On the \(p\)-length of some finite \(p\)-soluble groups.
All groups considered in this paper are finite and \(p\) is a prime number. The main result of the paper is the following: (Theorem A) Let \(\mathcal P\) be a subgroup-closed class of \(p\)-groups, and let \(\mathfrak D(\mathcal P)\) denote the class of all \(p\)-soluble groups whose Sylow \(p\)-subgroups are in \(\mathcal P\). Also, let \(\mathfrak L_p\) be the class of all groups of \(p\)-length at most one. Suppose that \(\mathfrak D(\mathcal P)\) is not contained in \(\mathfrak L_p\), and let \(G\) be a \(p\)-soluble group of minimal order in \(\mathfrak D(\mathcal P)\setminus\mathfrak L_p\). If \(P\) is a Sylow \(p\)-subgroup of \(G\), then \(\Phi(G)\), the Frattini subgroup of \(G\), is contained in \(P\) and one of the following holds: (1) If \(p\) is not a Fermat prime or the Hall \(p'\)-subgroups of \(G\) are abelian, then the nilpotence class of \(P/\Phi(G)\) is greater than or equal to \(p\). (2) If \(p\) is a Fermat prime, then the nilpotence class of \(P/\Phi(G)\) is greater than or equal to \(p-1\). As a consequence of this theorem, alternative proofs and improvements of recent results of \textit{J. González-Sánchez} and \textit{T. S. Weigel} concerning sufficient conditions either for a finite group to be \(p\)-nilpotent (Theorem D), or for a \(p\)-soluble group to have \(p\)-length at most 1 (Theorem E) [in Isr. J. Math. 181, 125-143 (2011; Zbl 1234.20024)], are obtained.
scientific article; zbMATH DE number 6379543
scientific article