Theorem of Cayley

In Theory of the groups, the theorem of Cayley is an elementary result affirming that any group is carried out like sub-group of a group of permutations: This theorem was shown in 1854 in an article entitled On the theory of the groups like dependence of the equation symbolic system $\ theta^n=1$ .

Remarks

This theorem is not always as powerful as it does not appear to with it. For example, in the case of the finished cardinal, it plunges a group in another group of cardinal higher and equal to.

This theorem is used for the theory of Représentation of the groups. That is to say a finished group of cardinal and $(e_1,…, e_n)$ a bases of a vector Space of dimension. The theorem of Caley indicates that is isomorphous with a group of permutation of the elements of the base. Each permutation can be prolonged in a Endomorphisme which here by definition is a Automorphisme. That defines a representation of the group, one then speaks about regular Représentation.

Demonstration

That is to say a group and an element of the group. One defines the $t_g$ application $G$ in as being the translation on the left:

\ forall X \ in G \, t_g (X) =gx

The elementary properties of the groups show that $t_g$ is well a permutation. That is to say $\ varphi \,$ the application of as a whole of arrival, part of $\ mathfrak {S} (G)$ , defined by:

\ forall G \ in G \ quad \ varphi (G) =t_g \;

$\ varphi \,$ is a morphism, indeed:

\ forall G \ H \ X \ in G \, \ varphi (gh) (X) =t_g \ t_h (X) = ghx= \ varphi (G) \ varphi (H) (X) \;

By construction, $\ varphi \,$ is surjective.
Let us show whereas $\ varphi \,$ is injective. For that let us consider and two elements of the group. If $t_g$ and $t_h$ are equal, then their images of the neutral element are also equal and is equal to. What shows that the application is injective.

The theorem is thus shown.

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