Let \(G\) be a finite group, let \(p\) be a prime dividing \(|G|\), let \(P\in\text{Syl}_p(G)\), \(O=[P,O^p(G)]\), \(U=P\cap O\) and \(V=\Omega(U)\), where \(\Omega=\Omega_1\) for \(p>2\) and \(\Omega=\Omega_2\) for \(p=2\).
The authors show that the \(p\)-nilpotency of \(G\) is determined to a large extent by the subgroup \(V\).
The paper contains a large number of characterizations of \(p\)-nilpotency. For example, the following are equivalent: 1) \(G\) is \(p\)-nilpotent; 2) \(V\) lies in \(Z(N_G(P))\); 3) \(N_G(H)/C_G(H)\) is a \(p\)-group for every nontrivial subgroup \(H\) of \(V\); 4) \(N_G(H)\) is \(p\)-nilpotent for every nontrivial subgroup \(H\) of \(V\); 5) \(N_G(P)\) is \(p\)-nilpotent and every minimal subgroup of \(U\) is complemented in \(P\).
Theorem 3.3 ensures that, given a saturated formation \(\mathcal F\) containing the class of supersolvable groups, and given a normal subgroup \(H\) of \(G\) such that \(G/H\in{\mathcal F}\), then \(G\in{\mathcal F}\) provided that for every prime \(p\in\pi(H)\) and every \(P\in\text{Syl}_p(H)\) the minimal subgroups of \(P\cap[P,O^p(G)]\) are complemented in \(N_G(P)\).