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A study of the set N_p of positive integers which occur as orders ofnonsingular derivations of finite-dimensional non-nilpotent Lie algebras ofcharacteristic 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 N_p. Ourmain result shows that any divisor n of q-1, where q is a power of p, such that$n\ge (p-1)^{1/p} (q-1)^{1-1/(2p)}$, belongs to N_p. This extends its specialcase for p=2 which was proved in a previous paper by a different method.

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