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Encyclopedia > Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one. Examples are the product of sets (see Cartesian product), groups (described below), the product of rings and of other algebraic structures. The product of topological spaces is another instance. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ... In abstract algebra, it is possible to combine several rings into one large product ring. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...

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Group direct product

In group theory one defines the direct product of two groups (G, *) and (H, o), denoted by G×H, as follows: In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

  • as set of the elements of the new group, take the cartesian product of the sets of elements of G and H, that is {(g, h): g in G, h in H};
  • on these elements put an operation, defined elementwise:
    (g, h) × (g' , h' ) = (g * g' , h o h' )

(Note the operation * may be the same as o.) In mathematics, a set can be thought of as any well-defined collection of distinct things considered as a whole. ... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ...


This construction gives a new group. It has a normal subgroup isomorphic to G (given by the elements of the form (g, 1)), and one isomorphic to H (comprising the elements (1, h)). In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...


The reverse also holds, there is the following recognition theorem: If a group K contains two normal subgroups G and H, such that K= GH and the intersection of G and H contains only the identity, then K = G x H. A relaxation of these conditions gives the semidirect product. In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ...


As an example, take as G and H two copies of the unique (up to isomorphisms) group of order 2, C2: say {1, a} and {1, b}. Then C2×C2 = {(1,1), (1,b), (a,1), (a,b)}, with the operation element by element. For instance, (1,b)*(a,1) = (1*a, b*1) = (a,b), and (1,b)*(1,b) = (1,b2) = (1,1).


With a direct product, we get some natural group homomorphisms for free: the projection maps Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...

,

called the coordinate functions.


Also, every homomorphism f on the direct product is totally determined by its component functions .


For any group (G, *), and any integer n ≥ 0, multiple application of the direct product gives the group of all n-tuples Gn (for n=0 the trivial group). Examples: In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects (an infinite sequence is a family). ...

A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

Vector space direct product

The direct product for vector spaces (not to be confused with the tensor product) is very similar to the one defined for groups above, using the cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components. Starting from R we get Euclidean space Rn, the prototypical example of a real n-dimensional vector space. The vector space direct product of Rm and Rn is Rm + n. A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...


Note that a direct product for a finite index is identical to the direct sum . The direct sum and direct product differ only for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...


Topological space direct product

The direct product for a collection of topological spaces Xi for i in I, some index set, once again makes use of the cartesian product Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

Defining the topology is a little tricky. For finitely many factors, this is the obvious and natural thing to do: simply take as a basis of open sets to be the collection of all cartesian products of open subsets from each factor: Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases...

This topology is called the product topology. For example, directly defining the product topology on R2 by the open sets of R (disjoint unions of open intervals), the basis for this topology would consist of all disjoint unions of open rectangles in the plane (as it turns out, it coincides with the usual metric topology). In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...


The product topology for infinite products has a twist, and this has to do with being able to make all the projection maps continuous and to make all functions into the product continuous if and only if all its component functions are continuous (i.e. to satisfy the categorical definition of product: the morphisms here are continuous functions): we take as a basis of open sets to be the collection of all cartesian products of open subsets from each factor, as before, with the proviso that all but finitely many factors are the entire space:

(Not a very pretty sight!). The more natural-sounding topology would be, in this case, to take products of infinitely many open subsets as before, and this does yield a somewhat interesting topology, the box topology. However it is not too difficult to find an example of bunch of continuous component functions whose product function is not continuous (see the separate entry box topology for an example and more). The problem which makes the twist necessary is ultimately rooted in the fact that the intersection of open sets is only guaranteed to be open for finitely many sets in the definition of topology. In topology, the cartesian product of topological spaces is can be topologized in several ways. ...


Products (with the product topology) are nice with respect to preserving properties of their factors; for example, the product of Hausdorff spaces is Hausdorff; the product of connected spaces is connected, and the product of compact spaces is compact. That last one, called Tychonoff's theorem, is yet another equivalence to the axiom of choice. In mathematics, Tychonoffs theorem states that the product of any collection of compact topological spaces is compact. ... In mathematics, the axiom of choice is an axiom of set theory. ...


For more properties and equivalent formulations, see the separate entry product topology. In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...


Categorical product

Main article: Product (category theory) In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. ...


The direct product can be abstracted to an arbitrary category. In a general category, given a collection of objects Ai and a collection of morphisms pi from A to Ai with i ranging in some index set I, an object A is said to be a categorical product in the category if, for any object B and any collection of morphisms fi from B to Ai, there exists a unique morphism f from B to A such that fi = pi f and this object A is unique. This not only works for two factors, but arbitrarily (possibly infinitely) many. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...


For groups we similarly define the direct product of a more general, arbitrary collection of groups Gi for i in I, I an index set. Denoting the cartesian product of the groups by G we define multiplication on G with the operation of componentwise multiplication; and corresponding to the pi in the definition above are the projection maps

,

the functions that take g to its ith component (gi).


Related topics

In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ... In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... In abstract algebra, the free product of groups constructs a group from two or more given ones. ...

References

  • Lang, S. Algebra. New York: Springer-Verlag, 2002.

  Results from FactBites:
 
Direct product - Wikipedia, the free encyclopedia (955 words)
The direct product for vector spaces (not to be confused with the tensor product) is very similar to the one defined for groups above, using the cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components.
The direct sum and direct product differ only for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries.
Products (with the product topology) are nice with respect to preserving properties of their factors; for example, the product of Hausdorff spaces is Hausdorff; the product of connected spaces is connected, and the product of compact spaces is compact.
  More results at FactBites »

 
 

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