• Fuchs, László (1973).Infinite Abelian Groups.Pure and Applied Mathematics. Vol. 36-II.Academic Press.MR0349869.
• Griffith, Phillip A. (1970).Infinite Abelian group theory.Chicago Lectures in Mathematics.University of Chicago Press.ISBN0-226-30870-7.
• Herstein, I. N.(1975).Topics in Algebra(2nd ed.).John Wiley & Sons.ISBN0-471-02371-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.JSTOR27642365.S2CID1038507.
• Jacobson, Nathan(2009).Basic Algebra I(2nd ed.).Dover Publications.ISBN978-0-486-47189-1.
• Rose, John S. (2012).A Course on Group Theory.Dover Publications.ISBN978-0-486-68194-8.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.MR0072131.Zbl0248.02049.
• 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.JSTOR1993461.Archived(PDF)from the original on 2022-10-09.

External links

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