The orders of nonsingular derivations of modular Lie algebras.
In the connection with the coclass conjectures for pro-\(p\) groups and coclass theory of Lie algebras \textit{A. Shalev} [J. Aust. Math. Soc., Ser. A 67, 254--260 (1999; Zbl 1015.17020)] formulated the following problem. Which are the possible orders \(n\) of nonsingular derivations of finite-deminsional non-nilpotent Lie algebras of prime characteristic \(p\)? Let \(p\) be a prime number and let \(n\) be a positive integer, prime to \(p\). In the present paper it is proved that the following statements are equivalent: 1) there exists a non-nilpotent Lie algebra of characteristic \(p\) with a nonsingular derivation of order \(n\); 2) there exists an element \(\alpha \in \overline {\mathbb F}_p\) such that \((\alpha +\lambda )^n=1\) for all \(\lambda \in \mathbb F_p\); 3) there exists an element \(c\in \mathbb F_p^*\) such that \(x^p-x-c\) divides \(x^n-1\) in \(\overline {\mathbb F}_p[x]\). It gives an affirmative answer to the final question of Shalev's paper, namely, there exists a finite-dimensional non-nilpotent Lie algebra of characteristic \(p\) which admits a nonsingular derivation of order \(\frac{p^p-1}{p-1}\) if \(p\) is odd and of order 73 if \(p=2\). It was shown by \textit{G. Benkart}, \textit{A. I. Kostrikin} and \textit{M. I. Kuznetsov} [J. Algebra 171, 894--916 (1995; Zbl 0815.17018)] that some class of simple Hamiltonian Lie algebras has a nonsingular derivation of order \(p^k-1\). Thus \(n=p^k-1\) is admissible in view of the problem. In the paper it is proved that the only numbers less than \(p^3-1\) which are admissible, are multiples of \(p^2-1\) (for \(p>3\)). Given a polynomal \(f(x)\in \overline {\mathbb F}_p[x]\) with nonzero constant term, the smallest positive integer \(n\) such that \(f(x)\) divides \(x^n-1\), is called the order of a polynomial \(f(x)\). The author proposes the following question. Given an integer \(r>1\), is it true for all sufficiently large primes \(p\) that all possible values \(n<p^r\) for the order of polynomial \(x^p-x-c\in \overline {\mathbb F}_p[x]\) have the form \(n=p^i-1\)?
scientific article; zbMATH DE number 1874398