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In mathematics, an 'abelian group', also called a 'commutative group', is a group (''G'',
★ ) such that ''a''
★ ''b'' = ''b''
★ ''a'' for all ''a'' and ''b'' in ''G''. In other words, the ''order'' in which the binary operation is performed doesn't matter. Such groups are generally easier to understand, although infinite abelian groups remain a subject of current research.
Groups that are not commutative are called ''non-abelian'' (or ''non-commutative''). Abelian groups are named after Niels Henrik Abel.

Contents

Notation

Examples

Multiplication table

Properties

Finite abelian groups

Automorphisms of finite abelian groups

Relation to other mathematical topics

A note on the typography

See also

References

Notation

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

Convention	Operation	Identity	Powers	Inverse	Direct sum/product
Addition	''x'' + ''y''	0	''nx''	−''x''	''G'' ⊕ ''H''
Multiplication	''x'' ★ ''y'' or ''xy''	''e'' or 1	''x''''n''	''x'' −1	''G'' × ''H''

The multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules. When studying abelian groups apart from other groups, the additive notation is usually used.

Examples

Every cyclic group ''G'' is abelian, because if ''x'', ''y'' are in ''G'', then ''xy'' = ''a''^''m''''a''^''n'' = ''a''^{''m'' + ''n''} = ''a''^{''n'' + ''m''} = ''a''^''n''''a''^''m'' = ''yx''. Thus the integers, 'Z', form an abelian group under addition, as do the integers modulo ''n'', 'Z'/''n'''Z'.
Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.
Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian.
Matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative.

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 operation ⋅, the (''i'', ''j'')'th entry of this table contains the product ''g''_''i'' ⋅ ''g''_''j''. The group is abelian if and only if this table is symmetric about the main diagonal (i.e. if the matrix is a symmetric matrix).
This is true since if the group is abelian, then ''g''_''i'' ⋅ ''g''_''j'' = ''g''_''j'' ⋅ ''g''_''i''. This implies that the (''i'', ''j'')'th entry of the table equals the (''j'', ''i'')'th entry - i.e. the table is symmetric about the main diagonal.

Properties

If ''n'' is a natural number and ''x'' is an element of an abelian group ''G'' written additively, then ''nx'' can be defined as ''x'' + ''x'' + ... + ''x'' (''n'' summands) and (−''n'')''x'' = −(''nx''). In this way, ''G'' becomes a module over the ring 'Z' of integers. In fact, the modules over 'Z' can be identified with the abelian groups.
Theorems about abelian groups (i.e. modules over the principal ideal domain 'Z') can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups.
If ''f'', ''g'' : ''G''  →  ''H'' are two group homomorphisms between 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 Hom(''G'', ''H'') of all group homomorphisms from ''G'' to ''H'' thus turns into 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 cardinality of the largest set of linearly independent elements of the group. The integers and the rational numbers have rank one, as well as every subgroup of the rationals. While the rank one torsion-free abelian groups are well understood, even finite-rank abelian groups are not well understood. Infinite-rank abelian groups can be extremely complex and many open questions exist, often intimately connected to questions of set theory.

Finite abelian groups

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. This is a special application of the fundamental theorem of finitely generated abelian groups in the case when ''G'' has torsion-free rank equal to 0.
'Z'_''mn'' is isomorphic to the direct product of 'Z'_''m'' and 'Z'_''n'' if and only if ''m'' and ''n'' are coprime.
Therefore we can write any finite abelian group ''G'' as a direct product of the form
:$mathbb\{Z\}\_\{k\_1\}\; oplus\; cdots\; oplus\; mathbb\{Z\}\_\{k\_u\}$
in two unique ways:

★ where the numbers ''k''₁,...,''k''_''u'' are powers of primes

★ where ''k''₁ divides ''k''₂, which divides ''k''₃ and so on up to ''k''_''u''.
Thus we have 3 2 or 6, 5 2 or 10, 4 3 or 12, 3 2 2 or 6 2, 7 2 or 14, and 5 3 or 15, but anyway 2 2, 4 2, 2 2 2, 3 3, 8 2, 4 4, 4 2 2, and 2 2 2 2.
For example, 'Z'/15'Z' = 'Z'/15 can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: 'Z'/15 = {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 'Z'/8 (the integers 0 to 7 under addition modulo 8), 'Z'/4  ⊕ 'Z'/2 (the odd integers 1 to 15 under multiplication modulo 16), or 'Z'/2  ⊕  'Z'/2  ⊕  'Z'/2.
See also list of small groups for finite abelian groups of order 16 or less.

Automorphisms of finite abelian groups

One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group ''G''. To do this, one uses the fact (which will not be proved here) that if ''G'' splits as a direct sum ''H'' ⊕ ''K'' of subgroups of coprime order, then Aut(''H'' ⊕ ''K'') ≅ Aut(''H'') ⊕ Aut(''K'').
Given this, the fundamental theorem shows that to compute the automorphism group of ''G'' it suffices to compute the automorphism groups of the Sylow ''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 ≤ ''e''₂ ≤ … ≤ ''e''_''n''
for some ''n'' > 0. One needs to find the automorphisms of
:''Z''_''p''^''e''1 ⊕ ... ⊕ ''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 finite cyclic group can be used. Another special case is when ''n'' is arbitrary but ''e''_''i'' = 1 for 1 ≤ ''i'' ≤ ''n''. Here, one is considering ''P'' to be of the form
:'Z'_''p'' ⊕ … ⊕ '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 'F'_''p''. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so
:Aut(''P'') ≅ GL(''n'', 'F'_''p''),
which is easily shown to have order
:|Aut(''P'')| = (''p''^''n'' − 1)…(''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 | ''e''_''r'' = ''e''_''k''}
and
:''c''_''k'' = min{r | ''e''_''r'' = ''e''_''k''}
then one has in particular ''d''_''k'' ≥ ''k'', ''c''_''k'' ≤ ''k'', and
:$|mathrm\{Aut\}(P)|\; =\; left(prod\_\{k=1\}^n\{p^\{d\_k\}\; -\; p^\{k-1\}\}$
ight)left(prod_{j=1}^n{(p^{e_j})^{n-d_j}}
ight)left(prod_{i=1}^n{(p^{e_i-1})^{n-c_i+1}}
ight).
One can check that this yields the orders in the previous examples as special cases (see [Hillar,Rhea]).

Relation to other mathematical topics

The collection of all abelian groups, together with the homomorphisms between them, forms a category, the prototype of an abelian category. We denote this category 'Ab'. See category of abelian groups for a list of its properties.
Many large abelian groups carry a natural topology, turning them into topological groups.

A note on the typography

Among mathematical adjectives derived from the proper name of a mathematician, the word "abelian" is rare in being expressed with a lowercase 'a', rather than 'A' (cf. Riemannian). Contrary to what one might expect, naming a concept in this way is considered one of the highest honours in mathematics for the namesake.

See also

★ Pontryagin duality

★ Class field theory

★ Abelianization

References

★ László Fuchs, ''Infinite abelian groups''. Vol. I. Pure and Applied Mathematics, Vol. 36 Academic Press, New York-London 1970 xi+290 pp.

★ László Fuchs, ''Infinite abelian groups''. Vol. II. Pure and Applied Mathematics. Vol. 36-II. Academic Press, New York-London, 1973. ix+363 pp.

★ Christopher Hillar and Darren Rhea. ''Automorphisms of Finite Abelian Groups''.