We give an efficient algorithm for Lang's Theorem in split connected reductive groups defined over finite fields of characteristic greater than 3. This algorithm can be used to construct many important structures in finite groups of Lie type. We use an algorithm for computing a Chevalley basis for a split reductive Lie algebra, which is of independent interest.

For a complete cartesian-closed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable Beck-Chevalley-type condition, it is shown that the category of lax reflexive (T,V)-algebras is a quasitopos. This result encompasses many known and new examples of quasitopoi. Mathematics Subject Classification: 18C20, 18D15, 18A05, 18B30, 18B35.

After a presentation of Brou e's conjecture for principal blocks with an abelian defect group, we describe a Rickard complex for GL 2 (q) arising from thè-adic cohomology of a Deligne-Lusztig variety, in accordance with the explicit form given by Brou e to his conjecture in the case of Chevalley groups in non natural characteristic.

We show that for integers k ≥ 2 and n ≥ 3, the diameter of the Cayley graph of SLn(Z/kZ) associated to a standard two-element generating set, is at most a constant times n ln k. This answers a question of A. Lubotzky concerning SLn(Fp) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing element...

We introduce the notion of a braid group parametrized by a ring, which is defined by generators and relations and based on the geometric idea of painted braids. We show that the parametrized braid group is isomorphic to the semi-direct product of the Steinberg group (of the ring) with the classical braid group. The technical heart of the proof is the Pure Braid Lemma 2.1.1, which asserts that c...

the present note arises from the author's talk at the conference ``ischia group theory 2014''. for subgroups $fle n$ of a group $g$ denote by $lat(f,n)$ the set of all subgroups of $n$, containing $f$. let $d$ be a subgroup of $g$. in this note we study the lattice $ll=lat(d,g)$ and the lattice $ll'$ of subgroups of $g$, normalized by $d$. we say that $ll$ satisfies sandwich classification theo...

The most widely known result of Chevalley–Warning type states that if one has a polynomial over a finite field of characteristic p, and the number of variables exceeds the degree, then the number of zeros is a multiple of p. In particular, if the polynomial is homogeneous, there is at least one non-trivial zero. There are a number of related results in the literature. Generally, let Fq be a fin...

In his famous paper [C1] Chevalley associated to any root datum of adjoint type and to any field k a certain group (now known as a Chevalley group) which in the case where k = C was the usual adjoint Lie group over C and which in the case where k is finite led to some new families of finite simple groups. Let O′ be the coordinate ring of a connected semisimple group G over C attached to a fixed...

The modules for a Chevalley group arising from admissible lattices in an irreducible module for the associated complex semisimple Lie algebra are studied. It is proved that the transpose of such a module is still in this collection and generically the cohomology modules of line bundles on the flag varieties are in this collection also. In the rank 1 case, all modules in this collection are inde...

For a complete cartesian-closed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable Beck-Chevalley-type condition, it is shown that the category of lax reflexive (T,V)-algebras is a quasitopos. This result encompasses many known and new examples of quasitopoi.