Lie algebras admitting a hypercentrally regular automorphism (Q1316915)

Let \(L\) be a Lie algebra, \(H(L)\) be its hypercenter, \(\sigma\) be an automorphism of \(L\) and \(L^{(\sigma)}\) denote the set of \(\sigma\)-fixed elements of \(L\). An automorphism \(\sigma\) of \(L\) is called hypercentrally regular if \(L^{(\sigma)}\subseteq H(L)\). The author proves that if a finite-dimensional Lie algebra \(L\) over a field of characteristic \(p>0\) admits a hypercentrally regular automorphism \(\sigma\) of order \(p^ k\) for some integer \(k\), then \(L\) is nilpotent.