Group nilpotent

In theory of the groups, a group is known as nilpotent when it has a certain property: intuitively, one can think that one can make the group abelian by the repeated use of the switch $=xyx^ {- 1} y^ {- 1}$ . The nilpotents groups appear naturally in Théorie of Welshman or for the classification of the groups of Dregs or the linear algebraic groups.

Definition

That is to say G a group noted multiplicativement, of neutral element E. If has and B are two sub-groups of G, one notes the sub-group generated by the switch of the form for X in has and there in B.

One then defines by recurrence a succession of sub-groups of G, noted $C^n (G)$ , by

$C^1 (G) =G$
$C^ {n+1} (G) =$ .

It is said that G east nilpotent if there exists an entirety N such as $C^n (G) = \ {E \}$ . Moreover, if G is a group nilpotent, its class of nilpotence is smallest entirety N such as $C^ {n+1} (G) = \ {E \}$ .

Examples

Any abelian group east nilpotent of class 1.

the sub-group $U_n (\ mathbb K)$ of $GL_n (\ mathbb K)$ formed of the higher triangular matrices with 1 on the diagonal east nilpotent of class $n-1$ .

a P-group east nilpotent.

the group of discrete Heisenberg east nilpotent of class 2.

Properties

a sub-group of a group nilpotent east nilpotent. The image of a group nilpotent by a morphism of group is a group nilpotent.

Is Z (G) the center of G. If G is not the commonplace group, then Z (G) is not either commonplace.

If G/Z (G) east nilpotent, then G east nilpotent.

If H is a clean sub-group of G nilpotent, then H is strictly included in its Normalisateur.

Any group nilpotent is resolvable.

Group linear nilpotents

It was already seen that $U_n (\ mathbb K)$ is a group nilpotent. It has the property interesting to be formed of elements unipotents , i.e. form $I_n+N$ , where NR is a matrix Nilpotent E. Theorems related on the reduction of the endomorphisms and the representations of the groups make it possible to show the reciprocal one. This can be seen as an analog of the Théorème of Engel on the algebras of Dregs.

Let us be more precise. It is shown first of all that if $G \ subset GL_n (\ mathbb K)$ is a sub-group formed only of elements unipotents, then all the elements of G are simultaneously trigonalisables. In other words, it is obtained that G is combined with a sub-group of $U_n (\ mathbb K)$ . In particular, G east nilpotent.

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