Direct product (groups)

In Mathematical and more precisely within the framework of the Theory of the groups, the produces direct is a method allowing, starting from several groups, to build new.

The produces direct corresponds to an extension in the direction where the structure obtained contains isomorphous substructures with those defining the product. This method, applied to the groups makes it possible to classify the three great cases of abelian groups: those finished, those of finished type and the groups of Dregs of finished size and having a number finished of component related.

In all this article, N is a strictly positive entirety, ( G ₁, *₁), ( G ₂, *₂)… ( G _n, *_n) groups, respective neutral elements indicate E ₁, E ₂,…, E _n.

Motivation

The direct product corresponds to the operation of the Cartesian produced with the transfer of the operation of the groups. A generalization of the Cartesian product to make compatible an algebraic structure is frequent, one can quote for example the vector Space, the Module on a ring, the Algèbre on a body or to a lesser extent (because the integrity, if it existed, disappears) the ring.

This approach offers a first service: the direct product is a tool making it possible to build new groups, it extends the number of easily accessible examples thus to study the theory.

There exists another major interest, that of the classification of the elements of the studied structure. Classification makes it possible to know, for each element the exact properties of which it profits as well as the other elements of the structure profiting from the same properties. An example of classification is that of the vector spaces on a body K of finished Dimension. Dimension classifies all these vector spaces exactly.

A frequent approach is that of the extension, it consists, to leave for example a group, to build a new group containing the precedent. In this direction, the direct product is an extension of a group. This method, of extension by product, is that used for the spaces vector and any vector space on K of finished size is isomorphous with a product of the vector space K (of dimension a ). The vaster space of the objects built thanks to the extension is, the more the field of application of the extension is broad.

In the case of the groups, the direct product is a technique of extension covering very largely the case Commutatif. If the group is abelian and has sufficient good properties, then it is produced direct abelian groups of the same category and its structure is particularly simple. The three most important cases are quoted in this article.

In the nonabelian case, other methods of extension is necessary, it is the raison d'être, for example semi-direct Produit.

Examples

Group of Klein

See also: Group of Klein

That is to say G the single group of a nature two , isomorphous with the cyclic Group of order two Z /2 Z . Its table is the following one:

|}

The group produced G X G is a group of four elements whose table is the following one:

|} The group obtained is isomorphous with the group of Klein, the only abelian group of order four , of which all the elements have them-even for reverse.

Commonplace product

Either {E} the group reduced to an element, G X {E} is a product of isomorphous group to G . One speaks then about produced commonplace .

Cyclic group

See also: cyclic Group

It is possible to raise the opposite question, that is to say G a group, for example a cyclic group of order G , G is it a noncommonplace product of some of its Sous-groupe S. the answer is given by the Théorème of the Chinese remainders.

* G is isomorphous with a noncommonplace product of sub-groups if and only if there exist two whole has and B first between them such as their product is equal to G .

In this case, G is isomorphous with the direct product of the two groups Z / has . Z and Z / B . Z .

Note: The demonstration is given in the associated article.

Algebraic fence of F ₂

F ₂ is single the Corps finished except for two elements (with an isomorphism). The algebraic Clôture K of F ₂ corresponds to smallest the subfield (except for an isomorphism) container F ₂ and such as all Polynôme with coefficients in K of degree equal to or higher than a admits at least a root. ( K , +) by definition of the structure of body is a group. It is shown that it is of an infinite nature and that: $\backslash \; mathbb\; K\; \backslash \; approx\; \backslash \; prod\_\; \{I\; \backslash \; in\; \backslash \; mathbb\; NR\}\; \backslash \; mathbb\; Z/2.\; \backslash \; mathbb\; Z\; \backslash ;$ The sign $\backslash \; approx$ indicates an isomorphism of group here. K is an example of infinite direct product of groups, it corresponds to the only countable group such as any element is its clean opposite.

Properties

Elementary properties

If G ₁ and G ₂ is finished cardinals, the properties of the Cartesian product show that:

* the order of G is equal to the product of the orders of G ₁ and G ₂.

* If G ₁ and G ₂ is two abelian groups then their direct product is too.

This property is a direct consequence of the definition. Indeed, if G ₁ and G ₂ is two abelian groups: $\backslash \; forall\; (x\_1,\; x\_2),\; (y\_1,\; y\_2)\; \backslash \; in\; G\; \backslash \; quad\; (x\_1,\; x\_2)\; *\; (y\_1,\; y\_2)\; =\; (x\_1*\_1y\_1,\; x\_2*\_2y\_2)\; =\; (y\_1*\_1x\_1,\; y\_2*\_2x\_2)\; =\; (y\_1,\; y\_2)\; *\; (x\_1,\; x\_2)$ In the general case, let us note I ₁ (resp. I ₂) the application of G ₁ (resp. G ₂) in G which with X associates ( X , E ₂) (resp. ( E ₁, X ). The sub-group image of I ₁ (resp. I ₂) is noted H ₁ (resp. H ₂). Finally the application S ₁ (resp. S ₂) of G in G ₁ (resp. G ₂) is defined by S ₁ ( X ₁, X ₂) = X ₁ (resp. S ₂ ( X ₁, X ₂) = X ₂).

* the applications I ₁ and I ₂ is injective morphisms .

* the applications S ₁ and S ₂ is surjective morphisms .

* the two following continuations is exact continuations.

$e\_1\; \backslash \; xrightarrow\; \{\}\; G\_1\; \backslash \; xrightarrow\; \{i\_1\}\; G=G\_1\; \backslash \; times\; G\_2\; \backslash \; xrightarrow\; \{s\_2\}\; G\_2\; \backslash \; xrightarrow\; \{\}\; e\_2\; \backslash \; quad\; and\; \backslash \; quad\; e\_2\backslash xrightarrow\{\}G\_2\backslash xrightarrow\{i\_2\}G=G\_1\backslash times\; G\_2\backslash xrightarrow\{s\_1\}G\_1\backslash xrightarrow\{\}e\_1$

Sub-groups H ₁ and H ₂

Here, H ₁ and H ₂ indicate the sub-groups G ₁ X { E } and { E } X G ₂. The sub-groups H ₁ and H ₂ have index properties of a product of groups:

* H ₁ (resp. H ₂) is a isomorphous Sous-groupe distinguished with G ₁ and (resp. G ₂).

Indeed, H ₁ is the image of G ₁ by I ₁ an injective morphism, the two structures are thus isomorphous. Moreover H ₁ is the core of S ₂, it is thus a sub-group distinguished. A similar reasoning shows the equivalent result for H ₂.

* Any element of G is written in a single way like product of an element of H ₁ and of an element of H ₂.

This proposal is an immediate consequence of the definition of direct product: $\backslash \; forall\; X\; \backslash \; in\; G\; \backslash \; quad\; \backslash \; exists!\; (x\_1,\; x\_2)\; \backslash \; in\; G=G\_1\; \backslash \; times\; G\_2\; \backslash \; quad\; thus\; \backslash \; quad\; x=\; (x\_1,\; e\_2)\; *\; (e\_1,\; x\_2)$

* Any element of H ₁ commutates with any element of H ₂.

By definition an element of H ₁ (resp. H ₂) is form ( X ₁, E ₂) (resp. ( E ₁, X ₂)), consequently: $\backslash \; forall\; (x\_1,\; e\_2)\; \backslash \; in\; H\_1\; \backslash ;\; \backslash \; forall\; (e\_1,\; x\_2)\; \backslash \; in\; H\_2\; \backslash \; quad\; (x\_1,\; e\_2)\; *\; (e\_1,\; x\_2)\; =\; (e\_1,\; x\_2)\; *\; (x\_1,\; e\_2)\; =\; (x\_1,\; x\_2)$

Reciprocal

Do the reciprocal problems are the following one, either G a group, exist two sub-groups H ₁ and H ₂ of G noncommonplace (i.e. different from { E } and G ) such as G or isomorph with the direct product of H ₁ and H ₂?

The answer is sometimes positive, as the example of the cyclic groups shows it. It is not always the case, the symmetrical Groupe of order three, containing the Permutation S of a whole of three elements, contains six elements. Only the sub-groups noncommonplace have as a cardinal two or three . However the only groups of cardinal two or three are cyclic groups thus abelian. As the direct product of two abelian groups is abelian and that the symmetrical group of order three is noncommutative, it is not produced direct noncommonplace of two sub-groups.

Requirement and sufficient

That is to say H ₁ and H ₂ two sub-groups of a group G , the elementary properties offer a requirement so that they correspond to a direct product. Any element of G is written like product of an element of H ₁ and of an element of H ₂ and any element of H ₁ commutates with any element of H ₂. This condition is not only necessary, but also sufficient:

* the φ application of H ₁x H ₂ in G which, with ( H ₁, H ₂) associates H ₁* H ₂ is an isomorphism of group, if and only if:

$(I)\; \backslash \; quad\; \backslash \; forall\; (h\_1,\; h\_2)\; \backslash \; in\; H\_1\; \backslash \; times\; H\_2\; \backslash \; quad\; h\_1*h\_2=h\_2*h\_1$

$(II)\; \backslash \; quad\; \backslash \; forall\; G\; \backslash \; in\; G\; \backslash \; quad\; \backslash \; exists!\; (h\_1,\; h\_2)\; \backslash \; in\; H\_1\; \backslash \; times\; H\_2\; \backslash \; quad\; such\; \backslash ;\; that\; \backslash \; quad\; g=h\_1*h\_2$

The fact that φ is a morphism comes directly from the condition (I) , indeed: $\backslash \; forall\; (h\_1,\; h\_2)\; (k\_1,\; k\_2)\; \backslash \; in\; H\_1\; \backslash \; times\; H\_2\; \backslash \; quad\; \backslash \; varphi\; \backslash \; Big\; ((h\_1,\; h\_2)\; (k\_1,\; k\_2)\; \backslash \; Big)\; =\; \backslash \; varphi\; \backslash \; Big\; (h\_1*k\_1,\; h\_2*k\_2\; \backslash \; Big)\; =h\_1*k\_1*h\_2*k\_2\; =h\_1*h\_2*k\_1*k\_2=\; \backslash \; varphi\; (h\_1,\; h\_2)\; *\; \backslash \; varphi\; (k\_1,\; k\_2)$ The condition (II) expresses the fact exactly that φ is bijective. The demonstration is thus completed.

Direct sum

Are H ₁ and H ₂ two sub-groups of G . It is said that the nap of H ₁ and H ₂ is equal to G if and only if the group generated by the elements of H ₁ and H ₂ is equal to the whole group G and if any element of H ₁ commutates with any element of H ₂.

If, moreover the intersection of H ₁ and H ₂ is tiny room to the neutral element, then the nap is known as direct and the following notation is used:

$G=H\_1\backslash oplus\; H\_2$

* the direct sum of H ₁ and H ₂ is equal to G if and only if any element of G is written in a single way like summons of an element of H ₁ and of an element of H ₂.

This definition spreads with N sub-groups.

The direct sum has many analogies with its counterpart in Linear algebra (cf direct Somme). That is to say H ₁ and H ₂ two sub-groups of G , then:

* the direct sum of H ₁ and H ₂ is equal to G , if G is isomorphous with the direct product of H ₁ and H ₂.

Projector

An approach, a little similar to that of the spaces vector, gives an equivalence between a direct product and a particular morphism called projector . Either G a group, H ₁ and H ₂ two sub-groups of G such as the φ application of the paragraph a preceding or isomorphism. Then any element G of G is written in a single way H ₁* H ₂ where H _i is element of H _i. That is to say p the application of G in G which with G associates H ₁. It profits from the following properties:

* the function p is a morphism of group, any element of its image commutates with any element of its core and p O p is equal to p .

Here O indicates the composition of the functions.

The analogy with the vector spaces gives place to the following definition:

* a projector p of G is a morphism of G in G such as any element of its image commutates with any element of its core and p O p is equal to p .

The data of a projector allows a decomposition of G in direct product:

* Is p a projector of G , then G is isomorphous with the direct product of the image of p and the core of p .

If G is abelian, any morphism whose square is equal to itself is a projector, indeed any element of the group commutates with any element of the group.

This property can be reformulated in the following way. Any exact continuation:

$1\; \backslash \; to\; H\; \backslash \; to\; G\; \backslash \; to\; G/H\; \backslash \; to1$

such as G is abelian, and that there exists a section G / H in G which factorizes in a direct product $\backslash \; scriptstyle\; G\; \backslash \; simeq\; H\; \backslash \; times\; G/H$.

Abelian group

See also: abelian Group

The general case cannot be treated, it is too vast, it is thus necessary to bring additional assumptions. These assumptions correspond primarily to three cases, treaties here.

abelian Group finished

The simplest case is that or G groups it is finished. A first example is given by the cyclic groups, they are enough to generate, using the direct product all the finished abelian groups.

* There exists a succession of strictly positive entireties ( has ₁, has ₂,…, has _k) such as G is isomorphous with the direct product of the cyclic groups of cardinal the various elements of the continuation.

What is written in the following way: $G\; \backslash \; simeq\; \backslash \; mathbb\; \{Z\}\; /a\_1\; \backslash \; mathbb\; \{Z\}\; \backslash \; times\; \backslash \; mathbb\; \{Z\}\; /a\_2\; \backslash \; mathbb\; \{Z\}\; \backslash \; times\; \backslash \; cdots\; \backslash \; times\; \backslash \; mathbb\; \{Z\}\; /a\_k\; \backslash \; mathbb\; \{Z\}$

Abelian group of finished type

See also: abelian Group of type finished

The second case is of a nature close to the preceding case. It corresponds to the groups containing a generating part finished. There exists thus at least a group which is not element of the preceding unit, that of the whole Z . One shows (in the associated article) which it is the single generator to add to obtain all the abelian groups of finished type.

* For any abelian group G of the finished type, there exists an entirety N and a group finished F such as G is isomorphous with the direct product of F and Z ⁿ.

Group of Dregs commutative

See also: commutative Group of Dregs

The two preceding categories are countable. There however exist important groups which are not it, one can quote for example the case of the linear isométries of the plan used previously. It is then necessary to associate three assumptions: the group lays out of a structure of differential Variété compatible with the group (one speaks about group of Dregs) , the tangent Espace is of finished size and the related component number of of the group is finished. The following property is then checked:

* Any group of Dregs of finished size and having a finished number of related components is isomorphous with a direct product of a finished group, a vector space of finished size and of a maximum Tore.

See too

External bonds

• Group produced by the mathématiques.net

References

• S. Lang, Algebra , Dunod, 2004.
• J.F. Labarre, the Theory of the groups, University Presses of France, 1978.
• G. Pichon linear Groups of Dregs, representations and applications, Hermann, 1973.

Random links:

Programming by contract | Vampiro | Germany-France in football | Assemble Romano | University of Augsburg | Basin of Bedford | Karl_Friedrich_Bahrdt

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org