Abelian group

An abelian group ( after the Norwegian mathematician Niels Henrik Abel ) is in group theory, a group for which the commutative

Applies. If a group is abelian, then to write their linking mostly as an addition to the zero element as a neutral element and as the negative or opposite of a, rare as multiplication by the unit element as a neutral element and as the inverse or reciprocal of

Examples

Every cyclic group is abelian; Examples are the additive group of integers or the residue class ring with the addition.

The real numbers form an abelian group under addition; without the zero they form an abelian group with multiplication.

General supplies every body in the same way two abelian groups and.

Another example is the factor group is isomorphic to the ( multiplicative ) group of complex roots of unity. The factor group is isomorphic to the group of all complex numbers with sum 1

In contrast, the group of invertible matrices over a body for an example of a non-Abelian group, the smallest non-Abelian group is S3.

Properties

For a small finite group can be easily seen if it is Abelian. The following applies:

Is a natural number and an element of the abelian group, then you can define it as the sum of exactly summands, 0x than 0 ( the neutral element of the group), and ( n ) as x - (nx). In this way, G becomes a module over the ring. Since each module is an abelian group, so you can identify the moduli with the abelian groups. Theorems about abelian groups can often be generalized to sets of modules over principal ideal rings. An example is the classification of finitely generated abelian groups ( see below).

Every subgroup of an abelian group is a normal subgroup, so you can create a group factor for each subgroup. Subgroups, factor groups, products and direct sums of abelian groups are abelian again.

If two group homomorphisms between abelian groups, then their sum f g, is defined by

Also a homomorphism. (This does not generally, if H is not abelian. ) The set of all group homomorphisms is with this addition itself into an abelian group.

The abelian groups and their homomorphisms form a category. This is the prototype of an abelian category.

Many Abelian groups have a natural topology, by which they are to topological groups.

Additional Attributes

• An abelian group is finitely generated if and only if there is a finite set, so that each element in the form

• The properties of free and projective are equivalent, as well as torsion and flat.
• An abelian group is called divisible if for all cases: (that is, for every there is a such that applies ). The abelian group of rational numbers with the addition of a link is a divisible group.

Structure theory

• Fully classified are the finitely generated abelian groups. Namely, they are direct sums of finitely many cyclic groups. This way one can be chosen so that - at an appropriate order - the order of each of these groups (from the second, and so far last ) is a multiple of the order of the previous one, whereby any infinite of them are placed at the end. In addition, these group orders are then determined ( with this order) unique.
• For any finite abelian groups can be assigned their rank analogous to the notion of dimension of a vector space every abelian group. It is defined as the largest cardinality of a linearly - independent subset. The integers and the rational numbers have rank 1, as each subgroup of. The abelian groups of rank 1 are well understood, however, many questions are still open to higher ranks. Abelian groups of infinite rank can be extremely complex and their open-ended questions are often closely connected to questions of set theory.