Direct product

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

All direct products of algebraic structures have in common that they consist of a Cartesian product of the and the links are defined component by component. ${\displaystyle X_{i}}$ ${\displaystyle X_{i}}$

Direct product of groups

In principle, the following applies to any group. If the link is called addition, which is common with many commutative groups, the construct discussed here is usually called a direct sum.

Outside and inside direct product

A distinction is made between the so-called outer direct product of groups on the one hand and the inner direct product of subgroups of a given group on the other. The following statements describe the external direct product. A new group is constructed from two or more groups, which is called the direct product of the given groups. The inner direct product of subgroups is dealt with in the article normal subgroups .

Direct product of two groups

If and groups , a link can be defined on the Cartesian product : ${\displaystyle (G_{1},*)}$ ${\displaystyle (G_{2},\star )}$ ${\displaystyle G_{1}\times G_{2}}$

${\displaystyle (x_{1},x_{2})\odot (y_{1},y_{2}):=(x_{1}*y_{1},x_{2}\star y_{2})}$

The two first components and the two second components are linked to one another here. There is again a group that one writes as. ${\displaystyle (G_{1},*)\odot (G_{2},\star )}$

example

If and are groups with addition as an operation, then the Cartesian product consists of the elements . This leads to the link table ${\displaystyle G=Z_{2}=\{0,1\}}$ ${\displaystyle H=Z_{3}=\{0,1,2\}}$ ${\displaystyle Z_{2}\times Z_{3}=\{(x,y)\mid x\in Z_{2},y\in Z_{3}\}}$ ${\displaystyle (0,0),(0,1),(0,2),(1,0),(1,1),(1,2)}$

${\displaystyle +}$ (0.0) (0.1) (0.2) (1.0) (1.1) (1.2) (0.0) (0.0) (0.1) (0.2) (1.0) (1.1) (1.2) (0.1) (0.1) (0.2) (0.0) (1.1) (1.2) (1.0) (0.2) (0.2) (0.0) (0.1) (1.2) (1.0) (1.1) (1.0) (1.0) (1.1) (1.2) (0.0) (0.1) (0.2) (1.1) (1.1) (1.2) (1.0) (0.1) (0.2) (0.0) (1.2) (1.2) (1.0) (1.1) (0.2) (0.0) (0.1)

If, as is often the case, the description does not distinguish a group from its basic set , the simplified description is usually used instead of . ${\displaystyle (G,*)}$ ${\displaystyle G}$ ${\displaystyle (G_{1},*)\odot (G_{2},\star )}$ ${\displaystyle G_{1}\times G_{2}}$

If and denote the neutral elements of and , then the subsets and are two to or isomorphic subsets of . Regardless of whether the groups and are Abelian (commutative), the elements of and , i.e. pairs of the form or with each other , commute . It follows that each element can be clearly written as a product with and . In particular, and are normal divisors of . ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle G_{1}}$ ${\displaystyle G_{2}}$ ${\displaystyle G_{1}'=G_{1}\times \{e_{2}\}}$ ${\displaystyle G_{2}'=\{e_{1}\}\times G_{2}}$ ${\displaystyle G_{1}}$ ${\displaystyle G_{2}}$ ${\displaystyle G_{1}\times G_{2}}$ ${\displaystyle G_{1}}$ ${\displaystyle G_{2}}$ ${\displaystyle G_{1}'}$ ${\displaystyle G_{2}'}$ ${\displaystyle (x_{1},e_{2})}$ ${\displaystyle (e_{1},x_{2})}$ ${\displaystyle x=(x_{1},x_{2})\in G_{1}\times G_{2}}$ ${\displaystyle x=g_{1}'\odot g'_{2}}$ ${\displaystyle g_{1}'=(x_{1},e_{2})\in G_{1}'}$ ${\displaystyle g_{2}'=(e_{1},x_{2})\in G_{2}'}$ ${\displaystyle G_{1}'}$ ${\displaystyle G_{2}'}$ ${\displaystyle G_{1}\times G_{2}}$

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

Direct product of a finite number of groups

For any finite number of groups , the definition of their direct product is analogous: The direct product is the set with the link ${\displaystyle G_{1},\ldots ,G_{n}}$ ${\displaystyle G_{1}\times \ldots \times G_{n}}$

${\displaystyle (x_{1},\ldots ,x_{n})\odot (y_{1},\ldots ,y_{n}):=(x_{1}*_{1}y_{1},\ldots ,x_{n}*_{n}y_{n})}$, where the link denotes in each case . ${\displaystyle *_{i}}$ ${\displaystyle G_{i}}$

Here, too, there is a group.

Here, too, the direct product for each group contains a normal sub-division that is too isomorphic. It consists of the elements of form ${\displaystyle G_{i}}$ ${\displaystyle G_{i}'}$ ${\displaystyle G_{i}}$

${\displaystyle (e_{1},\ldots ,e_{i-1},x_{i},e_{i+1},\ldots ,e_{n})}$, . ${\displaystyle x_{i}\in G_{i}}$

The elements of different commutate and each element of the direct product has a clearly defined representation as a product of such elements:   with  . ${\displaystyle G_{i}'}$ ${\displaystyle x}$ ${\displaystyle x=\prod g_{i}'}$ ${\displaystyle g_{i}'\in G_{i}'}$

example

Every finite Abelian group is either cyclic or isomorphic to the direct product of cyclic groups of prime power order. Except for the order, these are clearly defined (main theorem about finitely generated Abelian groups ).

Direct product and direct sum of an infinite number of groups

Analogous to the case of finitely many groups, the direct product of an infinite number of groups is defined as their Cartesian product with a component-wise connection . ${\displaystyle \{G_{i}\mid i\in I\}}$ ${\displaystyle \prod _{i\in I}{G_{i}}}$ ${\displaystyle (x_{i})_{i\in I}\odot (y_{i})_{i\in I}:=(x_{i}*_{i}y_{i})_{i\in I}}$

The set of elements of the direct product, which can be written as a combination of tuples, which differ from the neutral element in only finitely many components, is generally a real subgroup of the entire 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 the neutral element of the respective factor group "stands".) ${\displaystyle (x_{i})_{i\in I}}$ ${\displaystyle J=\{i\in I\mid x_{i}\neq e_{i}\}}$ ${\displaystyle J}$ ${\displaystyle (x_{i})}$
• Each element of the direct sum lies at the core of all but a finite number of canonical projections . ${\displaystyle (\pi _{i})_{i\in I}}$

From these characterizations it becomes clear that for products with a finite number of nontrivial factors, the sum group and the product group are identical.

Direct product of rings, vector spaces and modules

Analogous to the direct product of groups, the direct product of rings can also be defined by defining addition and multiplication component by component. You get a ring again, but it is no longer an integrity ring because it contains zero divisors .

As with groups, the direct product of an infinite number of rings differs from 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 one) is also defined as a Cartesian product with component-wise 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) the direct product is correct ${\displaystyle V_{1},\ldots ,V_{n}}$

${\displaystyle \prod _{i=1}^{n}V_{i}}$

with the direct sum

${\displaystyle \bigoplus _{i=1}^{n}V_{i}}$

match. For infinitely many vector spaces (or R-modules) they differ in that the direct product consists of the entire Cartesian product, while the direct sum consists only of the tuples that differ from the zero vector in only finitely many places i . ${\displaystyle V_{i}}$

The direct product

${\displaystyle \prod _{i=1}^{\infty }\mathbb {Q} }$

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

The direct sum

${\displaystyle \bigoplus _{i=1}^{\infty }\mathbb {Q} }$

is the vector space of all rational number sequences that contain only finitely many non-zeros, i.e. H. 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 ${\displaystyle \{X_{i}|i\in I\}}$

${\displaystyle \prod _{i\in I}X_{i}}$,

but defining the new topology is more difficult.

For finitely many spaces the topology of the product is defined as the smallest topology (i.e. the one with the fewest open sets) that the set ${\displaystyle X_{1},\ldots ,X_{n}}$

${\displaystyle {\mathcal {B}}=\{U_{1}\times \ldots \times U_{n}|U_{i}\ {\textrm {offen\ in}}\ X_{i}\}}$

contains all "open cuboids". This amount thus forms a basis for the topology of the product. The topology obtained in this way is called the product topology . ${\displaystyle {\mathcal {B}}}$

The product topology that is generated on the Cartesian product when one chooses the ordinary topology (in which the open sets are generated by the open intervals) is precisely the ordinary topology of Euclidean space . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{n}}$

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

Web links

• Eric W. Weisstein et al .: Direct Product (from MathWorld - A Wolfram Web Resource)

literature

• K. Meyberg: Algebra, part 1. 2nd edition, Hanser Verlag, Munich 1980, ISBN 3-446-13079-9