Abelian group explained
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 nonabelian counterparts, and finite abelian groups are very well understood and fully classified.
Definition
An abelian group is a set,
, together with an
operation
that combines any two
elements
and
of
to form another element of
denoted
. The symbol
is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation,
, must satisfy five requirements known as the
abelian group axioms:
 Closure: For all
,
in
, the result of the operation
is also in
.
 Associativity: For all
,
, and
in
, the equation
holds.
 Identity element: There exists an element
in
, such that for all elements
in
, the equation
holds.
 Inverse element: For each
in
there exists an element
in
such that
, where
is the identity element.
 Commutativity: For all
,
in
,
.
A group in which the group operation is not commutative is called a "nonabelian group" or "noncommutative group".^{[2]}
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 
 0 



Multiplication 
or
 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 nonabelian groups are considered, some notable exceptions being nearrings and partially ordered groups, where an operation is written additively even when nonabelian.^{[3]}
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. If the group is
G=\{g_{1}=e,g_{2,}...,g_{n}\}
under the the entry of this table contains the product
.
g_{i} ⋅ g_{j}=g_{j} ⋅ g_{i}
for all
, which is iff the
entry of the table equals the
entry for all
, i.e. the table is symmetric about the main diagonal.
Examples
, denoted
, the operation + combines any two integers to form a third integer, addition is associative, zero is the
additive identity, every integer
has an
additive inverse,
, and the addition operation is commutative since
for any two integers
and
.
is abelian, because if
,
are in
, then
xy=a^{ma}^{n}=a^{m+n}=a^{na}^{m}=yx
. Thus the
integers,
, form an abelian group under addition, as do the
integers modulo
,
.
 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.^{[4]}
 The concepts of abelian group and

module agree. More specifically, every
module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers
in a unique way.
In general, matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. However, some groups of matrices are abelian groups under matrix multiplication – one example is the group of
rotation matrices.
Historical remarks
Camille Jordan named abelian groups after Norwegian mathematician Niels Henrik Abel, because Abel found that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated by using radicals.^{[5]}
Properties
If
is a
natural number and
is an element of an abelian group
written additively, then
can be defined as
(
summands) and
. In this way,
becomes a
module over the
ring
of integers. In fact, the modules over
can be identified with the abelian groups.
) 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
for
prime, and the latter is a direct sum of finitely many copies of
.
If
are two
group homomorphisms between abelian groups, then their sum
, defined by
, is again a homomorphism. (This is not true if
is a nonabelian group.) The set
of all group homomorphisms from
to
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.^{[6]} 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).
of a group
is the set of elements that commute with every element of
. A group
is abelian if and only if it is equal to its center
. The center of a group
is always a
characteristic abelian subgroup of
. If the quotient group
of a group by its center is cyclic then
is abelian.
^{[7]} Finite abelian groups
Cyclic groups of integers modulo
,
, were among the first examples of groups. It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The
automorphism group 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.^{[8]} In fact, for every prime number
there are (up to isomorphism) exactly two groups of order
, namely
and
.
Classification
The fundamental theorem of finite abelian groups states that every finite abelian group
can be expressed as the direct sum of cyclic subgroups of
primepower order; it is also known as the
basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups.
^{[9]} This is generalized by the fundamental theorem of finitely generated abelian groups, with finite groups being the special case when
G has zero
rank; this in turn admits numerous further generalizations.
The classification was proven by Leopold Kronecker in 1870, though it was not stated in modern grouptheoretic terms until later, and was preceded by a similar classification of quadratic forms by Carl Friedrich Gauss in 1801; see history for details.
The cyclic group
of order
is isomorphic to the direct sum of
and
if and only if
and
are
coprime. It follows that any finite abelian group
is isomorphic to a direct sum of the form
in either of the following canonical ways:
are powers of (not necessarily distinct) primes,
divides
, which divides
, and so on up to
.
For example,
can be expressed as the direct sum of two cyclic subgroups of order 3 and 5:
Z_{15}\cong\{0,5,10\} ⊕ \{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 are
isomorphic.
For another example, every abelian group of order 8 is isomorphic to either
(the integers 0 to 7 under addition modulo 8),
(the odd integers 1 to 15 under multiplication modulo 16), or
.
See also list of small groups 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
. To do this, one uses the fact that if
splits as a direct sum
of subgroups of
coprime order, then
\operatorname{Aut}(H ⊕ K)\cong\operatorname{Aut}(H) ⊕ \operatorname{Aut}(K).
Given this, the fundamental theorem shows that to compute the automorphism group of
it suffices to compute the automorphism groups of the
Sylow
subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of
). Fix a prime
and suppose the exponents
of the cyclic factors of the Sylow
subgroup are arranged in increasing order:
e_{1\leq}e_{2}\leq … \leqe_{n}
for some
. One needs to find the automorphisms of
One special case is when
, so that there is only one cyclic primepower factor in the Sylow
subgroup
. In this case the theory of automorphisms of a finite
cyclic group can be used. Another special case is when
is arbitrary but
for
. Here, one is considering
to be of the form
so elements of this subgroup can be viewed as comprising a vector space of dimension
over the finite field of
elements
. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so
\operatorname{Aut}(P)\congGL(n,F_{p),}
where
is the appropriate
general linear group. This is easily shown to have order
\left\operatorname{Aut}(P)\right=(p^{n1) … (p}^{np}^{n1}).
In the most general case, where the
and
are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines
d_{k=max\{r\mid}e_{r}=
\}
and
c_{k=min\{r\mid}e_{r=e}_{k\}}
then one has in particular
,
, and
\left\operatorname{Aut}(P)\right=
p^{k1})
.
One can check that this yields the orders in the previous examples as special cases (see Hillar, C., & Rhea, D.).
Finitely generated abelian groups
See main article: Finitely generated abelian group. An abelian group is finitely generated if it contains a finite set of elements (called generators)
such that every element of the group is a
linear combination with integer coefficients of elements of .
Let be a free abelian group with basis
B=\{b_{1,}\ldots,b_{n\}.}
There is a unique
group homomorphism
such that
p(b_{i)}=x_{i }fori=1,\ldots,n.
This homomorphism is
surjective, and its
kernel is finitely generated (since integers form a
Noetherian ring). 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 of linear map defined by . 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 is equivalent with multiplying 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 is equivalent with multiplying on the right by a unimodular matrix.
The Smith normal form of is a matrix
\Z^{r} ⊕ \Z/d_{1,1}\Z ⊕ … ⊕ \Z/d_{k,k}\Z,
where is the number of zero rows at the bottom of (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.
Infinite abelian groups
. Any
finitely generated abelian group
is isomorphic to the direct sum of
copies of
and a finite abelian group, which in turn is decomposable into a direct sum of finitely many
cyclic groups of
prime power orders. Even though the decomposition is not unique, the number
, called the
rank of
, and the prime powers giving the orders of finite cyclic summands are uniquely determined.
By contrast, classification of general infinitely generated abelian groups is far from complete. Divisible groups, i.e. abelian groups
in which the equation
admits a solution
for any natural number
and element
of
, constitute one important class of infinite abelian groups that can be completely characterized. Every divisible group is isomorphic to a direct sum, with summands isomorphic to
and
Prüfer groups
for various prime numbers
, and the cardinality of the set of summands of each type is uniquely determined.
^{[10]} Moreover, if a divisible group
is a subgroup of an abelian group
then
admits a direct complement: a subgroup
of
such that
. Thus divisible groups are
injective modules in the
category of abelian groups, and conversely, every injective abelian group is divisible (Baer's criterion). An abelian group without nonzero divisible subgroups is called
reduced.
Two important special classes of infinite abelian groups with diametrically opposite properties are torsion groups and torsionfree groups, exemplified by the groups
(periodic) and
(torsionfree).
Torsion groups
An abelian group is called periodic or torsion, if every element has finite order. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second Prüfer theorems state that if
is a periodic group, and it either has a
bounded exponent, i.e.,
for some natural number
, or is countable and the
heights of the elements of
are finite for each
, then
is isomorphic to a direct sum of finite cyclic groups.
^{[11]} The cardinality of the set of direct summands isomorphic to
in such a decomposition is an invariant of
.
^{[12]} 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
groups with elements of infinite height: those groups are completely classified by means of their Ulm invariants.
Torsionfree and mixed groups
An abelian group is called torsionfree if every nonzero element has infinite order. Several classes of torsionfree abelian groups have been studied extensively:
An abelian group that is neither periodic nor torsionfree is called mixed. If
is an abelian group and
is its
torsion subgroup, then the factor group
is torsionfree. However, in general the torsion subgroup is not a direct summand of
, so
is
not isomorphic to
. Thus the theory of mixed groups involves more than simply combining the results about periodic and torsionfree groups. The additive group
of integers is torsionfree
module.
^{[14]} Invariants and classification
One of the most basic invariants of an infinite abelian group
is its
rank: the cardinality of the maximal
linearly independent subset of
. Abelian groups of rank 0 are precisely the periodic groups, while
torsionfree abelian groups of rank 1 are necessarily subgroups of
and can be completely described. More generally, a torsionfree abelian group of finite rank
is a subgroup of
. On the other hand, the group of
adic integers
is a torsionfree abelian group of infinite
rank and the groups
with different
are nonisomorphic, 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 torsionfree 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 and basic subgroups. Introduction of various invariants of torsionfree abelian groups has been one avenue of further progress. See the books by Irving Kaplansky, László Fuchs, Phillip Griffith, and 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
 A. L. S. Corner's results on countable torsionfree 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 groups.
, the prototype of an
abelian category.
proved that the firstorder theory of abelian groups, unlike its nonabelian counterpart, is decidable. Most algebraic structures other than Boolean algebras are undecidable.
There are still many areas of current research:
 Amongst torsionfree 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 infiniterank torsionfree 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 firstorder 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 presentday 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 the 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.^{[15]}
See also
 , the smallest nonabelian group
References
 Book: Cox, David . David A. Cox. Galois Theory . 2004 . . 9781118031339 . 2119052 .
 Book: Fuchs, László. László Fuchs . 1970 . Infinite Abelian Groups . Pure and Applied Mathematics . 36I . . 0255673 .
 Book: Fuchs, László . 1973 . Infinite Abelian Groups . Pure and Applied Mathematics . 36II . . 0349869 .
 Book: Griffith, Phillip A. . 1970 . Infinite Abelian group theory . Chicago Lectures in Mathematics . . 0226308707.
 Book: Herstein, I. N. . Israel Nathan Herstein . 1975 . Topics in Algebra . registration . 2nd . . 047102371X.
 Hillar . Christopher . Rhea . Darren . 2007 . Automorphisms of finite abelian groups . . 114 . 10 . 917–923 . 10.1080/00029890.2007.11920485 . math/0605185. 27642365. 2006math......5185H . 1038507 .
 Book: Jacobson, Nathan. Nathan Jacobson . 2009. Basic Algebra I . 2nd . Dover Publications . 9780486471891.
 Book: Rose, John S. . 2012 . A Course on Group Theory . . 9780486681948. Unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978.
 Szmielew . Wanda. Wanda Szmielew . 1955 . Elementary properties of abelian groups . . 41 . 2. 203–271. 10.4064/fm412203271. 0072131. 0248.02049. free.
Notes and References
 p. 41
 Ramík, J., Pairwise Comparisons Method: Theory and Applications in Decision Making (Cham: Springer Nature Switzerland, 2020), p. 11.
 [Maurice AuslanderAuslander, M.]
 Rose 2012, p. 32.
 [David A. CoxCox, D. A.]
 Dixon, M. R., Kurdachenko, L. A., & Subbotin, I. Y., Linear Groups: The Accent on Infinite Dimensionality (Milton Park, AbingdononThames & Oxfordshire: Taylor & Francis, 2020), pp. 49–50.
 Rose 2012, p. 48.
 Rose 2012, p. 79.
 [:de:Hans KurzweilKurzweil, H.]
 For example,
Q/Z\cong\sum_{p}Q_{p/Z}_{p}
.
 Countability assumption in the second Prüfer theorem cannot be removed: the torsion subgroup of the direct product of the cyclic groups
for all natural
is not a direct sum of cyclic groups.
 Faith, C. C., Rings and Things and a Fine Array of Twentieth Century Associative Algebra (Providence: American Mathematical Society, 2004), p. 6.
 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 1117, 1985 (New York: Gordon & Breach, 1987), pp. 259–274.
 Lal, R., Algebra 2: Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier (Berlin, Heidelberg: Springer, 2017), p. 206.
 Web site: Abel Prize Awarded: The Mathematicians' Nobel. https://web.archive.org/web/20121231055255/http://www.maa.org/devlin/devlin_04_04.html . 31 December 2012. dead. 3 July 2016.