A sufficient condition for a number to be the order of a nonsingular derivation of a Lie algebra (Q836097)

The author deals with a topic suggested and initiated by Shalev and later continued by himself about the set \(\mathcal{N}_p\) of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic \(p > 0\). Shalev had proved that no nontrivial elements of \(\mathcal{N}_p\) (that is, those numbers in \(\mathcal{N}_p\) which are prime to \(p\) and not multiples of any \(p^k -1\), with \(k \leq 2\)) is smaller than \(p^2\), and it was extended by the author to conclude that no nontrivial element of \(\mathcal{N}_p\) is smaller than \(p^3\), except for \((3^3 - 3)/2 = 13\), when \(p = 3\). In the paper, the author extends the results by Shalev when \(p = 2\) and by himself to an arbitrary prime \(p\), by proving that a divisor \(n\) of \(q-1\), where \(q\) is a power of \(p\), belongs to \(\mathcal{N}_p\) provided it satisfies \(n \geq (p - 1)^{1/p} (q - 1)^{1 - 1/(2p)}\). He also collects several remarks on the set \(\mathcal{N}_p\), like the density of the set and a notion of relative size of its elements.