Abelian group

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 the Norwegian mathematician Niels Henrik Abel.
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].

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 four requirements known as the abelian group axioms :
;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 "non-abelian group" or "non-commutative group".

Facts

Notation

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 non-abelian groups are considered, with some notable exceptions being near-rings and partially ordered groups, where an operation is written additively even when non-abelian.

Multiplication table

To verify that a finite group is abelian, a table – known as a Cayley table – can be constructed in a similar fashion to a multiplication table. If the group is under the the entry of this table contains the product.
The group is abelian if and only if this table is symmetric about the main diagonal, which means that for all, the entry of the table equals the entry. Indeed, this equality expresses that

Examples

For the integers and the operation addition, 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.
Every cyclic group is abelian, because if, are in, then. 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.
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

named abelian groups after the 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 calculated by using radicals.

Properties

If is a natural number and is an element of an abelian group written additively, then can be defined as and. In this way, becomes a module over the ring of integers. In fact, the modules over can be identified with the abelian groups.
Theorems about 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. 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 elements of the group. Finite abelian groups and torsion groups have rank zero, and every abelian group of rank zero is a torsion group. The integers and the rational 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.
The center 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.

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. In fact, for every prime number there are 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 prime-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups. This is generalized by the fundamental theorem of finitely generated abelian groups, 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 group-theoretic 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:

the numbers are powers of primes,
or 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:. 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 , , 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 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
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. Fix a prime and suppose the exponents of the cyclic factors of the Sylow -subgroup are arranged in increasing order:
for some. One needs to find the automorphisms of
One special case is when, so that there is only one cyclic prime-power 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
where is the appropriate general linear group. This is easily shown to have order
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
and
then one has in particular,, and
One can check that this yields the orders in the previous examples as special cases.