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

TL;DR

This paper characterizes a broad class of positive integers that can serve as the order of nonsingular derivations in finite-dimensional non-nilpotent Lie algebras over fields of characteristic p, extending previous results for p=2.

Contribution

It provides a new sufficient condition for integers to be the order of nonsingular derivations, generalizing earlier findings and demonstrating the abundance of such integers for any prime characteristic p.

Findings

Any divisor n of q-1 with q a power of p and n ≥ (p-1)^{1/p} (q-1)^{1-1/(2p)} belongs to N_p.

Extends previous p=2 case to arbitrary prime p.

Shows the abundance of elements in N_p under the given condition.

Abstract

A study of the set N_p 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 show the abundance of elements of N_p. Our main 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 special case for p=2 which was proved in a previous paper by a different method.