Dimension subgroups and \(p\)-th powers in \(p\)-groups.
\textit{C. M. Scoppola} and \textit{A. Shalev} [Isr. J. Math. 73, No. 1, 45-56 (1991; Zbl 0734.20008)] have proved that if the finite \(p\)-group \(G\) has nilpotence class strictly less than \(p^k-p^{k-1}\), then every product of \(p^k\)-th powers in \(G\) can be written as the \(p\)-th power of a suitable element of \(G\). They also showed that the result does not generally hold if the class of \(G\) is \(p^k\). In the paper under review, the author gives the best possible estimate, by showing that the result holds for groups of class strictly less than \(p^k\). The main tool is provided by the dimension subgroups, for which results of independent interest are proved.
scientific article; zbMATH DE number 2060049