Jennings' theorem for \(p\)-split groups. (Q1772445)

\textit{S.~A.~Jennings} gave a description [Trans. Am. Math. Soc. 50, 175--185 (1941; Zbl 0025.24401)] of bases of the radical layers of the group ring of a finite \(p\)-group \(G\) over a field of characteristic \(p\), in terms of the descending central series of \(G\) given by the dimension subgroups. \textit{J.~L.~Alperin} later extended this [Q. J. Math., Oxf. (2) 39, 129--133 (1988; Zbl 0657.20004)] to permutation modules for finite \(p\)-groups. The goal of the paper under review is to give a natural extension of these results, covering also associated Hilbert polynomials, to \(p\)-split groups, that is, to those finite groups which have a normal \(p\)-Sylow subgroup.