Spreads and nilpotence class in nilpotent groups and Lie algebras. (Q468687)

For a non-abelian finite \(p\)-group \(G\), let \(p^{b(G)}\) be the maximum, and \(p^{s(G)}\) the minimum, of sizes of conjugacy classes of non-central elements of \(G\). The number \(\delta=\delta(G)=b(G)-s(G)\) is called the spread of \(G\). \textit{A. Jaikin-Zapirain} proved [Proc. Am. Math. Soc. 133, No. 10, 2817-2820 (2005; Zbl 1077.20032)] that the nilpotency class of \(G\) is at most \(2\delta^2+2\delta+3\). In the present paper this bound in improved to \(\delta p/(p-1)+3-1/(p-1)\leq 2\delta+2\). Similar parameters can be introduced for finite-dimensional nilpotent Lie algebras over a field, with codimensions of centralizers in place of exponents in class sizes, as well as for finitely generated torsion-free nilpotent groups, and previous bounds for the nilpotency class are also significantly improved in these settings.