Direct product

In mathematics, a direct product is a mathematical structure that is formed with the aid of the Cartesian product from existing mathematical structures. Important examples are the direct product of groups, rings, and other algebraic structures, as well as direct products of nichtalgebraischen structures such as topological spaces.

All direct products of algebraic structures in common is that they consist of a Cartesian product and the links are defined componentwise. Here we consider examples of such products.

Direct product of groups

Outer and inner product

In the illustrated in the following terms must be made between external and internal structure characterization of the direct product and the direct sum of set-theoretic reasons. The comments in this article focus on the external construction, the inner characterization is explained in Article normal subgroup.

Direct product of two groups

Are and groups, as may be defined on the Cartesian product link:

Here, therefore, each of the first two components and the two second components are combined. The result is again a group as you write.

Since it is often a group equates with their base set, one also easily be used for the direct product of the groups the same sign as for the Cartesian product of the basic quantities and namely.

The group includes a normal divider is isomorphic, it consists of the elements of the mold, and a normal to isomorphic divider which consists of the elements of the mold. This call and the neutral elements of and

Whether the Abelian groups and ( commutative ) are commutate the elements of the form with which the form. It follows that every element of clearly can write as a product

A generalization of the direct product of two groups is the semi- direct product.

Direct product of finitely many groups

For any finite number of groups to define their direct product is analogous: The direct product of the amount with the link

It also arises here again a group.

Again, the direct product of each group includes a normal divider is isomorphic, it consists of the elements of the form

Each element of the direct product can be represented as a product of elements of this form.

Every finite abelian group is either cyclic or isomorphic to the direct product of cyclic groups of prime power order. These are uniquely determined up to the order (main theorem on finitely generated abelian groups).

Direct product and direct sum of infinitely many groups

Similar to the case of finitely many groups you define the direct product of infinitely many groups as their Cartesian product with componentwise link.

The set of elements of the direct product, which can be written as a combination of tuples which differ in only one component of the neutral element in general is a proper subgroup of the total direct product. This subset is called the direct sum of the groups.

Equivalent characterizations of the direct sum as a subgroup of the direct product:

• It consists of those elements for which the index set is finite. ( Is the set of " positions " of where not the neutral element of each factor group " is ". )
• Each element of the direct sum is at the heart of all but finitely many canonical projections.

From these characterizations is clear that the sum and the product group are identical devices with a finite number of non-trivial factors.

Direct product of rings, vector spaces and modules

Similar to the direct product of groups can also define the direct product of rings by addition and multiplication defined componentwise. This gives again a ring but no more integral domain because it contains zero divisors.

As with groups, also the direct product of infinitely many different rings of the direct sum of the rings.

The direct product of vector spaces over the same field K (or of R - modules over the same commutative ring R with unity) is also defined as a Cartesian product with componentwise addition and scalar multiplication (or multiplication by the ring elements ). The resulting vector space is then called the product space.

For a finite number of vector spaces (or R-modules ) is the direct product agrees

With the direct sum

Match. For an infinite number of vector spaces (or R-modules ), they differ in that the direct product of the entire Cartesian product is made, while the direct sum consists of only the tuples that are i the zero vector in different at only finitely many places.

The direct product of

Is the vector space of all rational sequences of numbers, it is uncountable.

The direct sum

Is the vector space of all rational numbers that contain only a finite number of non- zeros, ie the space of all terminating rational number sequences. He is countable.

Direct product of topological spaces

For the direct product of topological spaces, we again form a Cartesian product

But the definition of the new topology is more difficult.

For a finite number of spaces one defines the topology of the product as the smallest topology (ie, with the fewest open sets ), the amount the

All "open square " contains. This quantity thus forms a basis for the topology of the product. The topology thus obtained is called the product topology.

Product topology that is formed on the Cartesian product, if one chooses to the common topology (in which the open amount of the open intervals can be generated ), is just the topology of the ordinary Euclidean space.

For the definition of the product topology for an infinite number of rooms and other features, see the article product topology.