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. Abelian groups are named after Niels Henrik Abel.

Notation

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

Examples

Every cyclic group G is abelian, because if x, y are in G, then xy = a^maⁿ = a^{m + n} = a^{n + m} = aⁿa^m = yx. In particular, the integers Z (under addition) and the integers modulo n Z/nZ (also under addition).

The real numbers form an abelian group under addition, as do the non-zero real numbers under multiplication. Every commutative ring gives rise to two abelian groups in the same fashion -- the additive group of all elements, and the multiplicative group of invertible elements, or units.

Any subgroup of an abelian group is normal, and hence factor groups can be formed at will. Subgroups, factor groups, products and direct sums of abelian groups are again abelian.

Multiplication table

To verify that a certain finite group is indeed abelian, a table (matrix) can be drawn up in the similar fashion to a multiplication table, where, if the group is G = {g₁ = e, g₂, ..., 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, 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.

For example, Z/15Z = 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 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.

Relation to other mathematical topics

The abelian group, together with group homomorphisms, form a category, the prototype of an abelian category. In this encyclopedia, 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

The abelian group is rare in being expressed with a lowercase a, rather than A. Contrary to what one might expect, naming a concept in this way is considered one of the highest honors in mathematics. Mathematical productions such theorems, proofs, and lemmas are not named in this way, not even such great works as Maxwell's equations or the Laplace operator.