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Many numbers can be orders of nonsingular derivations of modular Lie algebras

A study of the set Np of positive integers which occur as orders of non-singular 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 show the abundance of elements of Np. 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) , belongs to Np. This extends its special case for p = 2 which was proved in a previous paper by a different method.