Divisible group

In mathematics, a group G is called divisible or divisible, if you can divide each group member by any natural number. What is meant is: for every group element and for each natural number a group element such that

Applies. Here, the group linkage has been written with a star.

If the link in the group written as addition ( as is usual in abelian groups ), it means the defining condition: for each and every natural number a with

Each group element is therefore divisible by.

If we write the link as general groups than usual multiplication, it means the condition: for each and every natural number a with

There is therefore from a - th root.

Background is the obvious question: When is a number by a natural number divisible or divisible? This is generalized to groups. Even Euclid described the problem: for which numbers the equation is solvable. What numbers are multiples of a given natural number.

Another topic discussed at first glance Euclid in the 10th book and proves that there is no fracture, which solves the equation. For what numbers is the equation

Solvable? Pressing both of these questions with the help of pictures, so the common background lights on.

  • Is, as is the Figure
  • The figure is not surjective. But the mapping is surjective.

This observation suggests of the integers and fractions to abstract.

Definition of the divisible group

For a group and a natural number following statements are equivalent:

If one of the statements and thus both to the group, it means the group divisible by. The group is called divisible if it is divisible by any natural number. In English literature it is called divisible such groups. Sometimes it is called such a group also divisible. The group is written additively, the condition is 1.

Examples

  • With the addition of a link is divisible by any natural number.
  • The set of real numbers with decimal fraction abbrechendem is a group together with the addition. This is by and divisible, but by no other prime number.
  • The most important example is the additive group of rational numbers. Here, the element sought is even unique.
  • The additive group of any vector space over the rational numbers is divisible, in particular this applies to
  • A group homomorphism is divisible groups divisible groups, in particular quotient divisible groups are divisible: for example,
  • A finite group G is divisible if | G | = 1, because otherwise the exponentiation is not surjective with n
  • For each prime p, the auditors group is divisible.
  • The unit group of quaternions is a nichtkommutatives example of a divisible group.
  • Another example is the three-dimensional nichtkommutatives special orthogonal group consisting of the rotations.

Divisible Abelian Groups

For an abelian group are the following statement equivalent.

The property 2 or 3 indicate that in the category of abelian groups is an injective object. The equivalence of 2 and 3 is the perches criterion (after Reinhold Baer).

Direct products of divisible - that is injective - Abelian groups are divisible. This is true in every module category. The direct sum of divisible groups is divisible. In general, the direct sum of injective modules is injective. The epimorphic image of a divisible group is divisible. So is divisible with also. This is a particularly important divisible Abelian group.

Injective hull

There are a sufficient number of injective groups

Is a subgroup of the abelian group. Every abelian group can be embedded in a divisible abelian monomorphic group. In the category of abelian groups, there are enough many injective. This results in:

For an abelian group the following are equivalent:

  • G is injective.
  • There is for every monomorphism with a homomorphism. This is the identity on G.

In particular, a divisible group is a direct summand in each upper group.

Injective hull

Is contained in a special way in the injective group. Is a monomorphism in any divisible group, so there is a. It is therefore. Therefore, it is a monomorphism. So up to isomorphism included in each divisible group containing. is the injective hull of. These are available at any abelian group G. To clarify, this is defined, the large subgroup.

Great subgroup

A subgroup is called great in G if the only subgroup of G, which U has made the cut. Thus, the following statements are equivalent:

  • Every homomorphism with is a monomorphism.
  • For all there is.

A monomorphism is called essential if large is in H.

Existence of an injective envelope

It is the following sentence:

There is a divisible group D and an essential monomorphism for every abelian group G. This D is unique up to isomorphism. It is injective hull of G and is sometimes referred to.

This statement applies to all categories of modules. Each module over a unitary ring has an injective hull. is the injective hull of. The auditor group for a prime p is injective hull of each group of species.

Structure theorem of divisible abelian groups

Each divisible abelian group is isomorphic to a ( possibly infinite ) direct sum of vector spaces and auditor groups.

The abelian group

  • Is as epimorphic image of the divisible group is divisible itself and therefore injective.
  • Is isomorphic to the group of roots of unity in. This is the set of complex numbers, for which there is a natural number n is with.
  • Contains a copy of each cyclic torsion group. This means that for every natural number a monomorphism.
  • So also contains the injective hull of each simple cyclic group. In this case, p is a prime number. This is the group of auditors. The Endomorphismering of is isomorphic to the ring of p- adic numbers.
  • There is an index set and a monomorphism for every abelian group. One says is an injective cogenerator in the category of abelian groups.
  • The functor receives not only exact consequences, but they also discovered. This means that if a homomorphism of abelian groups and is an epimorphism, so is a monomorphism. It follows, for example, the following interesting relationship between divisible and torsion-free abelian groups: A group is torsion-free if and only if it is divisible.
  • Every finitely generated torsion group is isomorphic to its dual in this sense group.
Torsion (Algebra)
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