Group finished

Introduction

That is to say $G$ a group. One notes his law and 1 multiplicativement his neutral element, except if $G$ is abelian, where the law (resp. the neutral element) is noted additivement (respectively 0).

It is said that $G$ is a group finished if its cardinal is finished. The cardinal is then noted $|G|$ and is called order group.

One will suppose in the continuation that $G$ is a finished group.

That is to say $g\; \backslash \; in\; G$. As $G$ is finished, the principle of Dirichlet (or of the pigeon cages) makes it possible to show that the unit $E\_g=\; \backslash \; \{K\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; |\backslash \; g^k=1\; \backslash \}$ is nonempty. He thus admits a smaller element, than one calls the order $g$.

Attention with the risk of confusion, here the term order indicates successively two completely different concepts .

The order $d$ of an element $g$ has a very useful arithmetic property (which comes directly from Euclidean division):

Is $n$ an entirety not no one such as $g^n=1$ then $d$ divides $n$.

A subset $S$ of $G$ generates this group if all the elements of $G$ are written like a product of elements or reverse of elements of $S$. The $S$ unit is called a generating left $G$.

As $G$ is finished, the reverse of an element $g$ is a power of $g$ (more precisely, there is $g^\; \{-\; 1\}\; =g^\; \{d-1\}$, where $d$ indicates the order of $g$). It thus follows that a subset $S$ of $G$ is a generating part if and only if any element of $G$ is a product of elements of $S$.

A finished group generated by a singleton {$g$} is known as cyclic . By abuse language, one says that the element $g$ generates $G$, and one notes then $G=\; \backslash \; langle\; G\; \backslash \; rangle$. It is easy to check that such a group is necessarily abelian.

Let us notice that in this case, the order of $G$ is equal to the order of one of its generators.

All the elements of a group finished $G$ have an order lower or equal to $|G|$.

A fundamental result in the study of the finished groups is the theorem of Lagrange:

Is $G$ a finished group and $H$ a sub-group of $G$, then the order of $H$ divides the order of $G$.

An immediate consequence is that if G is a finished group, if m=|G|, then if $g\; \backslash \; in\; G,\; g^m=1$. (to consider the sub-group generated per G)

Examples

Here some traditional examples of finished groups:

• Group of the nth roots of the unit: $(\backslash \; mathbb\; \{U\}\; \_n,\; \backslash \; cdot)$ where $\backslash \; mathbb\; \{U\}\; \_n=\; \backslash \; \{e^\; \{\backslash \; frac\; \{2ki\; \backslash \; pi\}\; \{N\}\},\; 0\; \backslash \; Leq\; K\; \backslash \; Leq\; n-1\; \backslash \}$;

• residual classes modulo N: $(\backslash \; mathbb\; Z/n\; \backslash \; mathbb\; Z,\; +)$;

• the whole of the isométries which preserve the regular polygon at N sides (also called diédral group, noted $D\_n$);

• the symmetrical groups: $(\backslash \; mathfrak\; \{S\}\; \_n,\; \backslash \; circ)$groupes of the bijections of a unit finished in itself;

• the group of the quaternions;

• the Group of Klein.

The first two examples indicate in fact the even group. To include/understand what one understands by even , one will introduce the concept of morphism of group.

Morphisms of groups

To also consult the detailed article Homomorphism of group

When two groups are isomorphous, they are identical from the point of view of the theory of the groups.

The study of the morphisms of groups is thus important for the comprehension of the groups. Branches of the theory of the finished groups, like the theory of the Representations of a group finished, are devoted entirely to this activity.

For example, the groups $(\backslash \; mathbb\; \{Z\}\; /n\; \backslash \; mathbb\; \{Z\},\; +)$ and $(\backslash \; mathbb\; \{U\}\; \_n,\; \backslash \; cdot)$ are isomorphous.

To show it, it is enough to check that the $f\; application:\; (\backslash \; mathbb\; \{Z\}\; /n\; \backslash \; mathbb\; \{Z\},\; +)\; \backslash \; longrightarrow\; (\backslash \; mathbb\; \{U\}\; \_n,\; \backslash \; cdot)$ defined by

$f\; (\backslash \; overline\; \{K\})\; =e^\; \{\backslash \; frac\; \{2ki\; \backslash \; pi\}\; \{N\}\}$

is an isomorphism of group.

To say that these groups are isomorphous means that they are identical : all the properties of the one are found in the other. In the theory of the groups, one will thus study the properties only one of these two groups (that which one wants).

Direct product - semi-direct Product

From two finished groups $H$ and $K$, one can build a new group: the produces direct external $H$ by $K$; more generally, if one is given in more one morphism $f:\; K\; \backslash \; to\; \backslash \; hbox\; \{Aut\}\; H$, then one can build the semi-direct Produit external $H$ by $K$ according to $f$. They are finished groups, of cardinal $|H||K|$.

A natural question arises then: if $G$ is a group, in which condition $G$ is a direct product interns (or a semi-direct product interns) of two sub-groups $H$ and $K$?

Let us give a criterion answering this question. One poses $HK=\; \backslash \; \{HK\; \backslash \; |\backslash \; H\; \backslash \; in\; H,\; K\; \backslash \; in\; K\; \backslash \}$ (attention, this subset of $G$ is in general not a group).

Produces direct intern

A $G$ group is produced direct internal of two sub-groups $H$ and $K$ if and only if:

* the $H$ groups and $K$ are distinguished in $G$,

* $H\; \backslash \; course\; K=\; \backslash \; \{1\; \backslash \}$,

* $\backslash \; displaystyle\; G=HK$.

Produces semi-direct intern

A $G$ group is produced semi-direct internal of two sub-groups $H$ and $K$ if and only if:

* the $H$ group is distinguished in $G$,

* $H\; \backslash \; course\; K=\; \backslash \; \{1\; \backslash \}$,

* $\backslash \; displaystyle\; G=HK$.

When $G$ is finished, one can help oneself of the following combinative equality:

$|HK|=\; \backslash \; frac.$

Thus, if $G$ is finished, the two criteria are seen considerably simplified. Indeed if the first two points of the criterion are checked, then the third point can be replaced by $|G|=|H||K|$.

Among the examples given above, one can show that the group of Klein is isomorphous with the direct product $\backslash \; mathbb\; \{Z\}\; /2\; \backslash \; mathbb\; \{Z\}\; \backslash \; times\; \backslash \; mathbb\; \{Z\}\; /2\; \backslash \; mathbb\; \{Z\}$. The diédral group is as for him isomorphous with the semi-direct product of $\backslash \; mathbb\; \{Z\}\; /n\; \backslash \; mathbb\; \{Z\}$ by $\backslash \; mathbb\; \{Z\}\; /2\; \backslash \; mathbb\; \{Z\}$.

It was seen that any cyclic group is abelian. The group of Klein shows that the reciprocal one is false. There is however the following remarkable result:

Any abelian group finished is a direct product of cyclic groups.

Sub-groups of Sylow

Either $G$ a finished group of order $n$, or $p$ a prime factor of $n$, or $p^\; \{R\}$ more great power of $p$ which divides $n$, so that $n=p^\; \{R\}\; m$, $m$ being a nondivisible entirety by $p$. One calls $p$-sous-groupe of Sylow of $G$ any sub-group of order $p^\; \{R\}$ of $G$.
The following statements are shown:
1° all $p$-sous-groupe of $G$, i.e. any sub-group of $G$ whose order is a power of $p$, is contained in at least a $p$-sous-groupe of Sylow of $G$; it results from them that the $p$-sous-groupes of Sylow of $G$ are the maximum elements of the whole of the $p$-sous-groupe of $G$, it by what certain authors define them;
2° the $p$-sous-groupe of Sylow of $G$ are combined between them;
3° their number is adequate to 1 modulo $p$;
4° this number divides the definite factor $m$ higher.

The sub-groups of Sylow are an essential instrument of the study of the finished groups.

Classification of the finished groups

One meets many structures of finished groups of nature very different. For this reason, the study of the finished groups is rich and complicated. A natural approach to approach this theory would be to give a classification of the finished groups, i.e., a list of families of groups describing, except for isomorphism, all the finished groups. This problem is very difficult. One is currently unable besides to produce such a list.

Let us bring some brief replies all the same. If the group is abelian, the theory is perfectly known. It spreads even with the abelian groups of type finished. If not, one introduces groups of a particular type: the simple groups. One will try to seize the central role which they have and to include/understand how, to a certain extent, they make it possible to apprehend the classification of the finished groups. Previously, one needs to introduce some concepts.

Either $G$ a finished group, one calls normal continuation $G$ any finished continuation strictly decreasing (within the meaning of inclusion) of sub-groups:

$\backslash \; \{1\; \backslash \}\; =G\_n\; \backslash \; subset\; G\_\; \{n-1\}\; \backslash \; subset\; \backslash \; ldots\; \backslash \; subset\; G\_\; \{1\}\; \backslash \; subset\; G\_0=G$

such as $G\_i$ is a Sous-groupe distinguished from $G\_\; \{i-1\}$.

A normal continuation is known as of decomposition if it is maximum. The $G\_i$ group being distinguished in $G\_\; \{i-1\}$, there is a direction to consider the Groupe quotient $G\_\; \{i-1\}\; /G\_i$ (noted $F\_i$ in the continuation). The groups $F\_i$ which appear in this construction are called the factors of decomposition of the continuation. The maximality of the continuation of decomposition involves immediately that they are simple.

In addition, a theorem of Jordan-Hölder affirms that two continuations of decomposition of $G$ have (except for isomorphism) same the factors of decomposition! (Attention, they can not appear in the same order).

Thus, with any group finished $G$, one is able to associate a succession of simple groups $(F\_1,\; \backslash \; ldots,\; F\_n)$.

This continuation does not characterize the group $G$ (what is damage, if not one would have completely brought back the study of the finished groups and their classification to that of the simple groups!). Let us take to be convinced the case of them of the cyclic group with 4 elements $\backslash \; mathbb\; \{Z\}\; /4\; \backslash \; mathbb\; \{Z\}$ and group of Klein (isomorphous with $\backslash \; mathbb\; \{Z\}\; /2\; \backslash \; mathbb\; \{Z\}\; \backslash \; times\; \backslash \; mathbb\; \{Z\}\; /2\; \backslash \; mathbb\; \{Z\}$). These two groups have the same continuation of factors of decomposition $(\backslash \; mathbb\; \{Z\}\; /2\; \backslash \; mathbb\; \{Z\},\; \backslash \; mathbb\; \{Z\}\; /2\; \backslash \; mathbb\; \{Z\})$ without to be isomorphous.

It however has a very strong influence on its structure. Let us quote for example the study of the resolvable groups (i.e., in the finished case, of the groups whose factors of decomposition are of the cyclic groups of order first).

One arrives quite naturally at a capital question in theory of the finished groups, known under the name of the Problème of the extension which is stated by:

Being given two finished groups $H$ and $K$, which are the finished groups $G$ such as

* $H$ isomorphous with a sub-group is distinguished from $G$ (which one notes wrongly always $H$).

* $G/H\; \backslash \; simeq\; K$.

The groups $G$ solutions of this problem are called extensions of $H$ by $K$ .

The direct and semi-direct products are examples of solutions to the problem of the extension. However, any solution of the problem of the extension does not arise unfortunately in the shape of a direct product or a semi-direct product. One can see it for example with the group of the quaternions, which is an extension of $\backslash \; mathbb\; \{Z\}\; /4\; \backslash \; mathbb\; \{Z\}$ by $\backslash \; mathbb\; \{Z\}\; /2\; \backslash \; mathbb\; \{Z\}$ without to be a direct product or a semi-direct product.

Let us suppose one moment that one can solve the problem of the extension in general. One would be then able to rebuild all the groups finished starting from the simple groups (by solving the problem of the extension gradually starting from a simple continuation of group - which would become then the continuation of factors of decomposition of the built group).

The problem of the extension thus seems a kind of reciprocal with that to associate with a group finished a succession of factors of decomposition.

This approach shows that the study of the finished groups returns to

* the study of the simple groups.

* the problem of the extension.

Thus the simple finished groups appear as the building blocs of the theory of the finished groups (one can make the analogy with the prime numbers in theory of the entireties! Attention all the same, the continuation of the factors first of an integer completely characterizes this number, what is not the case simple groups for the finished groups as one has just seen it!)

In 1981, after more than one half century of keen work and a few thousands of pages of demonstration, the mathematical community gives a classification of the simple finished groups. More precisely, any simple finished group belongs to the one of the following families:

* the cyclic groups of which the order is a prime number.

* the groups of Chevalley.

* the alternate groups $\backslash \; mathfrak\; \{has\}\; \_n$, with $n\; \backslash \; geq\; 5$.

* the sporadic groups, 26.

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