Let M∗(q) be the unique nonassociative finite simple Moufang loop constructed over GF (q). We prove that Aut(M∗(2)) is the Chevalley group G2(2), by extending multiplicative automorphism of M∗(2) into linear automorphisms of the unique split octonion algebra over GF (2). Many of our auxiliary results apply in the general case. In the course of the proof we show that every element of a split oct...

The classical Waring problem deals with expressing every natural number as a sum of g(k) k powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given word w 6= 1. In this paper we study this problem for Lie groups and Chevalley groups over infinite fields. We show that for a fixed word w 6= 1 and for a classical ...

Take G to be a split, simply-connected semisimple algebraic group (for example, SL(w, K) or Sp(2m, K)) over an algebraic closure K of the field of p elements, where p is prime. Denote by G(n) the finite subgroup consisting of elements of G defined over the field of p elements. The simple (rational) G-modules L(X) are parametrized by the set X of dominant weights, a subset of the character group...

Let A be a symmetrizable generalized Cartan matrix. Let g be the corresponding Kac-Moody algebra. Let G(R) be Tits’ Kac-Moody group functor over commutative rings R associated to g. Then G(R) is given by five axioms KMG1-KMG5 which are natural extensions of the properties of Chevalley-Demazure group schemes. We give a construction of the Tits functor for symmetrizable Kac-Moody groups G using i...