A sufficient condition for a number to be the order of a nonsingular derivation of a Lie algebra

A study of the set of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p > 0 was initiated by Shalev and continued by the present author. The main goal of this paper is to produce more elements of . Our main result shows that any divisor n of q − 1, where q is a power of p, such that n ≥ (p − 1)^1/p (q − 1)^1−1/(2p), necessarily belongs to . This extends its special case for p = 2 which was proved in a previous paper by a different method.