View source for Abelian group

{{short description|Commutative group (mathematics)}}
{{For|the group described by the archaic use of the related term "Abelian linear group"|Symplectic group}}

In [[mathematics]], an '''abelian group''', also called a '''commutative group''', is a [[group (mathematics)|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 [[integer]]s and the [[real number]]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 the Norwegian mathematician [[Niels Henrik Abel]].<ref>{{Harvard citation text|Jacobson|2009}}, [https://books.google.com/books?id=JHFpv0tKiBAC&pg=PA41&hl=cs&source=gbs_selected_pages&redir_esc=y#v=onepage&q&f=false p. 41].</ref>{{rp|41}}

The concept of an abelian group underlies many fundamental [[algebraic structure]]s, such as [[field (mathematics)|fields]], [[ring (mathematics)|rings]], [[vector space]]s, and [[algebra over a field|algebras]]. The theory of abelian groups is generally simpler than that of their [[non-abelian group|non-abelian]] counterparts, and finite abelian groups are very well understood and [[#Classification|fully classified]].

{{Group theory sidebar|Basics}}

== Definition ==
An abelian group is a [[set (mathematics)|set]] <math>A</math>, together with an [[Binary operation|operation]] ･ , that combines any two [[element (mathematics)|elements]] <math>a</math> and <math>b</math> of <math>A</math> to form another element of <math>A,</math> denoted <math>a \cdot b</math>. The symbol ･ is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, <math>(A, \cdot)</math>, 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 {{mvar|A}}, that the result is ''[[Well-defined expression|well-defined]]'', and that the result ''[[Element (mathematics)#Notation and terminology|belongs to]]'' {{mvar|A}}):

;Associativity: For all <math>a</math>, <math>b</math>, and <math>c</math> in <math>A</math>, the equation <math>(a \cdot b)\cdot c = a \cdot (b \cdot c)</math> holds.
;Identity element: There exists an element <math>e</math> in <math>A</math>, such that for all elements <math>a</math> in <math>A</math>, the equation <math>e \cdot a = a \cdot e = a</math> holds.
;Inverse element: For each <math>a</math> in <math>A</math> there exists an element <math>b</math> in <math>A</math> such that <math>a \cdot b = b \cdot a = e</math>, where <math>e</math> is the identity element.
;Commutativity: For all <math>a</math>, <math>b</math> in <math>A</math>, <math>a \cdot b = b \cdot a</math>.

A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".<ref>Ramík, J., ''Pairwise Comparisons Method: Theory and Applications in Decision Making'' ([[Cham, Switzerland|Cham]]: [[Springer Nature|Springer Nature Switzerland]], 2020), [https://books.google.com/books?id=2qDSDwAAQBAJ&pg=PA11&redir_esc=y#v=onepage&q&f=false p. 11].</ref>{{rp|11}}

== Facts ==
=== Notation ===
{{see also|Additive group|Multiplicative group}}
There are two main notational conventions for abelian groups – additive and multiplicative.

{| class="wikitable" style="margin: 1em auto 1em auto"
|-
! Convention
! Operation
! Identity
! Powers
! Inverse
|-
! Addition
| <math>x + y</math> || 0 || <math>nx</math> || <math>-x</math>
|-
! Multiplication
| <math>x \cdot y</math> or <math>xy</math> || 1
| <math>x^n</math>
| <math>x^{-1}</math>
|}
Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for [[module (mathematics)|module]]s and [[ring (mathematics)|ring]]s. The additive notation may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered, some notable exceptions being [[near-ring]]s and [[partially ordered group]]s, where an operation is written additively even when non-abelian.<ref>[[Maurice Auslander|Auslander, M.]], & [[David Buchsbaum|Buchsbaum, D.]], ''Groups, Rings, Modules'' ([[Mineola, New York|Mineola, NY]]: [[Dover Publications]], 1974), [https://books.google.com/books?id=MVEuBAAAQBAJ&pg=PA28&redir_esc=y#v=onepage&q&f=false pp. 28–29].</ref>{{rp|28–29}}<ref>Stanojkovski, M., ''Intense Automorphisms of Finite Groups'' ([[Providence, Rhode Island|Providence, RI]]: [[American Mathematical Society]], 2021) [https://books.google.com/books?id=d_NUEAAAQBAJ&pg=PA9&redir_esc=y#v=onepage&q&f=false pp. 9–14].</ref>{{rp|9–14}}

=== Multiplication table ===
To verify that a [[finite group]] is abelian, a table (matrix) – known as a [[Cayley table]] – can be constructed in a similar fashion to a [[multiplication table]].<ref>Isaev, A. P., & [[Valery Rubakov|Rubakov, V. A.]], ''Theory of Groups and Symmetries: Finite Groups, Lie Groups, and Lie Algebras'' ([[Singapore]]: [[World Scientific]], 2018), [https://books.google.com/books?id=7sFUDwAAQBAJ&pg=PA10&redir_esc=y#v=onepage&q&f=false p. 10].</ref>{{rp|10}} If the group is <math>G = \{g_1 = e, g_2, \dots, g_n \}</math> under the {{nowrap|operation <math>\cdot</math>,}} the {{nowrap|<math>(i, j)</math>-th}} entry of this table contains the product <math>g_i \cdot g_j</math>.

The group is abelian [[if and only if]] this table is [[symmetric]] about the main diagonal. This is true since the group is abelian [[If and only if|iff]] <math>g_i \cdot g_j = g_j \cdot g_i</math> for all <math>i, j = 1, ..., n</math>, which is iff the <math>(i, j)</math> entry of the table equals the <math>(j, i)</math> entry for all <math>i, j = 1, ..., n</math>, i.e. the table is symmetric about the main diagonal.

== Examples ==
* For the [[integer]]s and the operation [[addition]] <math>+</math>, denoted <math>(\mathbb{Z}, +)</math>, the operation + combines any two integers to form a third integer, addition is associative, zero is the [[additive identity]], every integer <math>n</math> has an [[additive inverse]], <math>-n</math>, and the addition operation is commutative since <math>n + m = m + n</math> for any two integers <math>m</math> and <math>n</math>.
* Every [[cyclic group]] <math>G</math> is abelian, because if <math>x</math>, <math>y</math> are in <math>G</math>, then <math>xy = a^ma^n = a^{m+n} = a^na^m = yx</math>. Thus the [[integer]]s, <math>\mathbb{Z}</math>, form an abelian group under addition, as do the [[modular arithmetic|integers modulo <math>n</math>]], <math>\mathbb{Z}/n \mathbb{Z}</math>.
* Every [[Ring (mathematics)|ring]] is an abelian group with respect to its addition operation. In a [[commutative ring]] the invertible elements, or [[unit (ring theory)|units]], form an abelian [[multiplicative group]]. In particular, the [[real number]]s 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 subgroup|normal]], so each subgroup gives rise to a [[quotient group]]. Subgroups, quotients, and [[Direct sum of groups|direct sums]] of abelian groups are again abelian. The finite [[simple group|simple]] abelian groups are exactly the cyclic groups of [[Prime number|prime]] [[Order (group theory)|order]].<ref>Rose 2012, [https://books.google.com/books?id=j-I7Zpq3GdIC&pg=PA32&redir_esc=y#v=onepage&q&f=false p. 32].</ref>{{rp|32}}
* The concepts of abelian group and <math>\mathbb{Z}</math>-[[Module (mathematics)|module]] agree.  More specifically, every <math>\mathbb{Z}</math>-module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers <math>\mathbb{Z}</math> in a unique way.

In general, [[matrix (mathematics)|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 <math>2 \times 2</math> [[rotation matrix|rotation matrices]].

== Historical remarks ==
<!-- This particular statement seems to be suspect, but the direction is right. Note: updated and corrected on Sept. 2, 2012 -->
[[Camille Jordan]] named abelian groups after the [[Norway|Norwegian]] [[mathematician]] [[Niels Henrik Abel]], who had found that the commutativity of the group of a [[polynomial]] implies that the roots of the polynomial can be [[solvability by radicals|calculated by using radicals]].<ref>[[David A. Cox|Cox, D. A.]], ''Galois Theory'' ([[Hoboken, New Jersey|Hoboken, NJ]]: [[Wiley (publisher)|John Wiley & Sons]], 2004), [https://books.google.com/books?id=96P8lsAF7fcC&pg=PA144&redir_esc=y#v=onepage&q&f=false pp. 144–145].</ref>{{rp|144–145}}<ref>Kepner, J., & H. Jananthan, ''Mathematics of Big Data'' ([[Cambridge, Massachusetts|Cambridge, MA]]: [[MIT Press]], 2018), [https://books.google.com/books?id=_rVNEAAAQBAJ&lpg=PA157&hl=cs&pg=PA157&redir_esc=y#v=onepage&q&f=false pp. 157–158].</ref>{{rp|157–158}}

== Properties ==
If <math>n</math> is a [[natural number]] and <math>x</math> is an element of an abelian group <math>G</math> written additively, then <math>nx</math> can be defined as <math>x + x + \cdots + x</math> (<math>n</math> summands) and <math>(-n)x = -(nx)</math>. In this way, <math>G</math> becomes a [[module (mathematics)|module]] over the [[ring (mathematics)|ring]] <math>\mathbb{Z}</math> of integers. In fact, the modules over <math>\mathbb{Z}</math> can be identified with the abelian groups.<ref>Eklof, Paul C., & Göbel, Rüdiger, eds., '' Abelian Groups and Modules: International Conference in Dublin, August 10–14, 1998'' ([[Basel]]: [[Springer Science+Business Media|Springer Basel AG]], 1999), [https://books.google.com/books?id=AMcACAAAQBAJ&pg=PA94&redir_esc=y#v=onepage&q&f=false pp. 94–97].</ref>{{rp|94–97}}

Theorems about abelian groups (i.e. [[module (mathematics)|module]]s over the [[principal ideal domain]] <math>\mathbb{Z}</math>) 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]]s which is a specialization of the [[structure theorem for finitely generated modules over a principal ideal domain]]. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a [[direct sum]] of a [[torsion group]] and a [[free abelian group]]. The former may be written as a direct sum of finitely many groups of the form <math>\mathbb{Z}/p^k\mathbb{Z}</math> for <math>p</math> prime, and the latter is a direct sum of finitely many copies of <math>\mathbb{Z}</math>.

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

Somewhat akin to the [[Dimension (vector space)|dimension]] of [[vector space]]s, every abelian group has a ''[[rank of an abelian group|rank]]''. It is defined as the maximal [[cardinal number|cardinality]] of a set of [[linearly independent]] (over the integers) elements of the group.<ref>Dixon, M. R., Kurdachenko, L. A., & Subbotin, I. Y., ''Linear Groups: The Accent on Infinite Dimensionality'' ([[Milton Park]], [[Abingdon-on-Thames]] & [[Oxfordshire]]: [[Taylor & Francis]], 2020), [https://books.google.com/books?id=fODaDwAAQBAJ&pg=PT49&redir_esc=y#v=onepage&q&f=false pp. 49–50].</ref>{{rp|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 number]]s have rank one, as well as every nonzero [[additive group|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 number]]s as a basis (this results from the [[fundamental theorem of arithmetic]]).

The [[center (group theory)|center]] <math>Z(G)</math> of a group <math>G</math> is the set of elements that commute with every element of <math>G</math>. A group <math>G</math> is abelian if and only if it is equal to its center <math>Z(G)</math>. The center of a group <math>G</math> is always a [[characteristic subgroup|characteristic]] abelian subgroup of <math>G</math>. If the quotient group <math>G/Z(G)</math> of a group by its center is cyclic then <math>G</math> is abelian.<ref>Rose 2012, [https://books.google.com/books?id=pYk6AAAAIAAJ&pg=PA48&redir_esc=y#v=onepage&q&f=false p. 48].</ref>

== Finite abelian groups ==
Cyclic groups of [[modular arithmetic|integers modulo <math>n</math>]], <math>\mathbb{Z}/n\mathbb{Z}</math>, 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.<ref>Rose 2012, [https://books.google.com/books?id=pYk6AAAAIAAJ&pg=PA79&redir_esc=y#v=onepage&q&f=false p. 79].</ref> In fact, for every prime number <math>p</math> there are (up to isomorphism) exactly two groups of order <math>p^2</math>, namely <math>\mathbb{Z}_{p^2}</math> and <math>\mathbb{Z}_p\times\mathbb{Z}_p</math>.

=== Classification ===
The '''fundamental theorem of finite abelian groups''' states that every finite abelian group <math>G</math> can be expressed as the direct sum of cyclic subgroups of [[prime number|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.<ref>[[:de:Hans Kurzweil|Kurzweil, H.]], & [[:de:Bernd Stellmacher|Stellmacher, B.]], ''The Theory of Finite Groups: An Introduction'' (New York, Berlin, Heidelberg: [[Springer Science+Business Media|Springer Verlag]], 2004), [https://books.google.com/books?id=iebSBwAAQBAJ&pg=PA43&redir_esc=y#v=onepage&q&f=false pp. 43–54].</ref> This is generalized by the [[fundamental theorem of finitely generated abelian groups]], with finite groups being the special case when ''G'' has zero [[rank of an abelian group|rank]]; this in turn admits numerous further generalizations.

The classification was proven by [[Leopold Kronecker]] 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]] in 1801; see [[Fundamental theorem of finitely generated abelian groups#History|history]] for details.

The cyclic group <math>\mathbb{Z}_{mn}</math> of order <math>mn</math> is isomorphic to the direct sum of <math>\mathbb{Z}_m</math> and <math>\mathbb{Z}_n</math> if and only if <math>m</math> and <math>n</math> are [[coprime]]. It follows that any finite abelian group <math>G</math> is isomorphic to a direct sum of the form

:<math>\bigoplus_{i=1}^{u}\ \mathbb{Z}_{k_i}</math>

in either of the following canonical ways:
* the numbers <math>k_1, k_2, \dots, k_u</math> are powers of (not necessarily distinct) primes,
* or <math>k_1</math> [[divisor|divides]] <math>k_2</math>, which divides <math>k_3</math>, and so on up to <math>k_u</math>.

For example, <math>\mathbb{Z}_{15}</math> can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: <math>\mathbb{Z}_{15} \cong  \{0,5,10\} \oplus \{0,3,6,9,12\}</math>. 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 [[group isomorphism|isomorphic]].

For another example, every abelian group of order 8 is isomorphic to either <math>\mathbb{Z}_8</math> (the integers 0 to 7 under addition modulo 8), <math>\mathbb{Z}_4\oplus \mathbb{Z}_2</math> (the odd integers 1 to 15 under multiplication modulo 16), or <math>\mathbb{Z}_2\oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2</math>.

See also [[list of small groups]] for finite abelian groups of order 30 or less.

=== Automorphisms ===
One can apply the [[#Classification|fundamental theorem]] to count (and sometimes determine) the [[Group isomorphism#Automorphisms|automorphisms]] of a given finite abelian group <math>G</math>. To do this, one uses the fact that if <math>G</math> splits as a direct sum <math>H\oplus K</math> of subgroups of [[coprime]] order, then
:<math>\operatorname{Aut}(H\oplus K) \cong \operatorname{Aut}(H)\oplus \operatorname{Aut}(K).</math>

Given this, the fundamental theorem shows that to compute the automorphism group of <math>G</math> it suffices to compute the automorphism groups of the [[Sylow theorems|Sylow]] <math>p</math>-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of <math>p</math>). Fix a prime <math>p</math> and suppose the exponents <math>e_i</math> of the cyclic factors of the Sylow <math>p</math>-subgroup are arranged in increasing order:

:<math>e_1\leq e_2 \leq\cdots\leq e_n</math>

for some <math>n > 0</math>. One needs to find the automorphisms of

:<math>\mathbf{Z}_{p^{e_1}} \oplus \cdots \oplus \mathbf{Z}_{p^{e_n}}.</math>

One special case is when <math>n = 1</math>, so that there is only one cyclic prime-power factor in the Sylow <math>p</math>-subgroup <math>P</math>. In this case the theory of automorphisms of a finite [[cyclic group]] can be used. Another special case is when <math>n</math> is arbitrary but <math>e_i = 1</math> for <math> 1 \le i  \le n</math>. Here, one is considering <math>P</math> to be of the form

:<math>\mathbf{Z}_p \oplus \cdots \oplus \mathbf{Z}_p,</math>

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

:<math>\operatorname{Aut}(P)\cong\mathrm{GL}(n,\mathbf{F}_p),</math>

where <math>\mathrm{GL}</math> is the appropriate [[general linear group]]. This is easily shown to have order

:<math> \left|\operatorname{Aut}(P)\right|=(p^n-1)\cdots(p^n-p^{n-1}).</math>

In the most general case, where the <math>e_i</math> and <math>n</math> are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines

:<math>d_k=\max\{r\mid e_r = e_k\}</math>

and

:<math>c_k=\min\{r\mid e_r=e_k\}</math>

then one has in particular <math>k \le d_k</math>, <math>c_k \le k</math>, and

:<math> \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}. </math>

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

==Finitely generated abelian groups==
{{main|Finitely generated abelian group}}
An abelian group {{mvar|A}} is finitely generated if it contains a finite set of elements (called ''generators'') <math>G=\{x_1, \ldots, x_n\}</math> such that every element of the group is a [[linear combination]] with integer coefficients of elements of {{mvar|G}}. 

Let {{mvar|L}} be a [[free abelian group]] with basis <math>B=\{b_1, \ldots, b_n\}.</math> 
There is a unique [[group homomorphism]]
<math>p\colon L \to A,</math> such that 
:<math>p(b_i) = x_i\quad \text{for } i=1,\ldots, n.</math>
This homomorphism is [[Surjective function|surjective]], and its [[kernel (linear algebra)|kernel]] is finitely generated (since integers form a [[Noetherian ring]]). Consider the matrix {{mvar|M}} with integer entries, such that the entries of its {{mvar|j}}th column are the coefficients of the {{mvar|j}}th generator of the kernel. Then, the abelian group is isomorphic to the [[cokernel]] of linear map defined by {{mvar|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 {{mvar|A}} is equivalent with multiplying {{mvar|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 {{mvar|M}} is equivalent with multiplying {{mvar|M}} on the right by a unimodular matrix.

The [[Smith normal form]] of {{mvar|M}} is a matrix
:<math>S=UMV,</math>
where {{mvar|U}} and {{mvar|V}} are unimodular, and {{mvar|S}} is a matrix such that all non-diagonal entries are zero, the non-zero diagonal entries {{tmath|d_{1,1}, \ldots, d_{k,k} }} are the first ones, and {{tmath|d_{j,j} }} is a divisor of {{tmath|d_{i,i} }} for {{math|''i'' > ''j''}}. The existence and the shape of the Smith normal form proves that the finitely generated abelian group {{mvar|A}} is the [[Direct sum of groups|direct sum]] 
:<math>\Z^r \oplus \Z/d_{1,1}\Z \oplus \cdots \oplus \Z/d_{k,k}\Z,</math> 
where {{mvar|r}} is the number of zero rows at the bottom of {{mvar|S}} (and also the [[Rank of an abelian group|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.<ref>Finkelstein, L., & [[William Kantor|Kantor, W. M.]], eds., ''Groups and Computation II: Workshop on Groups and Computation, June 7–10, 1995'' ([[Providence, Rhode Island|Providence]]: [[American Mathematical Society|AMS]], 1997), [https://books.google.com/books?id=lyL8Ui36FKsC&pg=PA26&redir_esc=y#v=onepage&q&f=false pp. 26–27].</ref>{{rp|26–27}}

== Infinite abelian groups ==
The simplest infinite abelian group is the [[infinite cyclic group]] <math>\mathbb{Z}</math>. Any [[finitely generated abelian group]] <math>A</math> is isomorphic to the direct sum of <math>r</math> copies of <math>\mathbb{Z}</math> and a finite abelian group, which in turn is decomposable into a direct sum of finitely many [[cyclic group]]s of [[prime power]] orders. Even though the decomposition is not unique, the number <math>r</math>, called the [[Rank of an abelian group|rank]] of <math>A</math>, and the prime powers giving the orders of finite cyclic summands are uniquely determined.

By contrast, classification of general infinitely generated abelian groups is far from complete. [[Divisible group]]s, i.e. abelian groups <math>A</math> in which the equation <math>nx = a</math> admits a solution <math>x \in A</math> for any natural number <math>n</math> and element <math>a</math> of <math>A</math>, 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 <math>\mathbb{Q}</math> and [[Prüfer group]]s <math>\mathbb{Q}_p/Z_p</math> for various prime numbers <math>p</math>, and the cardinality of the set of summands of each type is uniquely determined.<ref>For example, <math>\mathbb{Q}/\mathbb{Z} \cong \sum_p \mathbb{Q}_p/\mathbb{Z}_p</math>.</ref> Moreover, if a divisible group <math>A</math> is a subgroup of an abelian group <math>G</math> then <math>A</math> admits a direct complement: a subgroup <math>C</math> of <math>G</math> such that <math>G = A \oplus C</math>. Thus divisible groups are [[injective module]]s in the [[category of abelian groups]], and conversely, every injective abelian group is divisible ([[Baer's criterion]]). An abelian group without non-zero divisible subgroups is called '''reduced'''.

Two important special classes of infinite abelian groups with diametrically opposite properties are ''torsion groups'' and ''torsion-free groups'', exemplified by the groups <math>\mathbb{Q}/\mathbb{Z}</math> (periodic) and <math>\mathbb{Q}</math> (torsion-free).

=== Torsion groups ===
An abelian group is called '''[[periodic group|periodic]]''' or '''[[torsion (algebra)|torsion]]''', if every element has finite [[order (group theory)|order]]. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second [[Prüfer theorems]] state that if <math>A</math> is a periodic group, and it either has a '''bounded exponent''', i.e., <math>nA = 0</math> for some natural number <math>n</math>, or is countable and the [[height (abelian group)|<math>p</math>-heights]] of the elements of <math>A</math> are finite for each <math>p</math>, then <math>A</math> is isomorphic to a direct sum of finite cyclic groups.<ref>Countability assumption in the second Prüfer theorem cannot be removed: the torsion subgroup of the [[direct product]] of the cyclic groups <math>\mathbb{Z}/p^m\mathbb{Z}</math> for all natural <math>m</math> is not a direct sum of cyclic groups.</ref> 
The cardinality of the set of direct summands isomorphic to <math>\mathbb{Z}/p^m\mathbb{Z}</math> in such a decomposition is an invariant of <math>A</math>.<ref>[[:de:Carl Faith|Faith, C. C.]], ''Rings and Things and a Fine Array of Twentieth Century Associative Algebra'' (Providence: AMS, 2004), [https://books.google.com/books?id=H1TzBwAAQBAJ&pg=PA6&redir_esc=y#v=onepage&q&f=false p. 6].</ref>{{rp|6}} 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 <math>p</math>-groups with elements of infinite height: those groups are completely classified by means of their [[Ulm invariant]]s.<ref>Gao, S., ''Invariant Descriptive Set Theory'' ([[Boca Raton, Florida|Boca Raton, FL]]: [[CRC Press]], 2008), [https://google.com/books?id=VfiOZWY282QC&pg=PT317 p. 317].</ref>{{rp|317}}

=== Torsion-free and mixed groups ===
An abelian group is called '''torsion-free''' if every non-zero element has infinite order. Several classes of [[torsion-free abelian group]]s have been studied extensively:

* [[Free abelian group]]s, i.e. arbitrary direct sums of <math>\mathbb{Z}</math>
* [[Cotorsion group|Cotorsion]] and [[algebraically compact module|algebraically compact]] torsion-free groups such as the [[p-adic integer|<math>p</math>-adic integers]]
* [[Slender group]]s<ref>Albrecht, U., "Products of Slender Abelian Groups", in Göbel, R., & Walker, E., eds., ''Abelian Group Theory: Proceedings of the Third Conference Held on Abelian Group Theory at Oberwolfach, August 11-17, 1985'' (New York: [[Taylor & Francis|Gordon & Breach]], 1987), [https://books.google.com/books?id=kwvGp9_4lOMC&pg=PT259&redir_esc=y#v=onepage&q&f=false pp. 259–274].</ref>{{rp|259–274}}

An abelian group that is neither periodic nor torsion-free is called '''mixed'''. If <math>A</math> is an abelian group and <math>T(A)</math> is its [[torsion subgroup]], then the factor group  <math>A/T(A)</math> is torsion-free. However, in general the torsion subgroup is not a direct summand of <math>A</math>, so  <math>A</math> is ''not'' isomorphic to <math>T(A) \oplus A/T(A)</math>. Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group <math>\mathbb{Z}</math> of integers is torsion-free <math>\mathbb{Z}</math>-module.<ref>Lal, R., ''Algebra 2: Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier'' (Berlin, Heidelberg: Springer, 2017), [https://books.google.com/books?id=FwPNDgAAQBAJ&pg=PA206&redir_esc=y#v=onepage&q&f=false p. 206].</ref>{{rp|206}}

=== Invariants and classification ===
One of the most basic invariants of an infinite abelian group <math>A</math> is its [[rank of an abelian group|rank]]: the cardinality of the maximal [[linearly independent]] subset of <math>A</math>. Abelian groups of rank 0 are precisely the periodic groups, while [[torsion-free abelian groups of rank 1]] are necessarily subgroups of <math>\mathbb{Q}</math> and can be completely described. More generally, a torsion-free abelian group of finite rank <math>r</math> is a subgroup of <math>\mathbb{Q}_r</math>. On the other hand, the group of [[p-adic integer|<math>p</math>-adic integers]] <math>\mathbb{Z}_p</math> is a torsion-free abelian group of infinite <math>\mathbb{Z}</math>-rank and the groups <math>\mathbb{Z}_p^n</math> with different <math>n</math> 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 (mathematics)|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]]
* A.L.S. Corner's results on countable torsion-free groups
* Shelah's work to remove cardinality restrictions
* [[Burnside ring]]

== Relation to other mathematical topics ==
Many large abelian groups possess a natural [[topology]], which turns them into [[topological group]]s.

The collection of all abelian groups, together with the [[Group homomorphism|homomorphisms]] between them, forms the [[category of abelian groups|category]] <math>\textbf{Ab}</math>, the prototype of an [[abelian category]].

{{harvs|first=Wanda|last=Szmielew|authorlink=Wanda Szmielew|year=1955|txt}} proved that the first-order theory of abelian groups, unlike its non-abelian counterpart, is decidable. Most [[algebraic structure]]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]]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 typography ==
Among mathematical [[adjective]]s 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.<ref>{{cite web|url=https://www.maa.org/external_archive/devlin/devlin_04_04.html|archive-url=https://web.archive.org/web/20121231055255/http://www.maa.org/devlin/devlin_04_04.html |archive-date=31 December 2012|url-status=dead|access-date=3 July 2016|title=Abel Prize Awarded: The Mathematicians' Nobel}}</ref>

== See also ==
{{Algebraic structures}}
*{{annotated link|Commutator subgroup}}
*{{annotated link|Abelianization}}
*{{annotated link|Dihedral group of order 6}}, the smallest non-abelian group
*{{annotated link|Elementary abelian group}}
*{{annotated link|Grothendieck group}}
*{{annotated link|Pontryagin duality}}

== Notes ==
{{Reflist}}

== References ==
* {{cite book |last=Cox |first=David |author-link=David A. Cox|url=https://books.google.com/books?id=96P8lsAF7fcC |date=2004 |title=Galois Theory |publisher=[[Wiley-Interscience]] |isbn=9781118031339 |mr=2119052 }}
* {{cite book|last=Fuchs|first=László|author-link=László Fuchs |date=1970 |title=Infinite Abelian Groups |series=Pure and Applied Mathematics |volume=36-I |publisher=[[Academic Press]] |mr=0255673 }}
* {{cite book |last=Fuchs |first=László |date=1973 |title=Infinite Abelian Groups |series=Pure and Applied Mathematics |volume=36-II |publisher=[[Academic Press]] |mr=0349869 }}
* {{cite book |first=Phillip A. |last=Griffith  |date=1970 |title=Infinite Abelian group theory |series=Chicago Lectures in Mathematics |publisher=[[University of Chicago Press]] |isbn=0-226-30870-7}}
* {{cite book |last=Herstein |first=I. N. |author-link=Israel Nathan Herstein |date=1975 |title=Topics in Algebra |url=https://archive.org/details/topicsinalgebra00hers |url-access=registration |edition=2nd |publisher=[[John Wiley & Sons]] |isbn=0-471-02371-X}}
* {{cite journal |last1=Hillar |first1=Christopher |last2=Rhea |first2=Darren |date=2007 |title=Automorphisms of finite abelian groups |url=https://web.archive.org/web/20060629091951id_/http://www.math.tamu.edu:80/~chillar/files/autabeliangrps.pdf|journal=[[American Mathematical Monthly]] |volume=114 |issue=10 |pages=917–923 |doi=10.1080/00029890.2007.11920485 |arxiv=math/0605185|jstor=27642365|bibcode=2006math......5185H |s2cid=1038507 }}
* {{Cite book| last=Jacobson| first=Nathan|author-link= Nathan Jacobson | date=2009| title=Basic Algebra I |url=https://books.google.com/books?id=JHFpv0tKiBAC| edition=2nd | publisher=[[Dover Publications]] | isbn = 978-0-486-47189-1}}
* {{cite book |last=Rose |first=John S. |date=2012 |title=A Course on Group Theory |url=https://books.google.com/books?id=pYk6AAAAIAAJ|publisher=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.
* {{cite journal |last=Szmielew |first=Wanda|author-link= Wanda Szmielew |date=1955 |title=Elementary Properties of Abelian Groups |journal=[[Fundamenta Mathematicae]] |volume=41 |issue=2|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm41/fm41122.pdf|pages= 203–271|doi=10.4064/fm-41-2-203-271|mr=0072131|zbl=0248.02049|doi-access=free}}
* {{cite journal |last1=Robinson |first1=Abraham|author-link1= Abraham Robinson |last2=Zakon |first2=Elias|date=1960 |title=Elementary Properties of Ordered Abelian Groups |journal=[[Transactions of the American Mathematical Society]] |volume=96 |issue=2|url=https://www.ams.org/journals/tran/1960-096-02/S0002-9947-1960-0114855-0/S0002-9947-1960-0114855-0.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/tran/1960-096-02/S0002-9947-1960-0114855-0/S0002-9947-1960-0114855-0.pdf |archive-date=2022-10-09 |url-status=live|pages= 222–236|doi=10.2307/1993461|jstor=1993461 |doi-access=free}}

==External links==
* {{springer|title=Abelian group|id=p/a010230|mode=cs1}}

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