Search

Let G be a linear algebraic group over a field k of characteristic p > 0, and suppose that the unipotent radical R of G is defined and split over k. By a Levi factor of G, one means a closed subgroup M which is a complement to R in G. In this paper, we give two results related to the descent of Levi factors. First, suppose l is a finite Galois extension of k for which the extension degree [l : k] is relatively prime to p. If G /l has a Levi decomposition, we show that G has a Levi decomposition. Second, suppose that there is a G-equivariant isomorphism of algebraic groups $${R \simeq Lie(R)}$$ – i.e. R is a vector group with a linear action of the reductive quotient G/R. If $${G_{{/k}_{sep}}}$$ has a Levi decomposition for a separable closure k sep of k, then G has a Levi decomposition. Finally, we give an example of a disconnected, abelian, linear algebraic group G for which $${G_{{/k}_{sep}}}$$ has a Levi decomposition, but G itself has no Levi decomposition.

The goal of this paper is to extend some previous results on abelian ideals of Borel subalgebras to so-called spherical ideals of b. These are ideals c of b such that their G-saturation G c is a spherical G-variety. We classify all maximal spherical ideals of b for all simple G. Let G be a connected reductive complex algebraic group with Lie algebra Lie G = g. Let B be a Borel subgroup of G with unipotent radical Bu. We denote the Lie algebras of B and Bu by b and bu, respectively. The group B acts on any ideal of b by means of the adjoint representation. There has been quite a lot of activity recently in the study of various aspects of ad-nilpotent ideals and, in particular, abelian ideals of b , for instance, see (5), (6), (9), (12), (14), and (16), and the additional references therein. After a preliminary section, we recall our main finiteness results on abelian ideals from (14). The goal of this paper is to extend these results to so-called spherical ideals of b in the next section. These are ideals c of b such that their G-saturation G c is a spherical G-variety. Our aim is to classify all maximal spherical ideals of b for all simple G.

The set of all elements of an associative ring R, not necessarily with a unit element, forms a semigroup R ad under the circle operation \({r\circ s}={r+s+rs}\) on R. The ring R is called radical if R ad is a group. It is proved that the semigroup R ad is nilpotent of class n (in sense of A. Mal'cev or B. H. Neumann and T. Taylor) if and only if the ring R is Lie-nilpotent of class n. This yields a positive answer to a question posed by A. Krasil'nikov and independently considered by D. Riley and V. Tasic. It is also shown that the adjoint group of a radical ring R is locally nilpotent if and only if R is locally Lie-nilpotent.

In [3] A. Bialostocki has introduced a special class of maximal nilpotent subgroups in an arbitrary finite group. ¶In this paper we want to investigate the relationship between this special class of maximal nilpotent subgroups and the more classical $\cal N $ -injectors in an arbitrary finite group. ¶To do this we prove the following theorem which is of fundamental importance in this investigation: ¶Let U be a nilpotent subgroup of G containing every U-invariant nilpotent subgroup of G. Then U is an $\cal N $ -injector of G.

In a yet to be published work R. E. Phillips and J. G. Rainbolt prove that every image of a periodic linear group (of finite degree) with trivial unipotent radical is isomorphic to a linear group over the same field and of bounded degree. Here we offer an alternative proof that is both quite short and delivers a little more. ¶¶ Our basic theorem, from which follow a number of corollaries, is the following. There is an integer-valued function f (n) of n only such that if G is any linear group of finite degree n and characteristic p≥ 0 and if N is any periodic normal subgroup of G, with $O_p(N) = \langle 1 \rangle$ if $p \neq 0$ , then G/N is isomorphic to a linear group of degree f (n) and characteristic p. One corollary is Phillips and Rainbolt's Theorem. A second has the condition $O_p(N) = \langle 1 \rangle$ if $p \neq 0$ replaced by $O_p(G) \leq N$ if $p \neq 0$ , but with the same conclusion (and with the same function f (n).

Let G be a periodic subgroup of GL (n,K ) for some field K and let N be a normal subgroup of G. It is not, in general, true that G / N has a faithful K-linear representation. The following will show that if G is periodic and does not contain any non-trivial normal unipotent subgroups then G / N has a faithful K-linear representation. In addition, we show that, in this case, the degree of the representation is bounded by a function of n.

We determine the Brauer trees of the unipotent characters of the (twisted) Chevalley groups F 4 (q) and 2 E 6 (q) with one exception. The unipotent character tables of these groups are known to a large extent. This information is used in an essential way in the proofs of our results.

Obviously we have not covered aU possible ascendant and descendant nilpotent properties. Using (1.1), (1.2), or other methods, and Lemma 1.2 it is possible to provide many more such properties. For exampleCq∼(Cpm×Cpm) is a metabelian group having infinitely many non-isomorphic Sylow-p-subgroups [6]. Thus, having one isomorphism class of Sylow-p-subgroups is a descendant nilpotent property.